src/ZF/Constructible/Separation.thy
author paulson
Wed Oct 09 11:07:13 2002 +0200 (2002-10-09)
changeset 13634 99a593b49b04
parent 13628 87482b5e3f2e
child 13687 22dce9134953
permissions -rw-r--r--
Re-organization of Constructible theories
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(*  Title:      ZF/Constructible/Separation.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{*Early Instances of Separation and Strong Replacement*}
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theory Separation = L_axioms + WF_absolute:
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text{*This theory proves all instances needed for locale @{text "M_basic"}*}
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text{*Helps us solve for de Bruijn indices!*}
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lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
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by simp
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lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI
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lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats
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                   fun_plus_iff_sats
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lemma Collect_conj_in_DPow:
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     "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |]
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      ==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
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by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
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lemma Collect_conj_in_DPow_Lset:
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     "[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
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      ==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
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apply (frule mem_Lset_imp_subset_Lset)
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apply (simp add: Collect_conj_in_DPow Collect_mem_eq
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                 subset_Int_iff2 elem_subset_in_DPow)
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done
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lemma separation_CollectI:
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     "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
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apply (unfold separation_def, clarify)
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apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
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apply simp_all
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done
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text{*Reduces the original comprehension to the reflected one*}
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lemma reflection_imp_L_separation:
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      "[| \<forall>x\<in>Lset(j). P(x) <-> Q(x);
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          {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
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          Ord(j);  z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
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apply (rule_tac i = "succ(j)" in L_I)
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 prefer 2 apply simp
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apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
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 prefer 2
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 apply (blast dest: mem_Lset_imp_subset_Lset)
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apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
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done
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text{*Encapsulates the standard proof script for proving instances of 
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Separation.  Typically @{term u} is a finite enumeration.*}
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lemma gen_separation:
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 assumes reflection: "REFLECTS [P,Q]"
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     and Lu:         "L(u)"
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     and collI: "!!j. u \<in> Lset(j)
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                \<Longrightarrow> Collect(Lset(j), Q(j)) \<in> DPow(Lset(j))"
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 shows "separation(L,P)"
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apply (rule separation_CollectI)
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apply (rule_tac A="{u,z}" in subset_LsetE, blast intro: Lu)
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apply (rule ReflectsE [OF reflection], assumption)
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apply (drule subset_Lset_ltD, assumption)
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2, clarify)
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apply (rule collI)
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apply assumption;  
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done
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subsection{*Separation for Intersection*}
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lemma Inter_Reflects:
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     "REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y,
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               \<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y]"
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by (intro FOL_reflections)
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lemma Inter_separation:
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     "L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)"
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apply (rule gen_separation [OF Inter_Reflects], simp)
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apply (rule DPow_LsetI)
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apply (rule ball_iff_sats)
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apply (rule imp_iff_sats)
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apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
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apply (rule_tac i=0 and j=2 in mem_iff_sats)
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apply (simp_all add: succ_Un_distrib [symmetric])
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done
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subsection{*Separation for Set Difference*}
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lemma Diff_Reflects:
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     "REFLECTS[\<lambda>x. x \<notin> B, \<lambda>i x. x \<notin> B]"
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by (intro FOL_reflections)  
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lemma Diff_separation:
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     "L(B) ==> separation(L, \<lambda>x. x \<notin> B)"
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apply (rule gen_separation [OF Diff_Reflects], simp)
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apply (rule DPow_LsetI)
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apply (rule not_iff_sats) 
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apply (rule_tac env="[x,B]" in mem_iff_sats)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Cartesian Product*}
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lemma cartprod_Reflects:
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     "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
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                \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
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                                   pair(**Lset(i),x,y,z))]"
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by (intro FOL_reflections function_reflections)
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lemma cartprod_separation:
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     "[| L(A); L(B) |]
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      ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
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apply (rule gen_separation [OF cartprod_Reflects, of "{A,B}"], simp)
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apply (drule mem_Lset_imp_subset_Lset, clarsimp)
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats)
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apply (rule conj_iff_sats)
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apply (rule_tac i=0 and j=2 and env="[x,z,A,B]" in mem_iff_sats, simp_all)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Image*}
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lemma image_Reflects:
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     "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
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           \<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(**Lset(i),x,y,p))]"
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by (intro FOL_reflections function_reflections)
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lemma image_separation:
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     "[| L(A); L(r) |]
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      ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
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apply (rule gen_separation [OF image_Reflects, of "{A,r}"], simp)
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apply (drule mem_Lset_imp_subset_Lset, clarsimp)
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats)
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apply (rule conj_iff_sats)
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apply (rule_tac env="[p,y,A,r]" in mem_iff_sats)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Converse*}
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lemma converse_Reflects:
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  "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
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     \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
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                     pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z))]"
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by (intro FOL_reflections function_reflections)
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lemma converse_separation:
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     "L(r) ==> separation(L,
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         \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
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apply (rule gen_separation [OF converse_Reflects], simp)
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats)
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apply (rule conj_iff_sats)
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apply (rule_tac i=0 and j=2 and env="[p,z,r]" in mem_iff_sats, simp_all)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Restriction*}
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lemma restrict_Reflects:
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     "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
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        \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z))]"
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by (intro FOL_reflections function_reflections)
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lemma restrict_separation:
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   "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
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apply (rule gen_separation [OF restrict_Reflects], simp)
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats)
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apply (rule conj_iff_sats)
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apply (rule_tac i=0 and j=2 and env="[x,z,A]" in mem_iff_sats, simp_all)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Composition*}
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lemma comp_Reflects:
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     "REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
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                  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
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                  xy\<in>s & yz\<in>r,
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        \<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i).
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                  pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) &
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                  pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]"
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by (intro FOL_reflections function_reflections)
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lemma comp_separation:
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     "[| L(r); L(s) |]
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      ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
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                  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
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                  xy\<in>s & yz\<in>r)"
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apply (rule gen_separation [OF comp_Reflects, of "{r,s}"], simp)
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apply (drule mem_Lset_imp_subset_Lset, clarsimp)
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats)+
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apply (rule conj_iff_sats)
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apply (rule_tac env="[z,y,x,xz,r,s]" in pair_iff_sats)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Predecessors in an Order*}
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lemma pred_Reflects:
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     "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p),
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                    \<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(**Lset(i),y,x,p)]"
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by (intro FOL_reflections function_reflections)
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lemma pred_separation:
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     "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
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apply (rule gen_separation [OF pred_Reflects, of "{r,x}"], simp)
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apply (drule mem_Lset_imp_subset_Lset, clarsimp)
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats)
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apply (rule conj_iff_sats)
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apply (rule_tac env = "[p,y,r,x]" in mem_iff_sats)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for the Membership Relation*}
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lemma Memrel_Reflects:
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     "REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y,
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            \<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(**Lset(i),x,y,z) & x \<in> y]"
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by (intro FOL_reflections function_reflections)
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lemma Memrel_separation:
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     "separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
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apply (rule gen_separation [OF Memrel_Reflects nonempty])
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats conj_iff_sats)+
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apply (rule_tac env = "[y,x,z]" in pair_iff_sats)
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apply (rule sep_rules | simp)+
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done
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subsection{*Replacement for FunSpace*}
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lemma funspace_succ_Reflects:
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 "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
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            pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
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            upair(L,cnbf,cnbf,z)),
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        \<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i).
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              \<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i).
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                pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) &
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                is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]"
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by (intro FOL_reflections function_reflections)
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lemma funspace_succ_replacement:
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     "L(n) ==>
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      strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
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                pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
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                upair(L,cnbf,cnbf,z))"
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apply (rule strong_replacementI)
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apply (rule_tac u="{n,A}" in gen_separation [OF funspace_succ_Reflects], simp)
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apply (drule mem_Lset_imp_subset_Lset, clarsimp)
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats)
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apply (rule conj_iff_sats)
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apply (rule_tac env = "[p,z,n,A]" in mem_iff_sats)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for a Theorem about @{term "is_recfun"}*}
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lemma is_recfun_reflects:
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  "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
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                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
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                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
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                                   fx \<noteq> gx),
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   \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
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          pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r &
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                (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) &
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                  fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]"
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by (intro FOL_reflections function_reflections fun_plus_reflections)
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lemma is_recfun_separation:
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     --{*for well-founded recursion*}
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     "[| L(r); L(f); L(g); L(a); L(b) |]
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     ==> separation(L,
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            \<lambda>x. \<exists>xa[L]. \<exists>xb[L].
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                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
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                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
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                                   fx \<noteq> gx))"
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apply (rule gen_separation [OF is_recfun_reflects, of "{r,f,g,a,b}"], simp)
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apply (drule mem_Lset_imp_subset_Lset, clarsimp)
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apply (rule DPow_LsetI)
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apply (rule bex_iff_sats conj_iff_sats)+
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apply (rule_tac env = "[xa,x,r,f,g,a,b]" in pair_iff_sats)
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apply (rule sep_rules | simp)+
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done
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subsection{*Instantiating the locale @{text M_basic}*}
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text{*Separation (and Strong Replacement) for basic set-theoretic constructions
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such as intersection, Cartesian Product and image.*}
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lemma M_basic_axioms_L: "M_basic_axioms(L)"
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  apply (rule M_basic_axioms.intro)
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       apply (assumption | rule
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	 Inter_separation Diff_separation cartprod_separation image_separation
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	 converse_separation restrict_separation
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	 comp_separation pred_separation Memrel_separation
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	 funspace_succ_replacement is_recfun_separation)+
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  done
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theorem M_basic_L: "PROP M_basic(L)"
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by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L])
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lemmas cartprod_iff = M_basic.cartprod_iff [OF M_basic_L]
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  and cartprod_closed = M_basic.cartprod_closed [OF M_basic_L]
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   321
  and sum_closed = M_basic.sum_closed [OF M_basic_L]
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   322
  and M_converse_iff = M_basic.M_converse_iff [OF M_basic_L]
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  and converse_closed = M_basic.converse_closed [OF M_basic_L]
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   324
  and converse_abs = M_basic.converse_abs [OF M_basic_L]
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   325
  and image_closed = M_basic.image_closed [OF M_basic_L]
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  and vimage_abs = M_basic.vimage_abs [OF M_basic_L]
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   327
  and vimage_closed = M_basic.vimage_closed [OF M_basic_L]
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  and domain_abs = M_basic.domain_abs [OF M_basic_L]
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  and domain_closed = M_basic.domain_closed [OF M_basic_L]
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  and range_abs = M_basic.range_abs [OF M_basic_L]
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  and range_closed = M_basic.range_closed [OF M_basic_L]
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   332
  and field_abs = M_basic.field_abs [OF M_basic_L]
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  and field_closed = M_basic.field_closed [OF M_basic_L]
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   334
  and relation_abs = M_basic.relation_abs [OF M_basic_L]
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   335
  and function_abs = M_basic.function_abs [OF M_basic_L]
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  and apply_closed = M_basic.apply_closed [OF M_basic_L]
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   337
  and apply_abs = M_basic.apply_abs [OF M_basic_L]
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   338
  and typed_function_abs = M_basic.typed_function_abs [OF M_basic_L]
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  and injection_abs = M_basic.injection_abs [OF M_basic_L]
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   340
  and surjection_abs = M_basic.surjection_abs [OF M_basic_L]
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  and bijection_abs = M_basic.bijection_abs [OF M_basic_L]
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   342
  and M_comp_iff = M_basic.M_comp_iff [OF M_basic_L]
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  and comp_closed = M_basic.comp_closed [OF M_basic_L]
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  and composition_abs = M_basic.composition_abs [OF M_basic_L]
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   345
  and restriction_is_function = M_basic.restriction_is_function [OF M_basic_L]
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   346
  and restriction_abs = M_basic.restriction_abs [OF M_basic_L]
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   347
  and M_restrict_iff = M_basic.M_restrict_iff [OF M_basic_L]
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   348
  and restrict_closed = M_basic.restrict_closed [OF M_basic_L]
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   349
  and Inter_abs = M_basic.Inter_abs [OF M_basic_L]
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   350
  and Inter_closed = M_basic.Inter_closed [OF M_basic_L]
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   351
  and Int_closed = M_basic.Int_closed [OF M_basic_L]
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   352
  and is_funspace_abs = M_basic.is_funspace_abs [OF M_basic_L]
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   353
  and succ_fun_eq2 = M_basic.succ_fun_eq2 [OF M_basic_L]
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   354
  and funspace_succ = M_basic.funspace_succ [OF M_basic_L]
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  and finite_funspace_closed = M_basic.finite_funspace_closed [OF M_basic_L]
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lemmas is_recfun_equal = M_basic.is_recfun_equal [OF M_basic_L]
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   358
  and is_recfun_cut = M_basic.is_recfun_cut [OF M_basic_L]
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   359
  and is_recfun_functional = M_basic.is_recfun_functional [OF M_basic_L]
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   360
  and is_recfun_relativize = M_basic.is_recfun_relativize [OF M_basic_L]
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   361
  and is_recfun_restrict = M_basic.is_recfun_restrict [OF M_basic_L]
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   362
  and univalent_is_recfun = M_basic.univalent_is_recfun [OF M_basic_L]
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   363
  and wellfounded_exists_is_recfun = M_basic.wellfounded_exists_is_recfun [OF M_basic_L]
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   364
  and wf_exists_is_recfun = M_basic.wf_exists_is_recfun [OF M_basic_L]
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   365
  and is_recfun_abs = M_basic.is_recfun_abs [OF M_basic_L]
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   366
  and irreflexive_abs = M_basic.irreflexive_abs [OF M_basic_L]
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   367
  and transitive_rel_abs = M_basic.transitive_rel_abs [OF M_basic_L]
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   368
  and linear_rel_abs = M_basic.linear_rel_abs [OF M_basic_L]
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   369
  and wellordered_is_trans_on = M_basic.wellordered_is_trans_on [OF M_basic_L]
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   370
  and wellordered_is_linear = M_basic.wellordered_is_linear [OF M_basic_L]
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   371
  and wellordered_is_wellfounded_on = M_basic.wellordered_is_wellfounded_on [OF M_basic_L]
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   372
  and wellfounded_imp_wellfounded_on = M_basic.wellfounded_imp_wellfounded_on [OF M_basic_L]
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   373
  and wellfounded_on_subset_A = M_basic.wellfounded_on_subset_A [OF M_basic_L]
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   374
  and wellfounded_on_iff_wellfounded = M_basic.wellfounded_on_iff_wellfounded [OF M_basic_L]
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   375
  and wellfounded_on_imp_wellfounded = M_basic.wellfounded_on_imp_wellfounded [OF M_basic_L]
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   376
  and wellfounded_on_field_imp_wellfounded = M_basic.wellfounded_on_field_imp_wellfounded [OF M_basic_L]
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   377
  and wellfounded_iff_wellfounded_on_field = M_basic.wellfounded_iff_wellfounded_on_field [OF M_basic_L]
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   378
  and wellfounded_induct = M_basic.wellfounded_induct [OF M_basic_L]
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   379
  and wellfounded_on_induct = M_basic.wellfounded_on_induct [OF M_basic_L]
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   380
  and linear_imp_relativized = M_basic.linear_imp_relativized [OF M_basic_L]
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   381
  and trans_on_imp_relativized = M_basic.trans_on_imp_relativized [OF M_basic_L]
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   382
  and wf_on_imp_relativized = M_basic.wf_on_imp_relativized [OF M_basic_L]
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   383
  and wf_imp_relativized = M_basic.wf_imp_relativized [OF M_basic_L]
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   384
  and well_ord_imp_relativized = M_basic.well_ord_imp_relativized [OF M_basic_L]
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   385
  and order_isomorphism_abs = M_basic.order_isomorphism_abs [OF M_basic_L]
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   386
  and pred_set_abs = M_basic.pred_set_abs [OF M_basic_L]
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   387
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   388
lemmas pred_closed = M_basic.pred_closed [OF M_basic_L]
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   389
  and membership_abs = M_basic.membership_abs [OF M_basic_L]
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   390
  and M_Memrel_iff = M_basic.M_Memrel_iff [OF M_basic_L]
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   391
  and Memrel_closed = M_basic.Memrel_closed [OF M_basic_L]
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   392
  and wellfounded_on_asym = M_basic.wellfounded_on_asym [OF M_basic_L]
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   393
  and wellordered_asym = M_basic.wellordered_asym [OF M_basic_L]
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   394
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   395
declare cartprod_closed [intro, simp]
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   396
declare sum_closed [intro, simp]
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   397
declare converse_closed [intro, simp]
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   398
declare converse_abs [simp]
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   399
declare image_closed [intro, simp]
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   400
declare vimage_abs [simp]
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   401
declare vimage_closed [intro, simp]
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   402
declare domain_abs [simp]
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   403
declare domain_closed [intro, simp]
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   404
declare range_abs [simp]
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   405
declare range_closed [intro, simp]
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   406
declare field_abs [simp]
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   407
declare field_closed [intro, simp]
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   408
declare relation_abs [simp]
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   409
declare function_abs [simp]
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   410
declare apply_closed [intro, simp]
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   411
declare typed_function_abs [simp]
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   412
declare injection_abs [simp]
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   413
declare surjection_abs [simp]
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   414
declare bijection_abs [simp]
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   415
declare comp_closed [intro, simp]
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   416
declare composition_abs [simp]
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   417
declare restriction_abs [simp]
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   418
declare restrict_closed [intro, simp]
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   419
declare Inter_abs [simp]
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   420
declare Inter_closed [intro, simp]
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   421
declare Int_closed [intro, simp]
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   422
declare is_funspace_abs [simp]
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   423
declare finite_funspace_closed [intro, simp]
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   424
declare membership_abs [simp] 
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   425
declare Memrel_closed  [intro,simp]
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   426
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   427
end