src/ZF/Constructible/Separation.thy
author wenzelm
Mon Jul 29 18:07:53 2002 +0200 (2002-07-29)
changeset 13429 2232810416fc
parent 13428 99e52e78eb65
child 13437 01b3fc0cc1b8
permissions -rw-r--r--
tuned;
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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_axioms"}*}
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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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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 separation_CollectI)
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apply (rule_tac A="{A,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF Inter_Reflects], 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 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 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 separation_CollectI)
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apply (rule_tac A="{A,B,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF cartprod_Reflects], 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 DPow_LsetI)
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apply (rename_tac u)
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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,u,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 separation_CollectI)
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apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF image_Reflects], 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 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 separation_CollectI)
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apply (rule_tac A="{r,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF converse_Reflects], 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 DPow_LsetI)
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apply (rename_tac u)
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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,u,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 separation_CollectI)
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apply (rule_tac A="{A,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF restrict_Reflects], 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 DPow_LsetI)
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apply (rename_tac u)
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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,u,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 separation_CollectI)
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apply (rule_tac A="{r,s,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF comp_Reflects], 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 DPow_LsetI)
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apply (rename_tac u)
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apply (rule bex_iff_sats)+
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apply (rename_tac x y z)
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apply (rule conj_iff_sats)
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apply (rule_tac env="[z,y,x,u,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 separation_CollectI)
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apply (rule_tac A="{r,x,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF pred_Reflects], 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 DPow_LsetI)
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apply (rename_tac u)
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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,u,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 separation_CollectI)
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apply (rule_tac A="{z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF Memrel_Reflects], 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)
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apply (rule DPow_LsetI)
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apply (rename_tac u)
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apply (rule bex_iff_sats conj_iff_sats)+
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apply (rule_tac env = "[y,x,u]" 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 rallI)
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apply (rule separation_CollectI)
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apply (rule_tac A="{n,A,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF funspace_succ_Reflects], 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)
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apply (rule DPow_LsetI)
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apply (rename_tac u)
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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,u,n,A]" in mem_iff_sats)
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   279
apply (rule sep_rules | simp)+
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   280
done
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   281
paulson@13306
   282
paulson@13316
   283
subsection{*Separation for Order-Isomorphisms*}
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   284
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   285
lemma well_ord_iso_Reflects:
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   286
  "REFLECTS[\<lambda>x. x\<in>A -->
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   287
                (\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r),
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   288
        \<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i).
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   289
                fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]"
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   290
by (intro FOL_reflections function_reflections)
paulson@13306
   291
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   292
lemma well_ord_iso_separation:
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   293
     "[| L(A); L(f); L(r) |]
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   294
      ==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L].
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   295
                     fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))"
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   296
apply (rule separation_CollectI)
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   297
apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast )
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   298
apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption)
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   299
apply (drule subset_Lset_ltD, assumption)
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   300
apply (erule reflection_imp_L_separation)
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   301
  apply (simp_all add: lt_Ord2)
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   302
apply (rule DPow_LsetI)
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   303
apply (rename_tac u)
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   304
apply (rule imp_iff_sats)
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   305
apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats)
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   306
apply (rule sep_rules | simp)+
paulson@13316
   307
done
paulson@13316
   308
paulson@13316
   309
paulson@13316
   310
subsection{*Separation for @{term "obase"}*}
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   311
paulson@13316
   312
lemma obase_reflects:
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  "REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
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   314
             ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
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   315
             order_isomorphism(L,par,r,x,mx,g),
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   316
        \<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i).
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   317
             ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
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   318
             order_isomorphism(**Lset(i),par,r,x,mx,g)]"
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   319
by (intro FOL_reflections function_reflections fun_plus_reflections)
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   320
paulson@13316
   321
lemma obase_separation:
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   322
     --{*part of the order type formalization*}
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   323
     "[| L(A); L(r) |]
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   324
      ==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
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   325
             ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
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   326
             order_isomorphism(L,par,r,x,mx,g))"
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   327
apply (rule separation_CollectI)
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   328
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
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   329
apply (rule ReflectsE [OF obase_reflects], assumption)
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   330
apply (drule subset_Lset_ltD, assumption)
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   331
apply (erule reflection_imp_L_separation)
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   332
  apply (simp_all add: lt_Ord2)
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   333
apply (rule DPow_LsetI)
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   334
apply (rename_tac u)
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   335
apply (rule bex_iff_sats)
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   336
apply (rule conj_iff_sats)
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   337
apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats)
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   338
apply (rule sep_rules | simp)+
paulson@13316
   339
done
paulson@13316
   340
paulson@13316
   341
paulson@13319
   342
subsection{*Separation for a Theorem about @{term "obase"}*}
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   343
paulson@13316
   344
lemma obase_equals_reflects:
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   345
  "REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
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   346
                ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
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   347
                membership(L,y,my) & pred_set(L,A,x,r,pxr) &
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   348
                order_isomorphism(L,pxr,r,y,my,g))),
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   349
        \<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i).
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   350
                ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i).
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   351
                membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) &
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   352
                order_isomorphism(**Lset(i),pxr,r,y,my,g)))]"
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   353
by (intro FOL_reflections function_reflections fun_plus_reflections)
paulson@13316
   354
paulson@13316
   355
paulson@13316
   356
lemma obase_equals_separation:
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   357
     "[| L(A); L(r) |]
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   358
      ==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
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   359
                              ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
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   360
                              membership(L,y,my) & pred_set(L,A,x,r,pxr) &
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   361
                              order_isomorphism(L,pxr,r,y,my,g))))"
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   362
apply (rule separation_CollectI)
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   363
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
paulson@13316
   364
apply (rule ReflectsE [OF obase_equals_reflects], assumption)
wenzelm@13428
   365
apply (drule subset_Lset_ltD, assumption)
paulson@13316
   366
apply (erule reflection_imp_L_separation)
paulson@13316
   367
  apply (simp_all add: lt_Ord2)
paulson@13385
   368
apply (rule DPow_LsetI)
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   369
apply (rename_tac u)
paulson@13316
   370
apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+
wenzelm@13428
   371
apply (rule_tac env = "[u,A,r]" in mem_iff_sats)
paulson@13316
   372
apply (rule sep_rules | simp)+
paulson@13316
   373
done
paulson@13316
   374
paulson@13316
   375
paulson@13316
   376
subsection{*Replacement for @{term "omap"}*}
paulson@13316
   377
paulson@13316
   378
lemma omap_reflects:
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   379
 "REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
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   380
     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
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   381
     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
wenzelm@13428
   382
 \<lambda>i z. \<exists>a \<in> Lset(i). a\<in>B & (\<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i).
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   383
        \<exists>par \<in> Lset(i).
wenzelm@13428
   384
         ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) &
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   385
         membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
paulson@13316
   386
         order_isomorphism(**Lset(i),par,r,x,mx,g))]"
paulson@13323
   387
by (intro FOL_reflections function_reflections fun_plus_reflections)
paulson@13316
   388
paulson@13316
   389
lemma omap_replacement:
wenzelm@13428
   390
     "[| L(A); L(r) |]
paulson@13316
   391
      ==> strong_replacement(L,
wenzelm@13428
   392
             \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
wenzelm@13428
   393
             ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
wenzelm@13428
   394
             pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
wenzelm@13428
   395
apply (rule strong_replacementI)
paulson@13316
   396
apply (rule rallI)
wenzelm@13428
   397
apply (rename_tac B)
wenzelm@13428
   398
apply (rule separation_CollectI)
wenzelm@13428
   399
apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast )
paulson@13316
   400
apply (rule ReflectsE [OF omap_reflects], assumption)
wenzelm@13428
   401
apply (drule subset_Lset_ltD, assumption)
paulson@13316
   402
apply (erule reflection_imp_L_separation)
paulson@13316
   403
  apply (simp_all add: lt_Ord2)
paulson@13385
   404
apply (rule DPow_LsetI)
wenzelm@13428
   405
apply (rename_tac u)
paulson@13316
   406
apply (rule bex_iff_sats conj_iff_sats)+
wenzelm@13428
   407
apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats)
paulson@13316
   408
apply (rule sep_rules | simp)+
paulson@13306
   409
done
paulson@13306
   410
paulson@13323
   411
paulson@13323
   412
subsection{*Separation for a Theorem about @{term "obase"}*}
paulson@13323
   413
paulson@13323
   414
lemma is_recfun_reflects:
wenzelm@13428
   415
  "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
wenzelm@13428
   416
                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
wenzelm@13428
   417
                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
paulson@13323
   418
                                   fx \<noteq> gx),
wenzelm@13428
   419
   \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
paulson@13323
   420
          pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r &
wenzelm@13428
   421
                (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) &
paulson@13323
   422
                  fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]"
paulson@13323
   423
by (intro FOL_reflections function_reflections fun_plus_reflections)
paulson@13323
   424
paulson@13323
   425
lemma is_recfun_separation:
paulson@13323
   426
     --{*for well-founded recursion*}
wenzelm@13428
   427
     "[| L(r); L(f); L(g); L(a); L(b) |]
wenzelm@13428
   428
     ==> separation(L,
wenzelm@13428
   429
            \<lambda>x. \<exists>xa[L]. \<exists>xb[L].
wenzelm@13428
   430
                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
wenzelm@13428
   431
                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
paulson@13323
   432
                                   fx \<noteq> gx))"
wenzelm@13428
   433
apply (rule separation_CollectI)
wenzelm@13428
   434
apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast )
paulson@13323
   435
apply (rule ReflectsE [OF is_recfun_reflects], assumption)
wenzelm@13428
   436
apply (drule subset_Lset_ltD, assumption)
paulson@13323
   437
apply (erule reflection_imp_L_separation)
paulson@13323
   438
  apply (simp_all add: lt_Ord2)
paulson@13385
   439
apply (rule DPow_LsetI)
wenzelm@13428
   440
apply (rename_tac u)
paulson@13323
   441
apply (rule bex_iff_sats conj_iff_sats)+
wenzelm@13428
   442
apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats)
paulson@13323
   443
apply (rule sep_rules | simp)+
paulson@13323
   444
done
paulson@13323
   445
paulson@13323
   446
paulson@13363
   447
subsection{*Instantiating the locale @{text M_axioms}*}
paulson@13363
   448
text{*Separation (and Strong Replacement) for basic set-theoretic constructions
paulson@13363
   449
such as intersection, Cartesian Product and image.*}
paulson@13363
   450
wenzelm@13429
   451
theorem M_axioms_L: "PROP M_axioms(L)"
wenzelm@13429
   452
  apply (rule M_axioms.intro)
wenzelm@13429
   453
   apply (rule M_triv_axioms_L)
wenzelm@13428
   454
  apply (rule M_axioms_axioms.intro)
wenzelm@13428
   455
               apply (assumption | rule
wenzelm@13428
   456
                 Inter_separation cartprod_separation image_separation
wenzelm@13428
   457
                 converse_separation restrict_separation
wenzelm@13428
   458
                 comp_separation pred_separation Memrel_separation
wenzelm@13428
   459
                 funspace_succ_replacement well_ord_iso_separation
wenzelm@13428
   460
                 obase_separation obase_equals_separation
wenzelm@13428
   461
                 omap_replacement is_recfun_separation)+
wenzelm@13428
   462
  done
paulson@13323
   463
wenzelm@13428
   464
lemmas cartprod_iff = M_axioms.cartprod_iff [OF M_axioms_L]
wenzelm@13428
   465
  and cartprod_closed = M_axioms.cartprod_closed [OF M_axioms_L]
wenzelm@13428
   466
  and sum_closed = M_axioms.sum_closed [OF M_axioms_L]
wenzelm@13428
   467
  and M_converse_iff = M_axioms.M_converse_iff [OF M_axioms_L]
wenzelm@13428
   468
  and converse_closed = M_axioms.converse_closed [OF M_axioms_L]
wenzelm@13428
   469
  and converse_abs = M_axioms.converse_abs [OF M_axioms_L]
wenzelm@13428
   470
  and image_closed = M_axioms.image_closed [OF M_axioms_L]
wenzelm@13428
   471
  and vimage_abs = M_axioms.vimage_abs [OF M_axioms_L]
wenzelm@13428
   472
  and vimage_closed = M_axioms.vimage_closed [OF M_axioms_L]
wenzelm@13428
   473
  and domain_abs = M_axioms.domain_abs [OF M_axioms_L]
wenzelm@13428
   474
  and domain_closed = M_axioms.domain_closed [OF M_axioms_L]
wenzelm@13428
   475
  and range_abs = M_axioms.range_abs [OF M_axioms_L]
wenzelm@13428
   476
  and range_closed = M_axioms.range_closed [OF M_axioms_L]
wenzelm@13428
   477
  and field_abs = M_axioms.field_abs [OF M_axioms_L]
wenzelm@13428
   478
  and field_closed = M_axioms.field_closed [OF M_axioms_L]
wenzelm@13428
   479
  and relation_abs = M_axioms.relation_abs [OF M_axioms_L]
wenzelm@13428
   480
  and function_abs = M_axioms.function_abs [OF M_axioms_L]
wenzelm@13428
   481
  and apply_closed = M_axioms.apply_closed [OF M_axioms_L]
wenzelm@13428
   482
  and apply_abs = M_axioms.apply_abs [OF M_axioms_L]
wenzelm@13428
   483
  and typed_function_abs = M_axioms.typed_function_abs [OF M_axioms_L]
wenzelm@13428
   484
  and injection_abs = M_axioms.injection_abs [OF M_axioms_L]
wenzelm@13428
   485
  and surjection_abs = M_axioms.surjection_abs [OF M_axioms_L]
wenzelm@13428
   486
  and bijection_abs = M_axioms.bijection_abs [OF M_axioms_L]
wenzelm@13428
   487
  and M_comp_iff = M_axioms.M_comp_iff [OF M_axioms_L]
wenzelm@13428
   488
  and comp_closed = M_axioms.comp_closed [OF M_axioms_L]
wenzelm@13428
   489
  and composition_abs = M_axioms.composition_abs [OF M_axioms_L]
wenzelm@13428
   490
  and restriction_is_function = M_axioms.restriction_is_function [OF M_axioms_L]
wenzelm@13428
   491
  and restriction_abs = M_axioms.restriction_abs [OF M_axioms_L]
wenzelm@13428
   492
  and M_restrict_iff = M_axioms.M_restrict_iff [OF M_axioms_L]
wenzelm@13428
   493
  and restrict_closed = M_axioms.restrict_closed [OF M_axioms_L]
wenzelm@13428
   494
  and Inter_abs = M_axioms.Inter_abs [OF M_axioms_L]
wenzelm@13428
   495
  and Inter_closed = M_axioms.Inter_closed [OF M_axioms_L]
wenzelm@13428
   496
  and Int_closed = M_axioms.Int_closed [OF M_axioms_L]
wenzelm@13428
   497
  and finite_fun_closed = M_axioms.finite_fun_closed [OF M_axioms_L]
wenzelm@13428
   498
  and is_funspace_abs = M_axioms.is_funspace_abs [OF M_axioms_L]
wenzelm@13428
   499
  and succ_fun_eq2 = M_axioms.succ_fun_eq2 [OF M_axioms_L]
wenzelm@13428
   500
  and funspace_succ = M_axioms.funspace_succ [OF M_axioms_L]
wenzelm@13428
   501
  and finite_funspace_closed = M_axioms.finite_funspace_closed [OF M_axioms_L]
paulson@13323
   502
wenzelm@13428
   503
lemmas is_recfun_equal = M_axioms.is_recfun_equal [OF M_axioms_L]
wenzelm@13428
   504
  and is_recfun_cut = M_axioms.is_recfun_cut [OF M_axioms_L]
wenzelm@13428
   505
  and is_recfun_functional = M_axioms.is_recfun_functional [OF M_axioms_L]
wenzelm@13428
   506
  and is_recfun_relativize = M_axioms.is_recfun_relativize [OF M_axioms_L]
wenzelm@13428
   507
  and is_recfun_restrict = M_axioms.is_recfun_restrict [OF M_axioms_L]
wenzelm@13428
   508
  and univalent_is_recfun = M_axioms.univalent_is_recfun [OF M_axioms_L]
wenzelm@13428
   509
  and exists_is_recfun_indstep = M_axioms.exists_is_recfun_indstep [OF M_axioms_L]
wenzelm@13428
   510
  and wellfounded_exists_is_recfun = M_axioms.wellfounded_exists_is_recfun [OF M_axioms_L]
wenzelm@13428
   511
  and wf_exists_is_recfun = M_axioms.wf_exists_is_recfun [OF M_axioms_L]
wenzelm@13428
   512
  and is_recfun_abs = M_axioms.is_recfun_abs [OF M_axioms_L]
wenzelm@13428
   513
  and irreflexive_abs = M_axioms.irreflexive_abs [OF M_axioms_L]
wenzelm@13428
   514
  and transitive_rel_abs = M_axioms.transitive_rel_abs [OF M_axioms_L]
wenzelm@13428
   515
  and linear_rel_abs = M_axioms.linear_rel_abs [OF M_axioms_L]
wenzelm@13428
   516
  and wellordered_is_trans_on = M_axioms.wellordered_is_trans_on [OF M_axioms_L]
wenzelm@13428
   517
  and wellordered_is_linear = M_axioms.wellordered_is_linear [OF M_axioms_L]
wenzelm@13428
   518
  and wellordered_is_wellfounded_on = M_axioms.wellordered_is_wellfounded_on [OF M_axioms_L]
wenzelm@13428
   519
  and wellfounded_imp_wellfounded_on = M_axioms.wellfounded_imp_wellfounded_on [OF M_axioms_L]
wenzelm@13428
   520
  and wellfounded_on_subset_A = M_axioms.wellfounded_on_subset_A [OF M_axioms_L]
wenzelm@13428
   521
  and wellfounded_on_iff_wellfounded = M_axioms.wellfounded_on_iff_wellfounded [OF M_axioms_L]
wenzelm@13428
   522
  and wellfounded_on_imp_wellfounded = M_axioms.wellfounded_on_imp_wellfounded [OF M_axioms_L]
wenzelm@13428
   523
  and wellfounded_on_field_imp_wellfounded = M_axioms.wellfounded_on_field_imp_wellfounded [OF M_axioms_L]
wenzelm@13428
   524
  and wellfounded_iff_wellfounded_on_field = M_axioms.wellfounded_iff_wellfounded_on_field [OF M_axioms_L]
wenzelm@13428
   525
  and wellfounded_induct = M_axioms.wellfounded_induct [OF M_axioms_L]
wenzelm@13428
   526
  and wellfounded_on_induct = M_axioms.wellfounded_on_induct [OF M_axioms_L]
wenzelm@13428
   527
  and wellfounded_on_induct2 = M_axioms.wellfounded_on_induct2 [OF M_axioms_L]
wenzelm@13428
   528
  and linear_imp_relativized = M_axioms.linear_imp_relativized [OF M_axioms_L]
wenzelm@13428
   529
  and trans_on_imp_relativized = M_axioms.trans_on_imp_relativized [OF M_axioms_L]
wenzelm@13428
   530
  and wf_on_imp_relativized = M_axioms.wf_on_imp_relativized [OF M_axioms_L]
wenzelm@13428
   531
  and wf_imp_relativized = M_axioms.wf_imp_relativized [OF M_axioms_L]
wenzelm@13428
   532
  and well_ord_imp_relativized = M_axioms.well_ord_imp_relativized [OF M_axioms_L]
wenzelm@13428
   533
  and order_isomorphism_abs = M_axioms.order_isomorphism_abs [OF M_axioms_L]
wenzelm@13428
   534
  and pred_set_abs = M_axioms.pred_set_abs [OF M_axioms_L]
paulson@13323
   535
wenzelm@13428
   536
lemmas pred_closed = M_axioms.pred_closed [OF M_axioms_L]
wenzelm@13428
   537
  and membership_abs = M_axioms.membership_abs [OF M_axioms_L]
wenzelm@13428
   538
  and M_Memrel_iff = M_axioms.M_Memrel_iff [OF M_axioms_L]
wenzelm@13428
   539
  and Memrel_closed = M_axioms.Memrel_closed [OF M_axioms_L]
wenzelm@13428
   540
  and wellordered_iso_predD = M_axioms.wellordered_iso_predD [OF M_axioms_L]
wenzelm@13428
   541
  and wellordered_iso_pred_eq = M_axioms.wellordered_iso_pred_eq [OF M_axioms_L]
wenzelm@13428
   542
  and wellfounded_on_asym = M_axioms.wellfounded_on_asym [OF M_axioms_L]
wenzelm@13428
   543
  and wellordered_asym = M_axioms.wellordered_asym [OF M_axioms_L]
wenzelm@13428
   544
  and ord_iso_pred_imp_lt = M_axioms.ord_iso_pred_imp_lt [OF M_axioms_L]
wenzelm@13428
   545
  and obase_iff = M_axioms.obase_iff [OF M_axioms_L]
wenzelm@13428
   546
  and omap_iff = M_axioms.omap_iff [OF M_axioms_L]
wenzelm@13428
   547
  and omap_unique = M_axioms.omap_unique [OF M_axioms_L]
wenzelm@13428
   548
  and omap_yields_Ord = M_axioms.omap_yields_Ord [OF M_axioms_L]
wenzelm@13428
   549
  and otype_iff = M_axioms.otype_iff [OF M_axioms_L]
wenzelm@13428
   550
  and otype_eq_range = M_axioms.otype_eq_range [OF M_axioms_L]
wenzelm@13428
   551
  and Ord_otype = M_axioms.Ord_otype [OF M_axioms_L]
wenzelm@13428
   552
  and domain_omap = M_axioms.domain_omap [OF M_axioms_L]
wenzelm@13428
   553
  and omap_subset = M_axioms.omap_subset [OF M_axioms_L]
wenzelm@13428
   554
  and omap_funtype = M_axioms.omap_funtype [OF M_axioms_L]
wenzelm@13428
   555
  and wellordered_omap_bij = M_axioms.wellordered_omap_bij [OF M_axioms_L]
wenzelm@13428
   556
  and omap_ord_iso = M_axioms.omap_ord_iso [OF M_axioms_L]
wenzelm@13428
   557
  and Ord_omap_image_pred = M_axioms.Ord_omap_image_pred [OF M_axioms_L]
wenzelm@13428
   558
  and restrict_omap_ord_iso = M_axioms.restrict_omap_ord_iso [OF M_axioms_L]
wenzelm@13428
   559
  and obase_equals = M_axioms.obase_equals [OF M_axioms_L]
wenzelm@13428
   560
  and omap_ord_iso_otype = M_axioms.omap_ord_iso_otype [OF M_axioms_L]
wenzelm@13428
   561
  and obase_exists = M_axioms.obase_exists [OF M_axioms_L]
wenzelm@13428
   562
  and omap_exists = M_axioms.omap_exists [OF M_axioms_L]
wenzelm@13428
   563
  and otype_exists = M_axioms.otype_exists [OF M_axioms_L]
wenzelm@13428
   564
  and omap_ord_iso_otype' = M_axioms.omap_ord_iso_otype' [OF M_axioms_L]
wenzelm@13428
   565
  and ordertype_exists = M_axioms.ordertype_exists [OF M_axioms_L]
wenzelm@13428
   566
  and relativized_imp_well_ord = M_axioms.relativized_imp_well_ord [OF M_axioms_L]
wenzelm@13428
   567
  and well_ord_abs = M_axioms.well_ord_abs [OF M_axioms_L]
wenzelm@13428
   568
wenzelm@13429
   569
declare cartprod_closed [intro, simp]
wenzelm@13429
   570
declare sum_closed [intro, simp]
wenzelm@13429
   571
declare converse_closed [intro, simp]
paulson@13323
   572
declare converse_abs [simp]
wenzelm@13429
   573
declare image_closed [intro, simp]
paulson@13323
   574
declare vimage_abs [simp]
wenzelm@13429
   575
declare vimage_closed [intro, simp]
paulson@13323
   576
declare domain_abs [simp]
wenzelm@13429
   577
declare domain_closed [intro, simp]
paulson@13323
   578
declare range_abs [simp]
wenzelm@13429
   579
declare range_closed [intro, simp]
paulson@13323
   580
declare field_abs [simp]
wenzelm@13429
   581
declare field_closed [intro, simp]
paulson@13323
   582
declare relation_abs [simp]
paulson@13323
   583
declare function_abs [simp]
wenzelm@13429
   584
declare apply_closed [intro, simp]
paulson@13323
   585
declare typed_function_abs [simp]
paulson@13323
   586
declare injection_abs [simp]
paulson@13323
   587
declare surjection_abs [simp]
paulson@13323
   588
declare bijection_abs [simp]
wenzelm@13429
   589
declare comp_closed [intro, simp]
paulson@13323
   590
declare composition_abs [simp]
paulson@13323
   591
declare restriction_abs [simp]
wenzelm@13429
   592
declare restrict_closed [intro, simp]
paulson@13323
   593
declare Inter_abs [simp]
wenzelm@13429
   594
declare Inter_closed [intro, simp]
wenzelm@13429
   595
declare Int_closed [intro, simp]
paulson@13323
   596
declare is_funspace_abs [simp]
wenzelm@13429
   597
declare finite_funspace_closed [intro, simp]
paulson@13323
   598
paulson@13306
   599
end