src/ZF/Constructible/Separation.thy
author paulson
Wed Jul 17 15:48:54 2002 +0200 (2002-07-17)
changeset 13385 31df66ca0780
parent 13363 c26eeb000470
child 13428 99e52e78eb65
permissions -rw-r--r--
Expressing Lset and L without using length and arity; simplifies Separation
proofs
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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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   278
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)+
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   307
done
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   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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	     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
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   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
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   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)))]"
paulson@13323
   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]. 
paulson@13316
   359
			      ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 
paulson@13316
   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))))"
paulson@13316
   362
apply (rule separation_CollectI) 
paulson@13316
   363
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 
paulson@13316
   364
apply (rule ReflectsE [OF obase_equals_reflects], assumption)
paulson@13316
   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)
paulson@13316
   369
apply (rename_tac u) 
paulson@13316
   370
apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+
paulson@13316
   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]. 
paulson@13316
   380
     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 
paulson@13316
   381
     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
paulson@13316
   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). 
paulson@13316
   383
        \<exists>par \<in> Lset(i). 
paulson@13316
   384
	 ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) & 
paulson@13316
   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:
paulson@13316
   390
     "[| L(A); L(r) |] 
paulson@13316
   391
      ==> strong_replacement(L,
paulson@13316
   392
             \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
paulson@13316
   393
	     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 
paulson@13316
   394
	     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
paulson@13316
   395
apply (rule strong_replacementI) 
paulson@13316
   396
apply (rule rallI)
paulson@13316
   397
apply (rename_tac B)  
paulson@13316
   398
apply (rule separation_CollectI) 
paulson@13316
   399
apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast ) 
paulson@13316
   400
apply (rule ReflectsE [OF omap_reflects], assumption)
paulson@13316
   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)
paulson@13316
   405
apply (rename_tac u) 
paulson@13316
   406
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13339
   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:
paulson@13323
   415
  "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. 
paulson@13323
   416
                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & 
paulson@13323
   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),
paulson@13323
   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 &
paulson@13323
   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*}
paulson@13323
   427
     "[| L(r); L(f); L(g); L(a); L(b) |] 
paulson@13323
   428
     ==> separation(L, 
paulson@13323
   429
            \<lambda>x. \<exists>xa[L]. \<exists>xb[L]. 
paulson@13323
   430
                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & 
paulson@13323
   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))"
paulson@13323
   433
apply (rule separation_CollectI) 
paulson@13323
   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)
paulson@13323
   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)
paulson@13323
   440
apply (rename_tac u) 
paulson@13323
   441
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13339
   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
paulson@13323
   451
ML
paulson@13323
   452
{*
paulson@13323
   453
val Inter_separation = thm "Inter_separation";
paulson@13323
   454
val cartprod_separation = thm "cartprod_separation";
paulson@13323
   455
val image_separation = thm "image_separation";
paulson@13323
   456
val converse_separation = thm "converse_separation";
paulson@13323
   457
val restrict_separation = thm "restrict_separation";
paulson@13323
   458
val comp_separation = thm "comp_separation";
paulson@13323
   459
val pred_separation = thm "pred_separation";
paulson@13323
   460
val Memrel_separation = thm "Memrel_separation";
paulson@13323
   461
val funspace_succ_replacement = thm "funspace_succ_replacement";
paulson@13323
   462
val well_ord_iso_separation = thm "well_ord_iso_separation";
paulson@13323
   463
val obase_separation = thm "obase_separation";
paulson@13323
   464
val obase_equals_separation = thm "obase_equals_separation";
paulson@13323
   465
val omap_replacement = thm "omap_replacement";
paulson@13323
   466
val is_recfun_separation = thm "is_recfun_separation";
paulson@13323
   467
paulson@13323
   468
val m_axioms = 
paulson@13323
   469
    [Inter_separation, cartprod_separation, image_separation, 
paulson@13323
   470
     converse_separation, restrict_separation, comp_separation, 
paulson@13323
   471
     pred_separation, Memrel_separation, funspace_succ_replacement, 
paulson@13323
   472
     well_ord_iso_separation, obase_separation,
paulson@13323
   473
     obase_equals_separation, omap_replacement, is_recfun_separation]
paulson@13323
   474
paulson@13363
   475
fun axioms_L th =
paulson@13363
   476
    kill_flex_triv_prems (m_axioms MRS (triv_axioms_L th));
paulson@13323
   477
paulson@13363
   478
bind_thm ("cartprod_iff", axioms_L (thm "M_axioms.cartprod_iff"));
paulson@13363
   479
bind_thm ("cartprod_closed", axioms_L (thm "M_axioms.cartprod_closed"));
paulson@13363
   480
bind_thm ("sum_closed", axioms_L (thm "M_axioms.sum_closed"));
paulson@13363
   481
bind_thm ("M_converse_iff", axioms_L (thm "M_axioms.M_converse_iff"));
paulson@13363
   482
bind_thm ("converse_closed", axioms_L (thm "M_axioms.converse_closed"));
paulson@13363
   483
bind_thm ("converse_abs", axioms_L (thm "M_axioms.converse_abs"));
paulson@13363
   484
bind_thm ("image_closed", axioms_L (thm "M_axioms.image_closed"));
paulson@13363
   485
bind_thm ("vimage_abs", axioms_L (thm "M_axioms.vimage_abs"));
paulson@13363
   486
bind_thm ("vimage_closed", axioms_L (thm "M_axioms.vimage_closed"));
paulson@13363
   487
bind_thm ("domain_abs", axioms_L (thm "M_axioms.domain_abs"));
paulson@13363
   488
bind_thm ("domain_closed", axioms_L (thm "M_axioms.domain_closed"));
paulson@13363
   489
bind_thm ("range_abs", axioms_L (thm "M_axioms.range_abs"));
paulson@13363
   490
bind_thm ("range_closed", axioms_L (thm "M_axioms.range_closed"));
paulson@13363
   491
bind_thm ("field_abs", axioms_L (thm "M_axioms.field_abs"));
paulson@13363
   492
bind_thm ("field_closed", axioms_L (thm "M_axioms.field_closed"));
paulson@13363
   493
bind_thm ("relation_abs", axioms_L (thm "M_axioms.relation_abs"));
paulson@13363
   494
bind_thm ("function_abs", axioms_L (thm "M_axioms.function_abs"));
paulson@13363
   495
bind_thm ("apply_closed", axioms_L (thm "M_axioms.apply_closed"));
paulson@13363
   496
bind_thm ("apply_abs", axioms_L (thm "M_axioms.apply_abs"));
paulson@13363
   497
bind_thm ("typed_function_abs", axioms_L (thm "M_axioms.typed_function_abs"));
paulson@13363
   498
bind_thm ("injection_abs", axioms_L (thm "M_axioms.injection_abs"));
paulson@13363
   499
bind_thm ("surjection_abs", axioms_L (thm "M_axioms.surjection_abs"));
paulson@13363
   500
bind_thm ("bijection_abs", axioms_L (thm "M_axioms.bijection_abs"));
paulson@13363
   501
bind_thm ("M_comp_iff", axioms_L (thm "M_axioms.M_comp_iff"));
paulson@13363
   502
bind_thm ("comp_closed", axioms_L (thm "M_axioms.comp_closed"));
paulson@13363
   503
bind_thm ("composition_abs", axioms_L (thm "M_axioms.composition_abs"));
paulson@13363
   504
bind_thm ("restriction_is_function", axioms_L (thm "M_axioms.restriction_is_function"));
paulson@13363
   505
bind_thm ("restriction_abs", axioms_L (thm "M_axioms.restriction_abs"));
paulson@13363
   506
bind_thm ("M_restrict_iff", axioms_L (thm "M_axioms.M_restrict_iff"));
paulson@13363
   507
bind_thm ("restrict_closed", axioms_L (thm "M_axioms.restrict_closed"));
paulson@13363
   508
bind_thm ("Inter_abs", axioms_L (thm "M_axioms.Inter_abs"));
paulson@13363
   509
bind_thm ("Inter_closed", axioms_L (thm "M_axioms.Inter_closed"));
paulson@13363
   510
bind_thm ("Int_closed", axioms_L (thm "M_axioms.Int_closed"));
paulson@13363
   511
bind_thm ("finite_fun_closed", axioms_L (thm "M_axioms.finite_fun_closed"));
paulson@13363
   512
bind_thm ("is_funspace_abs", axioms_L (thm "M_axioms.is_funspace_abs"));
paulson@13363
   513
bind_thm ("succ_fun_eq2", axioms_L (thm "M_axioms.succ_fun_eq2"));
paulson@13363
   514
bind_thm ("funspace_succ", axioms_L (thm "M_axioms.funspace_succ"));
paulson@13363
   515
bind_thm ("finite_funspace_closed", axioms_L (thm "M_axioms.finite_funspace_closed"));
paulson@13323
   516
*}
paulson@13323
   517
paulson@13323
   518
ML
paulson@13323
   519
{*
paulson@13363
   520
bind_thm ("is_recfun_equal", axioms_L (thm "M_axioms.is_recfun_equal"));  
paulson@13363
   521
bind_thm ("is_recfun_cut", axioms_L (thm "M_axioms.is_recfun_cut")); 
paulson@13363
   522
bind_thm ("is_recfun_functional", axioms_L (thm "M_axioms.is_recfun_functional"));
paulson@13363
   523
bind_thm ("is_recfun_relativize", axioms_L (thm "M_axioms.is_recfun_relativize"));
paulson@13363
   524
bind_thm ("is_recfun_restrict", axioms_L (thm "M_axioms.is_recfun_restrict"));
paulson@13363
   525
bind_thm ("univalent_is_recfun", axioms_L (thm "M_axioms.univalent_is_recfun"));
paulson@13363
   526
bind_thm ("exists_is_recfun_indstep", axioms_L (thm "M_axioms.exists_is_recfun_indstep"));
paulson@13363
   527
bind_thm ("wellfounded_exists_is_recfun", axioms_L (thm "M_axioms.wellfounded_exists_is_recfun"));
paulson@13363
   528
bind_thm ("wf_exists_is_recfun", axioms_L (thm "M_axioms.wf_exists_is_recfun")); 
paulson@13363
   529
bind_thm ("is_recfun_abs", axioms_L (thm "M_axioms.is_recfun_abs"));
paulson@13363
   530
bind_thm ("irreflexive_abs", axioms_L (thm "M_axioms.irreflexive_abs"));  
paulson@13363
   531
bind_thm ("transitive_rel_abs", axioms_L (thm "M_axioms.transitive_rel_abs"));  
paulson@13363
   532
bind_thm ("linear_rel_abs", axioms_L (thm "M_axioms.linear_rel_abs"));  
paulson@13363
   533
bind_thm ("wellordered_is_trans_on", axioms_L (thm "M_axioms.wellordered_is_trans_on")); 
paulson@13363
   534
bind_thm ("wellordered_is_linear", axioms_L (thm "M_axioms.wellordered_is_linear")); 
paulson@13363
   535
bind_thm ("wellordered_is_wellfounded_on", axioms_L (thm "M_axioms.wellordered_is_wellfounded_on")); 
paulson@13363
   536
bind_thm ("wellfounded_imp_wellfounded_on", axioms_L (thm "M_axioms.wellfounded_imp_wellfounded_on")); 
paulson@13363
   537
bind_thm ("wellfounded_on_subset_A", axioms_L (thm "M_axioms.wellfounded_on_subset_A"));
paulson@13363
   538
bind_thm ("wellfounded_on_iff_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_iff_wellfounded"));
paulson@13363
   539
bind_thm ("wellfounded_on_imp_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_imp_wellfounded"));
paulson@13363
   540
bind_thm ("wellfounded_on_field_imp_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_field_imp_wellfounded"));
paulson@13363
   541
bind_thm ("wellfounded_iff_wellfounded_on_field", axioms_L (thm "M_axioms.wellfounded_iff_wellfounded_on_field"));
paulson@13363
   542
bind_thm ("wellfounded_induct", axioms_L (thm "M_axioms.wellfounded_induct")); 
paulson@13363
   543
bind_thm ("wellfounded_on_induct", axioms_L (thm "M_axioms.wellfounded_on_induct")); 
paulson@13363
   544
bind_thm ("wellfounded_on_induct2", axioms_L (thm "M_axioms.wellfounded_on_induct2")); 
paulson@13363
   545
bind_thm ("linear_imp_relativized", axioms_L (thm "M_axioms.linear_imp_relativized")); 
paulson@13363
   546
bind_thm ("trans_on_imp_relativized", axioms_L (thm "M_axioms.trans_on_imp_relativized")); 
paulson@13363
   547
bind_thm ("wf_on_imp_relativized", axioms_L (thm "M_axioms.wf_on_imp_relativized")); 
paulson@13363
   548
bind_thm ("wf_imp_relativized", axioms_L (thm "M_axioms.wf_imp_relativized")); 
paulson@13363
   549
bind_thm ("well_ord_imp_relativized", axioms_L (thm "M_axioms.well_ord_imp_relativized")); 
paulson@13363
   550
bind_thm ("order_isomorphism_abs", axioms_L (thm "M_axioms.order_isomorphism_abs"));  
paulson@13363
   551
bind_thm ("pred_set_abs", axioms_L (thm "M_axioms.pred_set_abs"));  
paulson@13323
   552
*}
paulson@13323
   553
paulson@13323
   554
ML
paulson@13323
   555
{*
paulson@13363
   556
bind_thm ("pred_closed", axioms_L (thm "M_axioms.pred_closed"));  
paulson@13363
   557
bind_thm ("membership_abs", axioms_L (thm "M_axioms.membership_abs"));  
paulson@13363
   558
bind_thm ("M_Memrel_iff", axioms_L (thm "M_axioms.M_Memrel_iff"));
paulson@13363
   559
bind_thm ("Memrel_closed", axioms_L (thm "M_axioms.Memrel_closed"));  
paulson@13363
   560
bind_thm ("wellordered_iso_predD", axioms_L (thm "M_axioms.wellordered_iso_predD"));
paulson@13363
   561
bind_thm ("wellordered_iso_pred_eq", axioms_L (thm "M_axioms.wellordered_iso_pred_eq"));
paulson@13363
   562
bind_thm ("wellfounded_on_asym", axioms_L (thm "M_axioms.wellfounded_on_asym"));
paulson@13363
   563
bind_thm ("wellordered_asym", axioms_L (thm "M_axioms.wellordered_asym"));
paulson@13363
   564
bind_thm ("ord_iso_pred_imp_lt", axioms_L (thm "M_axioms.ord_iso_pred_imp_lt"));
paulson@13363
   565
bind_thm ("obase_iff", axioms_L (thm "M_axioms.obase_iff"));
paulson@13363
   566
bind_thm ("omap_iff", axioms_L (thm "M_axioms.omap_iff"));
paulson@13363
   567
bind_thm ("omap_unique", axioms_L (thm "M_axioms.omap_unique"));
paulson@13363
   568
bind_thm ("omap_yields_Ord", axioms_L (thm "M_axioms.omap_yields_Ord"));
paulson@13363
   569
bind_thm ("otype_iff", axioms_L (thm "M_axioms.otype_iff"));
paulson@13363
   570
bind_thm ("otype_eq_range", axioms_L (thm "M_axioms.otype_eq_range"));
paulson@13363
   571
bind_thm ("Ord_otype", axioms_L (thm "M_axioms.Ord_otype"));
paulson@13363
   572
bind_thm ("domain_omap", axioms_L (thm "M_axioms.domain_omap"));
paulson@13363
   573
bind_thm ("omap_subset", axioms_L (thm "M_axioms.omap_subset")); 
paulson@13363
   574
bind_thm ("omap_funtype", axioms_L (thm "M_axioms.omap_funtype")); 
paulson@13363
   575
bind_thm ("wellordered_omap_bij", axioms_L (thm "M_axioms.wellordered_omap_bij"));
paulson@13363
   576
bind_thm ("omap_ord_iso", axioms_L (thm "M_axioms.omap_ord_iso"));
paulson@13363
   577
bind_thm ("Ord_omap_image_pred", axioms_L (thm "M_axioms.Ord_omap_image_pred"));
paulson@13363
   578
bind_thm ("restrict_omap_ord_iso", axioms_L (thm "M_axioms.restrict_omap_ord_iso"));
paulson@13363
   579
bind_thm ("obase_equals", axioms_L (thm "M_axioms.obase_equals")); 
paulson@13363
   580
bind_thm ("omap_ord_iso_otype", axioms_L (thm "M_axioms.omap_ord_iso_otype"));
paulson@13363
   581
bind_thm ("obase_exists", axioms_L (thm "M_axioms.obase_exists"));
paulson@13363
   582
bind_thm ("omap_exists", axioms_L (thm "M_axioms.omap_exists"));
paulson@13363
   583
bind_thm ("otype_exists", axioms_L (thm "M_axioms.otype_exists"));
paulson@13363
   584
bind_thm ("omap_ord_iso_otype", axioms_L (thm "M_axioms.omap_ord_iso_otype"));
paulson@13363
   585
bind_thm ("ordertype_exists", axioms_L (thm "M_axioms.ordertype_exists"));
paulson@13363
   586
bind_thm ("relativized_imp_well_ord", axioms_L (thm "M_axioms.relativized_imp_well_ord")); 
paulson@13363
   587
bind_thm ("well_ord_abs", axioms_L (thm "M_axioms.well_ord_abs"));  
paulson@13323
   588
*}
paulson@13323
   589
paulson@13323
   590
declare cartprod_closed [intro,simp]
paulson@13323
   591
declare sum_closed [intro,simp]
paulson@13323
   592
declare converse_closed [intro,simp]
paulson@13323
   593
declare converse_abs [simp]
paulson@13323
   594
declare image_closed [intro,simp]
paulson@13323
   595
declare vimage_abs [simp]
paulson@13323
   596
declare vimage_closed [intro,simp]
paulson@13323
   597
declare domain_abs [simp]
paulson@13323
   598
declare domain_closed [intro,simp]
paulson@13323
   599
declare range_abs [simp]
paulson@13323
   600
declare range_closed [intro,simp]
paulson@13323
   601
declare field_abs [simp]
paulson@13323
   602
declare field_closed [intro,simp]
paulson@13323
   603
declare relation_abs [simp]
paulson@13323
   604
declare function_abs [simp]
paulson@13323
   605
declare apply_closed [intro,simp]
paulson@13323
   606
declare typed_function_abs [simp]
paulson@13323
   607
declare injection_abs [simp]
paulson@13323
   608
declare surjection_abs [simp]
paulson@13323
   609
declare bijection_abs [simp]
paulson@13323
   610
declare comp_closed [intro,simp]
paulson@13323
   611
declare composition_abs [simp]
paulson@13323
   612
declare restriction_abs [simp]
paulson@13323
   613
declare restrict_closed [intro,simp]
paulson@13323
   614
declare Inter_abs [simp]
paulson@13323
   615
declare Inter_closed [intro,simp]
paulson@13323
   616
declare Int_closed [intro,simp]
paulson@13323
   617
declare finite_fun_closed [rule_format]
paulson@13323
   618
declare is_funspace_abs [simp]
paulson@13323
   619
declare finite_funspace_closed [intro,simp]
paulson@13323
   620
paulson@13306
   621
end