src/ZF/Constructible/Separation.thy
author paulson
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instantiation of locales M_trancl and M_wfrank; proofs of list_replacement{1,2}
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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 DPowI2) 
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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 DPowI2)
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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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apply (simp_all add: succ_Un_distrib [symmetric])
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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 DPowI2)
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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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apply (simp_all add: succ_Un_distrib [symmetric])
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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 DPowI2)
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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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apply (simp_all add: succ_Un_distrib [symmetric])
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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 DPowI2)
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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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apply (simp_all add: succ_Un_distrib [symmetric])
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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 DPowI2)
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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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apply (simp_all add: succ_Un_distrib [symmetric])
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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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   221
  apply (simp_all add: lt_Ord2, clarify)
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apply (rule DPowI2)
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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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apply (simp_all add: succ_Un_distrib [symmetric])
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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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   242
apply (rule_tac A="{z}" in subset_LsetE, blast ) 
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   243
apply (rule ReflectsE [OF Memrel_Reflects], assumption)
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   244
apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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   246
  apply (simp_all add: lt_Ord2)
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   247
apply (rule DPowI2)
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   248
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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apply (simp_all add: succ_Un_distrib [symmetric])
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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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   267
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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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   272
                upair(L,cnbf,cnbf,z))"
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   273
apply (rule strong_replacementI) 
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   274
apply (rule rallI) 
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   275
apply (rule separation_CollectI) 
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   276
apply (rule_tac A="{n,A,z}" in subset_LsetE, blast ) 
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   277
apply (rule ReflectsE [OF funspace_succ_Reflects], assumption)
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   278
apply (drule subset_Lset_ltD, assumption) 
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   279
apply (erule reflection_imp_L_separation)
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   280
  apply (simp_all add: lt_Ord2)
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diff changeset
   281
apply (rule DPowI2)
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diff changeset
   282
apply (rename_tac u) 
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diff changeset
   283
apply (rule bex_iff_sats)
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   284
apply (rule conj_iff_sats)
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   285
apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats) 
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apply (rule sep_rules | simp)+
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   287
apply (simp_all add: succ_Un_distrib [symmetric])
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   288
done
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   289
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   291
subsection{*Separation for Order-Isomorphisms*}
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lemma well_ord_iso_Reflects:
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  "REFLECTS[\<lambda>x. x\<in>A --> 
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   295
                (\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r),
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   296
        \<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i). 
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   297
                fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]"
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   298
by (intro FOL_reflections function_reflections)
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   299
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lemma well_ord_iso_separation:
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     "[| L(A); L(f); L(r) |] 
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      ==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L]. 
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		     fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))"
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   304
apply (rule separation_CollectI) 
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paulson
parents:
diff changeset
   305
apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast ) 
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   306
apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption)
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paulson
parents:
diff changeset
   307
apply (drule subset_Lset_ltD, assumption) 
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paulson
parents:
diff changeset
   308
apply (erule reflection_imp_L_separation)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   309
  apply (simp_all add: lt_Ord2)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   310
apply (rule DPowI2)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   311
apply (rename_tac u) 
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   312
apply (rule imp_iff_sats)
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parents:
diff changeset
   313
apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats) 
13316
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parents: 13314
diff changeset
   314
apply (rule sep_rules | simp)+
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parents: 13314
diff changeset
   315
apply (simp_all add: succ_Un_distrib [symmetric])
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parents: 13314
diff changeset
   316
done
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parents: 13314
diff changeset
   317
d16629fd0f95 more and simpler separation proofs
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diff changeset
   318
d16629fd0f95 more and simpler separation proofs
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parents: 13314
diff changeset
   319
subsection{*Separation for @{term "obase"}*}
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parents: 13314
diff changeset
   320
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parents: 13314
diff changeset
   321
lemma obase_reflects:
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diff changeset
   322
  "REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
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parents: 13314
diff changeset
   323
	     ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
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diff changeset
   324
	     order_isomorphism(L,par,r,x,mx,g),
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parents: 13314
diff changeset
   325
        \<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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parents: 13314
diff changeset
   326
	     ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
d16629fd0f95 more and simpler separation proofs
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parents: 13314
diff changeset
   327
	     order_isomorphism(**Lset(i),par,r,x,mx,g)]"
13323
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diff changeset
   328
by (intro FOL_reflections function_reflections fun_plus_reflections)
13316
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parents: 13314
diff changeset
   329
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parents: 13314
diff changeset
   330
lemma obase_separation:
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parents: 13314
diff changeset
   331
     --{*part of the order type formalization*}
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parents: 13314
diff changeset
   332
     "[| L(A); L(r) |] 
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paulson
parents: 13314
diff changeset
   333
      ==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   334
	     ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   335
	     order_isomorphism(L,par,r,x,mx,g))"
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   336
apply (rule separation_CollectI) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   337
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   338
apply (rule ReflectsE [OF obase_reflects], assumption)
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paulson
parents: 13314
diff changeset
   339
apply (drule subset_Lset_ltD, assumption) 
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paulson
parents: 13314
diff changeset
   340
apply (erule reflection_imp_L_separation)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   341
  apply (simp_all add: lt_Ord2)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   342
apply (rule DPowI2)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   343
apply (rename_tac u) 
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   344
apply (rule bex_iff_sats)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   345
apply (rule conj_iff_sats)
13316
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paulson
parents: 13314
diff changeset
   346
apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats) 
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paulson
parents: 13314
diff changeset
   347
apply (rule sep_rules | simp)+
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paulson
parents: 13314
diff changeset
   348
apply (simp_all add: succ_Un_distrib [symmetric])
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   349
done
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   350
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   351
13319
23de7b3af453 More Separation proofs
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parents: 13316
diff changeset
   352
subsection{*Separation for a Theorem about @{term "obase"}*}
13316
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paulson
parents: 13314
diff changeset
   353
d16629fd0f95 more and simpler separation proofs
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parents: 13314
diff changeset
   354
lemma obase_equals_reflects:
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parents: 13314
diff changeset
   355
  "REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   356
		ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   357
		membership(L,y,my) & pred_set(L,A,x,r,pxr) &
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   358
		order_isomorphism(L,pxr,r,y,my,g))),
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   359
	\<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i). 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   360
		ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i). 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   361
		membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) &
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   362
		order_isomorphism(**Lset(i),pxr,r,y,my,g)))]"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   363
by (intro FOL_reflections function_reflections fun_plus_reflections)
13316
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   364
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   365
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   366
lemma obase_equals_separation:
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parents: 13314
diff changeset
   367
     "[| L(A); L(r) |] 
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paulson
parents: 13314
diff changeset
   368
      ==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   369
			      ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   370
			      membership(L,y,my) & pred_set(L,A,x,r,pxr) &
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   371
			      order_isomorphism(L,pxr,r,y,my,g))))"
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   372
apply (rule separation_CollectI) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   373
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   374
apply (rule ReflectsE [OF obase_equals_reflects], assumption)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   375
apply (drule subset_Lset_ltD, assumption) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   376
apply (erule reflection_imp_L_separation)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   377
  apply (simp_all add: lt_Ord2)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   378
apply (rule DPowI2)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   379
apply (rename_tac u) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   380
apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   381
apply (rule_tac env = "[u,A,r]" in mem_iff_sats) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   382
apply (rule sep_rules | simp)+
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   383
apply (simp_all add: succ_Un_distrib [symmetric])
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   384
done
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   385
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   386
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   387
subsection{*Replacement for @{term "omap"}*}
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   388
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   389
lemma omap_reflects:
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   390
 "REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   391
     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   392
     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   393
 \<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). 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   394
        \<exists>par \<in> Lset(i). 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   395
	 ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) & 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   396
         membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   397
         order_isomorphism(**Lset(i),par,r,x,mx,g))]"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   398
by (intro FOL_reflections function_reflections fun_plus_reflections)
13316
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   399
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   400
lemma omap_replacement:
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   401
     "[| L(A); L(r) |] 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   402
      ==> strong_replacement(L,
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   403
             \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   404
	     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   405
	     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   406
apply (rule strong_replacementI) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   407
apply (rule rallI)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   408
apply (rename_tac B)  
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   409
apply (rule separation_CollectI) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   410
apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast ) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   411
apply (rule ReflectsE [OF omap_reflects], assumption)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   412
apply (drule subset_Lset_ltD, assumption) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   413
apply (erule reflection_imp_L_separation)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   414
  apply (simp_all add: lt_Ord2)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   415
apply (rule DPowI2)
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   416
apply (rename_tac u) 
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   417
apply (rule bex_iff_sats conj_iff_sats)+
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13324
diff changeset
   418
apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats) 
13316
d16629fd0f95 more and simpler separation proofs
paulson
parents: 13314
diff changeset
   419
apply (rule sep_rules | simp)+
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   420
apply (simp_all add: succ_Un_distrib [symmetric])
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   421
done
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   422
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   423
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   424
subsection{*Separation for a Theorem about @{term "obase"}*}
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   425
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   426
lemma is_recfun_reflects:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   427
  "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   428
                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   429
                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   430
                                   fx \<noteq> gx),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   431
   \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i). 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   432
          pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r &
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   433
                (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) & 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   434
                  fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   435
by (intro FOL_reflections function_reflections fun_plus_reflections)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   436
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   437
lemma is_recfun_separation:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   438
     --{*for well-founded recursion*}
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   439
     "[| L(r); L(f); L(g); L(a); L(b) |] 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   440
     ==> separation(L, 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   441
            \<lambda>x. \<exists>xa[L]. \<exists>xb[L]. 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   442
                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   443
                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   444
                                   fx \<noteq> gx))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   445
apply (rule separation_CollectI) 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   446
apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast ) 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   447
apply (rule ReflectsE [OF is_recfun_reflects], assumption)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   448
apply (drule subset_Lset_ltD, assumption) 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   449
apply (erule reflection_imp_L_separation)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   450
  apply (simp_all add: lt_Ord2)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   451
apply (rule DPowI2)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   452
apply (rename_tac u) 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   453
apply (rule bex_iff_sats conj_iff_sats)+
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13324
diff changeset
   454
apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats) 
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   455
apply (rule sep_rules | simp)+
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   456
apply (simp_all add: succ_Un_distrib [symmetric])
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   457
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   458
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   459
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   460
subsection{*Instantiating the locale @{text M_axioms}*}
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   461
text{*Separation (and Strong Replacement) for basic set-theoretic constructions
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   462
such as intersection, Cartesian Product and image.*}
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   463
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   464
ML
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   465
{*
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   466
val Inter_separation = thm "Inter_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   467
val cartprod_separation = thm "cartprod_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   468
val image_separation = thm "image_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   469
val converse_separation = thm "converse_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   470
val restrict_separation = thm "restrict_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   471
val comp_separation = thm "comp_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   472
val pred_separation = thm "pred_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   473
val Memrel_separation = thm "Memrel_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   474
val funspace_succ_replacement = thm "funspace_succ_replacement";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   475
val well_ord_iso_separation = thm "well_ord_iso_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   476
val obase_separation = thm "obase_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   477
val obase_equals_separation = thm "obase_equals_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   478
val omap_replacement = thm "omap_replacement";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   479
val is_recfun_separation = thm "is_recfun_separation";
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   480
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   481
val m_axioms = 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   482
    [Inter_separation, cartprod_separation, image_separation, 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   483
     converse_separation, restrict_separation, comp_separation, 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   484
     pred_separation, Memrel_separation, funspace_succ_replacement, 
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   485
     well_ord_iso_separation, obase_separation,
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   486
     obase_equals_separation, omap_replacement, is_recfun_separation]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   487
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   488
fun axioms_L th =
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   489
    kill_flex_triv_prems (m_axioms MRS (triv_axioms_L th));
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   490
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   491
bind_thm ("cartprod_iff", axioms_L (thm "M_axioms.cartprod_iff"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   492
bind_thm ("cartprod_closed", axioms_L (thm "M_axioms.cartprod_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   493
bind_thm ("sum_closed", axioms_L (thm "M_axioms.sum_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   494
bind_thm ("M_converse_iff", axioms_L (thm "M_axioms.M_converse_iff"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   495
bind_thm ("converse_closed", axioms_L (thm "M_axioms.converse_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   496
bind_thm ("converse_abs", axioms_L (thm "M_axioms.converse_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   497
bind_thm ("image_closed", axioms_L (thm "M_axioms.image_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   498
bind_thm ("vimage_abs", axioms_L (thm "M_axioms.vimage_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   499
bind_thm ("vimage_closed", axioms_L (thm "M_axioms.vimage_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   500
bind_thm ("domain_abs", axioms_L (thm "M_axioms.domain_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   501
bind_thm ("domain_closed", axioms_L (thm "M_axioms.domain_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   502
bind_thm ("range_abs", axioms_L (thm "M_axioms.range_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   503
bind_thm ("range_closed", axioms_L (thm "M_axioms.range_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   504
bind_thm ("field_abs", axioms_L (thm "M_axioms.field_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   505
bind_thm ("field_closed", axioms_L (thm "M_axioms.field_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   506
bind_thm ("relation_abs", axioms_L (thm "M_axioms.relation_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   507
bind_thm ("function_abs", axioms_L (thm "M_axioms.function_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   508
bind_thm ("apply_closed", axioms_L (thm "M_axioms.apply_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   509
bind_thm ("apply_abs", axioms_L (thm "M_axioms.apply_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   510
bind_thm ("typed_function_abs", axioms_L (thm "M_axioms.typed_function_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   511
bind_thm ("injection_abs", axioms_L (thm "M_axioms.injection_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   512
bind_thm ("surjection_abs", axioms_L (thm "M_axioms.surjection_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   513
bind_thm ("bijection_abs", axioms_L (thm "M_axioms.bijection_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   514
bind_thm ("M_comp_iff", axioms_L (thm "M_axioms.M_comp_iff"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   515
bind_thm ("comp_closed", axioms_L (thm "M_axioms.comp_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   516
bind_thm ("composition_abs", axioms_L (thm "M_axioms.composition_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   517
bind_thm ("restriction_is_function", axioms_L (thm "M_axioms.restriction_is_function"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   518
bind_thm ("restriction_abs", axioms_L (thm "M_axioms.restriction_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   519
bind_thm ("M_restrict_iff", axioms_L (thm "M_axioms.M_restrict_iff"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   520
bind_thm ("restrict_closed", axioms_L (thm "M_axioms.restrict_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   521
bind_thm ("Inter_abs", axioms_L (thm "M_axioms.Inter_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   522
bind_thm ("Inter_closed", axioms_L (thm "M_axioms.Inter_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   523
bind_thm ("Int_closed", axioms_L (thm "M_axioms.Int_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   524
bind_thm ("finite_fun_closed", axioms_L (thm "M_axioms.finite_fun_closed"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   525
bind_thm ("is_funspace_abs", axioms_L (thm "M_axioms.is_funspace_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   526
bind_thm ("succ_fun_eq2", axioms_L (thm "M_axioms.succ_fun_eq2"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   527
bind_thm ("funspace_succ", axioms_L (thm "M_axioms.funspace_succ"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   528
bind_thm ("finite_funspace_closed", axioms_L (thm "M_axioms.finite_funspace_closed"));
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   529
*}
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   530
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   531
ML
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   532
{*
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   533
bind_thm ("is_recfun_equal", axioms_L (thm "M_axioms.is_recfun_equal"));  
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   534
bind_thm ("is_recfun_cut", axioms_L (thm "M_axioms.is_recfun_cut")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   535
bind_thm ("is_recfun_functional", axioms_L (thm "M_axioms.is_recfun_functional"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   536
bind_thm ("is_recfun_relativize", axioms_L (thm "M_axioms.is_recfun_relativize"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   537
bind_thm ("is_recfun_restrict", axioms_L (thm "M_axioms.is_recfun_restrict"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   538
bind_thm ("univalent_is_recfun", axioms_L (thm "M_axioms.univalent_is_recfun"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   539
bind_thm ("exists_is_recfun_indstep", axioms_L (thm "M_axioms.exists_is_recfun_indstep"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   540
bind_thm ("wellfounded_exists_is_recfun", axioms_L (thm "M_axioms.wellfounded_exists_is_recfun"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   541
bind_thm ("wf_exists_is_recfun", axioms_L (thm "M_axioms.wf_exists_is_recfun")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   542
bind_thm ("is_recfun_abs", axioms_L (thm "M_axioms.is_recfun_abs"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   543
bind_thm ("irreflexive_abs", axioms_L (thm "M_axioms.irreflexive_abs"));  
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   544
bind_thm ("transitive_rel_abs", axioms_L (thm "M_axioms.transitive_rel_abs"));  
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   545
bind_thm ("linear_rel_abs", axioms_L (thm "M_axioms.linear_rel_abs"));  
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   546
bind_thm ("wellordered_is_trans_on", axioms_L (thm "M_axioms.wellordered_is_trans_on")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   547
bind_thm ("wellordered_is_linear", axioms_L (thm "M_axioms.wellordered_is_linear")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   548
bind_thm ("wellordered_is_wellfounded_on", axioms_L (thm "M_axioms.wellordered_is_wellfounded_on")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   549
bind_thm ("wellfounded_imp_wellfounded_on", axioms_L (thm "M_axioms.wellfounded_imp_wellfounded_on")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   550
bind_thm ("wellfounded_on_subset_A", axioms_L (thm "M_axioms.wellfounded_on_subset_A"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   551
bind_thm ("wellfounded_on_iff_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_iff_wellfounded"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   552
bind_thm ("wellfounded_on_imp_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_imp_wellfounded"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   553
bind_thm ("wellfounded_on_field_imp_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_field_imp_wellfounded"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   554
bind_thm ("wellfounded_iff_wellfounded_on_field", axioms_L (thm "M_axioms.wellfounded_iff_wellfounded_on_field"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   555
bind_thm ("wellfounded_induct", axioms_L (thm "M_axioms.wellfounded_induct")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   556
bind_thm ("wellfounded_on_induct", axioms_L (thm "M_axioms.wellfounded_on_induct")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   557
bind_thm ("wellfounded_on_induct2", axioms_L (thm "M_axioms.wellfounded_on_induct2")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   558
bind_thm ("linear_imp_relativized", axioms_L (thm "M_axioms.linear_imp_relativized")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   559
bind_thm ("trans_on_imp_relativized", axioms_L (thm "M_axioms.trans_on_imp_relativized")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   560
bind_thm ("wf_on_imp_relativized", axioms_L (thm "M_axioms.wf_on_imp_relativized")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   561
bind_thm ("wf_imp_relativized", axioms_L (thm "M_axioms.wf_imp_relativized")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   562
bind_thm ("well_ord_imp_relativized", axioms_L (thm "M_axioms.well_ord_imp_relativized")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   563
bind_thm ("order_isomorphism_abs", axioms_L (thm "M_axioms.order_isomorphism_abs"));  
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   564
bind_thm ("pred_set_abs", axioms_L (thm "M_axioms.pred_set_abs"));  
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   565
*}
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   566
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   567
ML
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   568
{*
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   569
bind_thm ("pred_closed", axioms_L (thm "M_axioms.pred_closed"));  
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   570
bind_thm ("membership_abs", axioms_L (thm "M_axioms.membership_abs"));  
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   571
bind_thm ("M_Memrel_iff", axioms_L (thm "M_axioms.M_Memrel_iff"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   572
bind_thm ("Memrel_closed", axioms_L (thm "M_axioms.Memrel_closed"));  
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   573
bind_thm ("wellordered_iso_predD", axioms_L (thm "M_axioms.wellordered_iso_predD"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   574
bind_thm ("wellordered_iso_pred_eq", axioms_L (thm "M_axioms.wellordered_iso_pred_eq"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   575
bind_thm ("wellfounded_on_asym", axioms_L (thm "M_axioms.wellfounded_on_asym"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   576
bind_thm ("wellordered_asym", axioms_L (thm "M_axioms.wellordered_asym"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   577
bind_thm ("ord_iso_pred_imp_lt", axioms_L (thm "M_axioms.ord_iso_pred_imp_lt"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   578
bind_thm ("obase_iff", axioms_L (thm "M_axioms.obase_iff"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   579
bind_thm ("omap_iff", axioms_L (thm "M_axioms.omap_iff"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   580
bind_thm ("omap_unique", axioms_L (thm "M_axioms.omap_unique"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   581
bind_thm ("omap_yields_Ord", axioms_L (thm "M_axioms.omap_yields_Ord"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   582
bind_thm ("otype_iff", axioms_L (thm "M_axioms.otype_iff"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   583
bind_thm ("otype_eq_range", axioms_L (thm "M_axioms.otype_eq_range"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   584
bind_thm ("Ord_otype", axioms_L (thm "M_axioms.Ord_otype"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   585
bind_thm ("domain_omap", axioms_L (thm "M_axioms.domain_omap"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   586
bind_thm ("omap_subset", axioms_L (thm "M_axioms.omap_subset")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   587
bind_thm ("omap_funtype", axioms_L (thm "M_axioms.omap_funtype")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   588
bind_thm ("wellordered_omap_bij", axioms_L (thm "M_axioms.wellordered_omap_bij"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   589
bind_thm ("omap_ord_iso", axioms_L (thm "M_axioms.omap_ord_iso"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   590
bind_thm ("Ord_omap_image_pred", axioms_L (thm "M_axioms.Ord_omap_image_pred"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   591
bind_thm ("restrict_omap_ord_iso", axioms_L (thm "M_axioms.restrict_omap_ord_iso"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   592
bind_thm ("obase_equals", axioms_L (thm "M_axioms.obase_equals")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   593
bind_thm ("omap_ord_iso_otype", axioms_L (thm "M_axioms.omap_ord_iso_otype"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   594
bind_thm ("obase_exists", axioms_L (thm "M_axioms.obase_exists"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   595
bind_thm ("omap_exists", axioms_L (thm "M_axioms.omap_exists"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   596
bind_thm ("otype_exists", axioms_L (thm "M_axioms.otype_exists"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   597
bind_thm ("omap_ord_iso_otype", axioms_L (thm "M_axioms.omap_ord_iso_otype"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   598
bind_thm ("ordertype_exists", axioms_L (thm "M_axioms.ordertype_exists"));
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   599
bind_thm ("relativized_imp_well_ord", axioms_L (thm "M_axioms.relativized_imp_well_ord")); 
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   600
bind_thm ("well_ord_abs", axioms_L (thm "M_axioms.well_ord_abs"));  
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   601
*}
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   602
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   603
declare cartprod_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   604
declare sum_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   605
declare converse_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   606
declare converse_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   607
declare image_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   608
declare vimage_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   609
declare vimage_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   610
declare domain_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   611
declare domain_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   612
declare range_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   613
declare range_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   614
declare field_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   615
declare field_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   616
declare relation_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   617
declare function_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   618
declare apply_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   619
declare typed_function_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   620
declare injection_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   621
declare surjection_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   622
declare bijection_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   623
declare comp_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   624
declare composition_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   625
declare restriction_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   626
declare restrict_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   627
declare Inter_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   628
declare Inter_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   629
declare Int_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   630
declare finite_fun_closed [rule_format]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   631
declare is_funspace_abs [simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   632
declare finite_funspace_closed [intro,simp]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13319
diff changeset
   633
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents:
diff changeset
   634
end