src/ZF/Constructible/Separation.thy
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(*  Title:      ZF/Constructible/Separation.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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*)
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header{*Early Instances of Separation and Strong Replacement*}
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theory Separation imports L_axioms WF_absolute begin
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text{*This theory proves all instances needed for locale @{text "M_basic"}*}
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text{*Helps us solve for de Bruijn indices!*}
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lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
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by simp
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lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI
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lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats
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                   fun_plus_iff_sats
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lemma Collect_conj_in_DPow:
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     "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |]
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      ==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
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by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
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lemma Collect_conj_in_DPow_Lset:
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     "[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
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      ==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
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apply (frule mem_Lset_imp_subset_Lset)
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apply (simp add: Collect_conj_in_DPow Collect_mem_eq
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                 subset_Int_iff2 elem_subset_in_DPow)
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done
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lemma separation_CollectI:
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     "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
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apply (unfold separation_def, clarify)
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apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
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apply simp_all
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done
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text{*Reduces the original comprehension to the reflected one*}
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lemma reflection_imp_L_separation:
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      "[| \<forall>x\<in>Lset(j). P(x) \<longleftrightarrow> Q(x);
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          {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
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          Ord(j);  z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
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apply (rule_tac i = "succ(j)" in L_I)
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 prefer 2 apply simp
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apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
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 prefer 2
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 apply (blast dest: mem_Lset_imp_subset_Lset)
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apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
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done
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text{*Encapsulates the standard proof script for proving instances of 
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      Separation.*}
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lemma gen_separation:
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 assumes reflection: "REFLECTS [P,Q]"
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     and Lu:         "L(u)"
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     and collI: "!!j. u \<in> Lset(j)
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                \<Longrightarrow> Collect(Lset(j), Q(j)) \<in> DPow(Lset(j))"
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 shows "separation(L,P)"
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apply (rule separation_CollectI)
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apply (rule_tac A="{u,z}" in subset_LsetE, blast intro: Lu)
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apply (rule ReflectsE [OF reflection], assumption)
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apply (drule subset_Lset_ltD, assumption)
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2, clarify)
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apply (rule collI, assumption)
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done
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text{*As above, but typically @{term u} is a finite enumeration such as
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  @{term "{a,b}"}; thus the new subgoal gets the assumption
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  @{term "{a,b} \<subseteq> Lset(i)"}, which is logically equivalent to 
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  @{term "a \<in> Lset(i)"} and @{term "b \<in> Lset(i)"}.*}
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lemma gen_separation_multi:
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 assumes reflection: "REFLECTS [P,Q]"
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     and Lu:         "L(u)"
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     and collI: "!!j. u \<subseteq> Lset(j)
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                \<Longrightarrow> Collect(Lset(j), Q(j)) \<in> DPow(Lset(j))"
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 shows "separation(L,P)"
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apply (rule gen_separation [OF reflection Lu])
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apply (drule mem_Lset_imp_subset_Lset)
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apply (erule collI) 
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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 \<longrightarrow> x \<in> y,
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               \<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A \<longrightarrow> 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 \<longrightarrow> x\<in>y)"
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apply (rule gen_separation [OF Inter_Reflects], simp)
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apply (rule DPow_LsetI)
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 txt{*I leave this one example of a manual proof.  The tedium of manually
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      instantiating @{term i}, @{term j} and @{term env} is obvious. *}
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apply (rule ball_iff_sats)
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apply (rule imp_iff_sats)
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apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
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apply (rule_tac i=0 and j=2 in mem_iff_sats)
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apply (simp_all add: succ_Un_distrib [symmetric])
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done
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subsection{*Separation for Set Difference*}
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lemma Diff_Reflects:
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     "REFLECTS[\<lambda>x. x \<notin> B, \<lambda>i x. x \<notin> B]"
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by (intro FOL_reflections)  
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lemma Diff_separation:
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     "L(B) ==> separation(L, \<lambda>x. x \<notin> B)"
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apply (rule gen_separation [OF Diff_Reflects], simp)
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apply (rule_tac env="[B]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Cartesian Product*}
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lemma cartprod_Reflects:
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     "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
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                \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
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                                   pair(##Lset(i),x,y,z))]"
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by (intro FOL_reflections function_reflections)
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lemma cartprod_separation:
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     "[| L(A); L(B) |]
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      ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
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apply (rule gen_separation_multi [OF cartprod_Reflects, of "{A,B}"], auto)
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apply (rule_tac env="[A,B]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Image*}
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lemma image_Reflects:
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     "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
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           \<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(##Lset(i),x,y,p))]"
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by (intro FOL_reflections function_reflections)
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lemma image_separation:
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     "[| L(A); L(r) |]
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      ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
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apply (rule gen_separation_multi [OF image_Reflects, of "{A,r}"], auto)
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apply (rule_tac env="[A,r]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Converse*}
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lemma converse_Reflects:
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  "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
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     \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
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                     pair(##Lset(i),x,y,p) & pair(##Lset(i),y,x,z))]"
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by (intro FOL_reflections function_reflections)
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lemma converse_separation:
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     "L(r) ==> separation(L,
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         \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
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apply (rule gen_separation [OF converse_Reflects], simp)
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apply (rule_tac env="[r]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Restriction*}
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lemma restrict_Reflects:
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     "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
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        \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(##Lset(i),x,y,z))]"
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by (intro FOL_reflections function_reflections)
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lemma restrict_separation:
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   "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
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apply (rule gen_separation [OF restrict_Reflects], simp)
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apply (rule_tac env="[A]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Composition*}
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lemma comp_Reflects:
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     "REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
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                  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
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                  xy\<in>s & yz\<in>r,
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        \<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i).
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                  pair(##Lset(i),x,z,xz) & pair(##Lset(i),x,y,xy) &
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                  pair(##Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]"
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by (intro FOL_reflections function_reflections)
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lemma comp_separation:
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     "[| L(r); L(s) |]
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      ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
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                  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
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                  xy\<in>s & yz\<in>r)"
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apply (rule gen_separation_multi [OF comp_Reflects, of "{r,s}"], auto)
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txt{*Subgoals after applying general ``separation'' rule:
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     @{subgoals[display,indent=0,margin=65]}*}
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apply (rule_tac env="[r,s]" in DPow_LsetI)
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txt{*Subgoals ready for automatic synthesis of a formula:
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     @{subgoals[display,indent=0,margin=65]}*}
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for Predecessors in an Order*}
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lemma pred_Reflects:
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     "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p),
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                    \<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(##Lset(i),y,x,p)]"
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by (intro FOL_reflections function_reflections)
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lemma pred_separation:
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     "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
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apply (rule gen_separation_multi [OF pred_Reflects, of "{r,x}"], auto)
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apply (rule_tac env="[r,x]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for the Membership Relation*}
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lemma Memrel_Reflects:
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     "REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y,
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            \<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(##Lset(i),x,y,z) & x \<in> y]"
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by (intro FOL_reflections function_reflections)
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lemma Memrel_separation:
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     "separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
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apply (rule gen_separation [OF Memrel_Reflects nonempty])
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apply (rule_tac env="[]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Replacement for FunSpace*}
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lemma funspace_succ_Reflects:
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 "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
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            pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
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            upair(L,cnbf,cnbf,z)),
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        \<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i).
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              \<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i).
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                pair(##Lset(i),f,b,p) & pair(##Lset(i),n,b,nb) &
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                is_cons(##Lset(i),nb,f,cnbf) & upair(##Lset(i),cnbf,cnbf,z))]"
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by (intro FOL_reflections function_reflections)
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lemma funspace_succ_replacement:
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     "L(n) ==>
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      strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
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                pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
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                upair(L,cnbf,cnbf,z))"
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apply (rule strong_replacementI)
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apply (rule_tac u="{n,B}" in gen_separation_multi [OF funspace_succ_Reflects], 
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       auto)
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apply (rule_tac env="[n,B]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Separation for a Theorem about @{term "is_recfun"}*}
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lemma is_recfun_reflects:
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  "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
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                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
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                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
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                                   fx \<noteq> gx),
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   \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
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          pair(##Lset(i),x,a,xa) & xa \<in> r & pair(##Lset(i),x,b,xb) & xb \<in> r &
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                (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(##Lset(i),f,x,fx) &
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                  fun_apply(##Lset(i),g,x,gx) & fx \<noteq> gx)]"
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by (intro FOL_reflections function_reflections fun_plus_reflections)
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lemma is_recfun_separation:
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     --{*for well-founded recursion*}
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     "[| L(r); L(f); L(g); L(a); L(b) |]
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     ==> separation(L,
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            \<lambda>x. \<exists>xa[L]. \<exists>xb[L].
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                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
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                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
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                                   fx \<noteq> gx))"
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apply (rule gen_separation_multi [OF is_recfun_reflects, of "{r,f,g,a,b}"], 
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            auto)
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apply (rule_tac env="[r,f,g,a,b]" in DPow_LsetI)
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apply (rule sep_rules | simp)+
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done
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subsection{*Instantiating the locale @{text M_basic}*}
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text{*Separation (and Strong Replacement) for basic set-theoretic constructions
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such as intersection, Cartesian Product and image.*}
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lemma M_basic_axioms_L: "M_basic_axioms(L)"
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  apply (rule M_basic_axioms.intro)
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       apply (assumption | rule
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         Inter_separation Diff_separation cartprod_separation image_separation
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         converse_separation restrict_separation
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         comp_separation pred_separation Memrel_separation
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         funspace_succ_replacement is_recfun_separation)+
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  done
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theorem M_basic_L: "PROP M_basic(L)"
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by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L])
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interpretation L?: M_basic L by (rule M_basic_L)
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end