src/ZF/Constructible/Separation.thy
author paulson
Mon Jul 08 17:51:56 2002 +0200 (2002-07-08)
changeset 13316 d16629fd0f95
parent 13314 84b9de3cbc91
child 13319 23de7b3af453
permissions -rw-r--r--
more and simpler separation proofs
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header{*Proving instances of Separation using Reflection!*}
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theory Separation = L_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 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_reflection)  
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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 [simplified]:
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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_reflection function_reflection)  
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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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text{*No @{text simplified} here: it simplifies the occurrence of 
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      the predicate @{term pair}!*}
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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_reflection function_reflection)
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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_reflection function_reflection)
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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_reflection function_reflection)
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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_reflection function_reflection)
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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_reflection function_reflection)
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lemma pred_separation:
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     "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
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apply (rule separation_CollectI) 
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apply (rule_tac A="{r,x,z}" in subset_LsetE, blast ) 
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apply (rule ReflectsE [OF pred_Reflects], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2, clarify)
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apply (rule 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_reflection function_reflection)
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lemma Memrel_separation:
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     "separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
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apply (rule separation_CollectI) 
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apply (rule_tac A="{z}" in subset_LsetE, blast ) 
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apply (rule ReflectsE [OF Memrel_Reflects], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2)
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apply (rule DPowI2)
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apply (rename_tac u) 
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apply (rule bex_iff_sats conj_iff_sats)+
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apply (rule_tac env = "[y,x,u]" in pair_iff_sats) 
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apply (rule sep_rules | simp)+
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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_reflection function_reflection)
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lemma funspace_succ_replacement:
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     "L(n) ==> 
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      strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. 
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                pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
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                upair(L,cnbf,cnbf,z))"
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apply (rule strong_replacementI) 
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apply (rule rallI) 
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apply (rule separation_CollectI) 
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apply (rule_tac A="{n,A,z}" in subset_LsetE, blast ) 
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apply (rule ReflectsE [OF funspace_succ_Reflects], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2)
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apply (rule 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 = "[x,u,n,A]" 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 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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                (\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r),
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        \<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i). 
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                fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]"
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by (intro FOL_reflection function_reflection)
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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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apply (rule separation_CollectI) 
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apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast ) 
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apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2)
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apply (rule DPowI2)
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apply (rename_tac u) 
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apply (rule imp_iff_sats)
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apply (rule_tac env = "[u,A,f,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 @{term "obase"}*}
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lemma obase_reflects:
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  "REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
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	     ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
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	     order_isomorphism(L,par,r,x,mx,g),
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        \<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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	     ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
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	     order_isomorphism(**Lset(i),par,r,x,mx,g)]"
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by (intro FOL_reflection function_reflection fun_plus_reflection)
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lemma obase_separation:
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     --{*part of the order type formalization*}
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     "[| L(A); L(r) |] 
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      ==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
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	     ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
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	     order_isomorphism(L,par,r,x,mx,g))"
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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 obase_reflects], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2)
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apply (rule 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 = "[x,u,A,r]" in ordinal_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 @{term "well_ord_iso"}*}
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lemma obase_equals_reflects:
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  "REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. 
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		ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 
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		membership(L,y,my) & pred_set(L,A,x,r,pxr) &
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		order_isomorphism(L,pxr,r,y,my,g))),
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	\<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i). 
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		ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i). 
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		membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) &
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		order_isomorphism(**Lset(i),pxr,r,y,my,g)))]"
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by (intro FOL_reflection function_reflection fun_plus_reflection)
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lemma obase_equals_separation:
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     "[| L(A); L(r) |] 
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      ==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. 
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			      ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 
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			      membership(L,y,my) & pred_set(L,A,x,r,pxr) &
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			      order_isomorphism(L,pxr,r,y,my,g))))"
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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 obase_equals_reflects], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2)
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apply (rule DPowI2)
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apply (rename_tac u) 
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apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+
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apply (rule_tac env = "[u,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{*Replacement for @{term "omap"}*}
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lemma omap_reflects:
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 "REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
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     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 
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     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
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 \<lambda>i z. \<exists>a \<in> Lset(i). a\<in>B & (\<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). 
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        \<exists>par \<in> Lset(i). 
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	 ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) & 
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         membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & 
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         order_isomorphism(**Lset(i),par,r,x,mx,g))]"
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by (intro FOL_reflection function_reflection fun_plus_reflection)
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lemma omap_replacement:
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     "[| L(A); L(r) |] 
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      ==> strong_replacement(L,
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             \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
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   404
	     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 
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	     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
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apply (rule strong_replacementI) 
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   407
apply (rule rallI)
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   408
apply (rename_tac B)  
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   409
apply (rule separation_CollectI) 
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   410
apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast ) 
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   411
apply (rule ReflectsE [OF omap_reflects], assumption)
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   412
apply (drule subset_Lset_ltD, assumption) 
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   413
apply (erule reflection_imp_L_separation)
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   414
  apply (simp_all add: lt_Ord2)
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   415
apply (rule DPowI2)
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   416
apply (rename_tac u) 
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   417
apply (rule bex_iff_sats conj_iff_sats)+
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   418
apply (rule_tac env = "[x,u,A,B,r]" in mem_iff_sats) 
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   419
apply (rule sep_rules | simp)+
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   420
apply (simp_all add: succ_Un_distrib [symmetric])
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   421
done
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   423
end