src/ZF/Constructible/Datatype_absolute.thy
author paulson
Fri Jul 19 13:29:22 2002 +0200 (2002-07-19)
changeset 13397 6e5f4d911435
parent 13395 4eb948d1eb4e
child 13398 1cadd412da48
permissions -rw-r--r--
Absoluteness of the function "nth"
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header {*Absoluteness Properties for Recursive Datatypes*}
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theory Datatype_absolute = Formula + WF_absolute:
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subsection{*The lfp of a continuous function can be expressed as a union*}
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constdefs
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  directed :: "i=>o"
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   "directed(A) == A\<noteq>0 & (\<forall>x\<in>A. \<forall>y\<in>A. x \<union> y \<in> A)"
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  contin :: "(i=>i) => o"
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   "contin(h) == (\<forall>A. directed(A) --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))"
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lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D"
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apply (induct_tac n) 
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 apply (simp_all add: bnd_mono_def, blast) 
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done
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lemma bnd_mono_increasing [rule_format]:
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     "[|i \<in> nat; j \<in> nat; bnd_mono(D,h)|] ==> i \<le> j --> h^i(0) \<subseteq> h^j(0)"
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apply (rule_tac m=i and n=j in diff_induct, simp_all)
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apply (blast del: subsetI
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	     intro: bnd_mono_iterates_subset bnd_monoD2 [of concl: h] ) 
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done
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lemma directed_iterates: "bnd_mono(D,h) ==> directed({h^n (0). n\<in>nat})"
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apply (simp add: directed_def, clarify) 
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apply (rename_tac i j)
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apply (rule_tac x="i \<union> j" in bexI) 
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apply (rule_tac i = i and j = j in Ord_linear_le)
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apply (simp_all add: subset_Un_iff [THEN iffD1] le_imp_subset
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                     subset_Un_iff2 [THEN iffD1])
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apply (simp_all add: subset_Un_iff [THEN iff_sym] bnd_mono_increasing
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                     subset_Un_iff2 [THEN iff_sym])
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done
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lemma contin_iterates_eq: 
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    "[|bnd_mono(D, h); contin(h)|] 
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     ==> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))"
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apply (simp add: contin_def directed_iterates) 
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apply (rule trans) 
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apply (rule equalityI) 
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 apply (simp_all add: UN_subset_iff)
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 apply safe
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 apply (erule_tac [2] natE) 
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  apply (rule_tac a="succ(x)" in UN_I) 
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   apply simp_all 
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apply blast 
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done
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lemma lfp_subset_Union:
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     "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) <= (\<Union>n\<in>nat. h^n(0))"
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apply (rule lfp_lowerbound) 
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 apply (simp add: contin_iterates_eq) 
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apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff) 
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done
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lemma Union_subset_lfp:
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     "bnd_mono(D,h) ==> (\<Union>n\<in>nat. h^n(0)) <= lfp(D,h)"
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apply (simp add: UN_subset_iff)
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apply (rule ballI)  
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apply (induct_tac n, simp_all) 
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apply (rule subset_trans [of _ "h(lfp(D,h))"])
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 apply (blast dest: bnd_monoD2 [OF _ _ lfp_subset] )  
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apply (erule lfp_lemma2) 
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done
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lemma lfp_eq_Union:
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     "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) = (\<Union>n\<in>nat. h^n(0))"
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by (blast del: subsetI 
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          intro: lfp_subset_Union Union_subset_lfp)
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subsubsection{*Some Standard Datatype Constructions Preserve Continuity*}
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lemma contin_imp_mono: "[|X\<subseteq>Y; contin(F)|] ==> F(X) \<subseteq> F(Y)"
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apply (simp add: contin_def) 
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apply (drule_tac x="{X,Y}" in spec) 
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apply (simp add: directed_def subset_Un_iff2 Un_commute) 
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done
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lemma sum_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) + G(X))"
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by (simp add: contin_def, blast)
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lemma prod_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) * G(X))" 
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apply (subgoal_tac "\<forall>B C. F(B) \<subseteq> F(B \<union> C)")
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 prefer 2 apply (simp add: Un_upper1 contin_imp_mono) 
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apply (subgoal_tac "\<forall>B C. G(C) \<subseteq> G(B \<union> C)")
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 prefer 2 apply (simp add: Un_upper2 contin_imp_mono) 
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apply (simp add: contin_def, clarify) 
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apply (rule equalityI) 
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 prefer 2 apply blast 
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apply clarify 
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apply (rename_tac B C) 
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apply (rule_tac a="B \<union> C" in UN_I) 
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 apply (simp add: directed_def, blast)  
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done
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lemma const_contin: "contin(\<lambda>X. A)"
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by (simp add: contin_def directed_def)
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lemma id_contin: "contin(\<lambda>X. X)"
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by (simp add: contin_def)
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subsection {*Absoluteness for "Iterates"*}
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constdefs
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  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
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   "iterates_MH(M,isF,v,n,g,z) ==
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        is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
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                    n, z)"
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  iterates_replacement :: "[i=>o, [i,i]=>o, i] => o"
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   "iterates_replacement(M,isF,v) ==
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      \<forall>n[M]. n\<in>nat --> 
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         wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))"
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lemma (in M_axioms) iterates_MH_abs:
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  "[| relativize1(M,isF,F); M(n); M(g); M(z) |] 
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   ==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, \<lambda>m. F(g`m), n)"
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by (simp add: nat_case_abs [of _ "\<lambda>m. F(g ` m)"]
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              relativize1_def iterates_MH_def)  
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lemma (in M_axioms) iterates_imp_wfrec_replacement:
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  "[|relativize1(M,isF,F); n \<in> nat; iterates_replacement(M,isF,v)|] 
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   ==> wfrec_replacement(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n), 
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                       Memrel(succ(n)))" 
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by (simp add: iterates_replacement_def iterates_MH_abs)
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theorem (in M_trancl) iterates_abs:
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  "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
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      n \<in> nat; M(v); M(z); \<forall>x[M]. M(F(x)) |] 
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   ==> is_wfrec(M, iterates_MH(M,isF,v), Memrel(succ(n)), n, z) <->
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       z = iterates(F,n,v)" 
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apply (frule iterates_imp_wfrec_replacement, assumption+)
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apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
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                 relativize2_def iterates_MH_abs 
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                 iterates_nat_def recursor_def transrec_def 
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                 eclose_sing_Ord_eq nat_into_M
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         trans_wfrec_abs [of _ _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
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done
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lemma (in M_wfrank) iterates_closed [intro,simp]:
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  "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
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      n \<in> nat; M(v); \<forall>x[M]. M(F(x)) |] 
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   ==> M(iterates(F,n,v))"
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apply (frule iterates_imp_wfrec_replacement, assumption+)
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apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
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                 relativize2_def iterates_MH_abs 
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                 iterates_nat_def recursor_def transrec_def 
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                 eclose_sing_Ord_eq nat_into_M
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         trans_wfrec_closed [of _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
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done
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subsection {*lists without univ*}
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lemmas datatype_univs = Inl_in_univ Inr_in_univ 
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                        Pair_in_univ nat_into_univ A_into_univ 
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lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"
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apply (rule bnd_monoI)
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 apply (intro subset_refl zero_subset_univ A_subset_univ 
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	      sum_subset_univ Sigma_subset_univ) 
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apply (rule subset_refl sum_mono Sigma_mono | assumption)+
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done
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lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"
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by (intro sum_contin prod_contin id_contin const_contin) 
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text{*Re-expresses lists using sum and product*}
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lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"
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apply (simp add: list_def) 
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apply (rule equalityI) 
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 apply (rule lfp_lowerbound) 
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  prefer 2 apply (rule lfp_subset)
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 apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
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 apply (simp add: Nil_def Cons_def)
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 apply blast 
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txt{*Opposite inclusion*}
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apply (rule lfp_lowerbound) 
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 prefer 2 apply (rule lfp_subset) 
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apply (clarify, subst lfp_unfold [OF list.bnd_mono]) 
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apply (simp add: Nil_def Cons_def)
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apply (blast intro: datatype_univs
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             dest: lfp_subset [THEN subsetD])
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done
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text{*Re-expresses lists using "iterates", no univ.*}
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lemma list_eq_Union:
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     "list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"
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by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)
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constdefs
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  is_list_functor :: "[i=>o,i,i,i] => o"
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    "is_list_functor(M,A,X,Z) == 
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        \<exists>n1[M]. \<exists>AX[M]. 
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         number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)"
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lemma (in M_axioms) list_functor_abs [simp]: 
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     "[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) <-> (Z = {0} + A*X)"
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by (simp add: is_list_functor_def singleton_0 nat_into_M)
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subsection {*formulas without univ*}
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lemma formula_fun_bnd_mono:
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     "bnd_mono(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
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apply (rule bnd_monoI)
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 apply (intro subset_refl zero_subset_univ A_subset_univ 
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	      sum_subset_univ Sigma_subset_univ nat_subset_univ) 
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apply (rule subset_refl sum_mono Sigma_mono | assumption)+
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done
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lemma formula_fun_contin:
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     "contin(\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
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by (intro sum_contin prod_contin id_contin const_contin) 
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text{*Re-expresses formulas using sum and product*}
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lemma formula_eq_lfp2:
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    "formula = lfp(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
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apply (simp add: formula_def) 
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apply (rule equalityI) 
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 apply (rule lfp_lowerbound) 
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  prefer 2 apply (rule lfp_subset)
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 apply (clarify, subst lfp_unfold [OF formula_fun_bnd_mono])
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 apply (simp add: Member_def Equal_def Neg_def And_def Forall_def)
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 apply blast 
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txt{*Opposite inclusion*}
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apply (rule lfp_lowerbound) 
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 prefer 2 apply (rule lfp_subset, clarify) 
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apply (subst lfp_unfold [OF formula.bnd_mono, simplified]) 
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apply (simp add: Member_def Equal_def Neg_def And_def Forall_def)  
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apply (elim sumE SigmaE, simp_all) 
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apply (blast intro: datatype_univs dest: lfp_subset [THEN subsetD])+  
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done
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text{*Re-expresses formulas using "iterates", no univ.*}
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lemma formula_eq_Union:
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     "formula = 
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      (\<Union>n\<in>nat. (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))) ^ n (0))"
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by (simp add: formula_eq_lfp2 lfp_eq_Union formula_fun_bnd_mono 
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              formula_fun_contin)
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constdefs
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  is_formula_functor :: "[i=>o,i,i] => o"
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    "is_formula_functor(M,X,Z) == 
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        \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M]. \<exists>X4[M]. 
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          omega(M,nat') & cartprod(M,nat',nat',natnat) & 
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          is_sum(M,natnat,natnat,natnatsum) &
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          cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & is_sum(M,X,X3,X4) &
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          is_sum(M,natnatsum,X4,Z)"
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lemma (in M_axioms) formula_functor_abs [simp]: 
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     "[| M(X); M(Z) |] 
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      ==> is_formula_functor(M,X,Z) <-> 
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          Z = ((nat*nat) + (nat*nat)) + (X + (X*X + X))"
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by (simp add: is_formula_functor_def) 
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subsection{*@{term M} Contains the List and Formula Datatypes*}
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constdefs
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  list_N :: "[i,i] => i"
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    "list_N(A,n) == (\<lambda>X. {0} + A * X)^n (0)"
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lemma Nil_in_list_N [simp]: "[] \<in> list_N(A,succ(n))"
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by (simp add: list_N_def Nil_def)
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lemma Cons_in_list_N [simp]:
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     "Cons(a,l) \<in> list_N(A,succ(n)) <-> a\<in>A & l \<in> list_N(A,n)"
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by (simp add: list_N_def Cons_def) 
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text{*These two aren't simprules because they reveal the underlying
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list representation.*}
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lemma list_N_0: "list_N(A,0) = 0"
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by (simp add: list_N_def)
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lemma list_N_succ: "list_N(A,succ(n)) = {0} + A * (list_N(A,n))"
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by (simp add: list_N_def)
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lemma list_N_imp_list:
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  "[| l \<in> list_N(A,n); n \<in> nat |] ==> l \<in> list(A)"
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by (force simp add: list_eq_Union list_N_def)
paulson@13397
   294
paulson@13397
   295
lemma list_N_imp_length_lt [rule_format]:
paulson@13397
   296
     "n \<in> nat ==> \<forall>l \<in> list_N(A,n). length(l) < n"
paulson@13397
   297
apply (induct_tac n)  
paulson@13397
   298
apply (auto simp add: list_N_0 list_N_succ 
paulson@13397
   299
                      Nil_def [symmetric] Cons_def [symmetric]) 
paulson@13397
   300
done
paulson@13397
   301
paulson@13397
   302
lemma list_imp_list_N [rule_format]:
paulson@13397
   303
     "l \<in> list(A) ==> \<forall>n\<in>nat. length(l) < n --> l \<in> list_N(A, n)"
paulson@13397
   304
apply (induct_tac l)
paulson@13397
   305
apply (force elim: natE)+
paulson@13397
   306
done
paulson@13397
   307
paulson@13397
   308
lemma list_N_imp_eq_length:
paulson@13397
   309
      "[|n \<in> nat; l \<notin> list_N(A, n); l \<in> list_N(A, succ(n))|] 
paulson@13397
   310
       ==> n = length(l)"
paulson@13397
   311
apply (rule le_anti_sym) 
paulson@13397
   312
 prefer 2 apply (simp add: list_N_imp_length_lt) 
paulson@13397
   313
apply (frule list_N_imp_list, simp)
paulson@13397
   314
apply (simp add: not_lt_iff_le [symmetric]) 
paulson@13397
   315
apply (blast intro: list_imp_list_N) 
paulson@13397
   316
done
paulson@13397
   317
  
paulson@13397
   318
text{*Express @{term list_rec} without using @{term rank} or @{term Vset},
paulson@13397
   319
neither of which is absolute.*}
paulson@13397
   320
lemma (in M_triv_axioms) list_rec_eq:
paulson@13397
   321
  "l \<in> list(A) ==>
paulson@13397
   322
   list_rec(a,g,l) = 
paulson@13397
   323
   transrec (succ(length(l)),
paulson@13397
   324
      \<lambda>x h. Lambda (list_N(A,x),
paulson@13397
   325
             list_case' (a, 
paulson@13397
   326
                \<lambda>a l. g(a, l, h ` succ(length(l)) ` l)))) ` l"
paulson@13397
   327
apply (induct_tac l) 
paulson@13397
   328
apply (subst transrec, simp) 
paulson@13397
   329
apply (subst transrec) 
paulson@13397
   330
apply (simp add: list_imp_list_N) 
paulson@13397
   331
done
paulson@13397
   332
paulson@13397
   333
constdefs
paulson@13397
   334
  is_list_N :: "[i=>o,i,i,i] => o"
paulson@13397
   335
    "is_list_N(M,A,n,Z) == 
paulson@13395
   336
      \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M]. 
paulson@13395
   337
       empty(M,zero) & 
paulson@13395
   338
       successor(M,n,sn) & membership(M,sn,msn) &
paulson@13395
   339
       is_wfrec(M, iterates_MH(M, is_list_functor(M,A),zero), msn, n, Z)"
paulson@13395
   340
  
paulson@13395
   341
  mem_list :: "[i=>o,i,i] => o"
paulson@13395
   342
    "mem_list(M,A,l) == 
paulson@13395
   343
      \<exists>n[M]. \<exists>listn[M]. 
paulson@13397
   344
       finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l \<in> listn"
paulson@13395
   345
paulson@13395
   346
  is_list :: "[i=>o,i,i] => o"
paulson@13395
   347
    "is_list(M,A,Z) == \<forall>l[M]. l \<in> Z <-> mem_list(M,A,l)"
paulson@13395
   348
paulson@13395
   349
constdefs
paulson@13395
   350
  is_formula_n :: "[i=>o,i,i] => o"
paulson@13395
   351
    "is_formula_n(M,n,Z) == 
paulson@13395
   352
      \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M]. 
paulson@13395
   353
       empty(M,zero) & 
paulson@13395
   354
       successor(M,n,sn) & membership(M,sn,msn) &
paulson@13395
   355
       is_wfrec(M, iterates_MH(M, is_formula_functor(M),zero), msn, n, Z)"
paulson@13395
   356
  
paulson@13395
   357
  mem_formula :: "[i=>o,i] => o"
paulson@13395
   358
    "mem_formula(M,p) == 
paulson@13395
   359
      \<exists>n[M]. \<exists>formn[M]. 
paulson@13395
   360
       finite_ordinal(M,n) & is_formula_n(M,n,formn) & p \<in> formn"
paulson@13395
   361
paulson@13395
   362
  is_formula :: "[i=>o,i] => o"
paulson@13395
   363
    "is_formula(M,Z) == \<forall>p[M]. p \<in> Z <-> mem_formula(M,p)"
paulson@13395
   364
wenzelm@13382
   365
locale (open) M_datatypes = M_wfrank +
paulson@13353
   366
 assumes list_replacement1: 
paulson@13363
   367
   "M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)"
paulson@13353
   368
  and list_replacement2: 
paulson@13363
   369
   "M(A) ==> strong_replacement(M, 
paulson@13353
   370
         \<lambda>n y. n\<in>nat & 
paulson@13353
   371
               (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
paulson@13363
   372
               is_wfrec(M, iterates_MH(M,is_list_functor(M,A), 0), 
paulson@13353
   373
                        msn, n, y)))"
paulson@13386
   374
  and formula_replacement1: 
paulson@13386
   375
   "iterates_replacement(M, is_formula_functor(M), 0)"
paulson@13386
   376
  and formula_replacement2: 
paulson@13386
   377
   "strong_replacement(M, 
paulson@13386
   378
         \<lambda>n y. n\<in>nat & 
paulson@13386
   379
               (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
paulson@13386
   380
               is_wfrec(M, iterates_MH(M,is_formula_functor(M), 0), 
paulson@13386
   381
                        msn, n, y)))"
paulson@13350
   382
paulson@13395
   383
paulson@13395
   384
subsubsection{*Absoluteness of the List Construction*}
paulson@13395
   385
paulson@13348
   386
lemma (in M_datatypes) list_replacement2': 
paulson@13353
   387
  "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"
paulson@13353
   388
apply (insert list_replacement2 [of A]) 
paulson@13353
   389
apply (rule strong_replacement_cong [THEN iffD1])  
paulson@13353
   390
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]]) 
paulson@13363
   391
apply (simp_all add: list_replacement1 relativize1_def) 
paulson@13353
   392
done
paulson@13268
   393
paulson@13268
   394
lemma (in M_datatypes) list_closed [intro,simp]:
paulson@13268
   395
     "M(A) ==> M(list(A))"
paulson@13353
   396
apply (insert list_replacement1)
paulson@13353
   397
by  (simp add: RepFun_closed2 list_eq_Union 
paulson@13353
   398
               list_replacement2' relativize1_def
paulson@13353
   399
               iterates_closed [of "is_list_functor(M,A)"])
paulson@13397
   400
paulson@13397
   401
lemma (in M_datatypes) list_N_abs [simp]:
paulson@13395
   402
     "[|M(A); n\<in>nat; M(Z)|] 
paulson@13397
   403
      ==> is_list_N(M,A,n,Z) <-> Z = list_N(A,n)"
paulson@13395
   404
apply (insert list_replacement1)
paulson@13397
   405
apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M
paulson@13395
   406
                 iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"])
paulson@13395
   407
done
paulson@13268
   408
paulson@13397
   409
lemma (in M_datatypes) list_N_closed [intro,simp]:
paulson@13397
   410
     "[|M(A); n\<in>nat|] ==> M(list_N(A,n))"
paulson@13397
   411
apply (insert list_replacement1)
paulson@13397
   412
apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M
paulson@13397
   413
                 iterates_closed [of "is_list_functor(M,A)"])
paulson@13397
   414
done
paulson@13397
   415
paulson@13395
   416
lemma (in M_datatypes) mem_list_abs [simp]:
paulson@13395
   417
     "M(A) ==> mem_list(M,A,l) <-> l \<in> list(A)"
paulson@13395
   418
apply (insert list_replacement1)
paulson@13397
   419
apply (simp add: mem_list_def list_N_def relativize1_def list_eq_Union
paulson@13395
   420
                 iterates_closed [of "is_list_functor(M,A)"]) 
paulson@13395
   421
done
paulson@13395
   422
paulson@13395
   423
lemma (in M_datatypes) list_abs [simp]:
paulson@13395
   424
     "[|M(A); M(Z)|] ==> is_list(M,A,Z) <-> Z = list(A)"
paulson@13395
   425
apply (simp add: is_list_def, safe)
paulson@13395
   426
apply (rule M_equalityI, simp_all)
paulson@13395
   427
done
paulson@13395
   428
paulson@13395
   429
subsubsection{*Absoluteness of Formulas*}
paulson@13293
   430
paulson@13386
   431
lemma (in M_datatypes) formula_replacement2': 
paulson@13386
   432
  "strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0))"
paulson@13386
   433
apply (insert formula_replacement2) 
paulson@13386
   434
apply (rule strong_replacement_cong [THEN iffD1])  
paulson@13386
   435
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]]) 
paulson@13386
   436
apply (simp_all add: formula_replacement1 relativize1_def) 
paulson@13386
   437
done
paulson@13386
   438
paulson@13386
   439
lemma (in M_datatypes) formula_closed [intro,simp]:
paulson@13386
   440
     "M(formula)"
paulson@13386
   441
apply (insert formula_replacement1)
paulson@13386
   442
apply  (simp add: RepFun_closed2 formula_eq_Union 
paulson@13386
   443
                  formula_replacement2' relativize1_def
paulson@13386
   444
                  iterates_closed [of "is_formula_functor(M)"])
paulson@13386
   445
done
paulson@13386
   446
paulson@13395
   447
lemma (in M_datatypes) is_formula_n_abs [simp]:
paulson@13395
   448
     "[|n\<in>nat; M(Z)|] 
paulson@13395
   449
      ==> is_formula_n(M,n,Z) <-> 
paulson@13395
   450
          Z = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0)"
paulson@13395
   451
apply (insert formula_replacement1)
paulson@13395
   452
apply (simp add: is_formula_n_def relativize1_def nat_into_M
paulson@13395
   453
                 iterates_abs [of "is_formula_functor(M)" _ 
paulson@13395
   454
                        "\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))"])
paulson@13395
   455
done
paulson@13395
   456
paulson@13395
   457
lemma (in M_datatypes) mem_formula_abs [simp]:
paulson@13395
   458
     "mem_formula(M,l) <-> l \<in> formula"
paulson@13395
   459
apply (insert formula_replacement1)
paulson@13395
   460
apply (simp add: mem_formula_def relativize1_def formula_eq_Union
paulson@13395
   461
                 iterates_closed [of "is_formula_functor(M)"]) 
paulson@13395
   462
done
paulson@13395
   463
paulson@13395
   464
lemma (in M_datatypes) formula_abs [simp]:
paulson@13395
   465
     "[|M(Z)|] ==> is_formula(M,Z) <-> Z = formula"
paulson@13395
   466
apply (simp add: is_formula_def, safe)
paulson@13395
   467
apply (rule M_equalityI, simp_all)
paulson@13395
   468
done
paulson@13395
   469
paulson@13395
   470
paulson@13397
   471
subsection{*Absoluteness for Some List Operators*}
paulson@13397
   472
paulson@13395
   473
subsection{*Absoluteness for @{text \<epsilon>}-Closure: the @{term eclose} Operator*}
paulson@13395
   474
paulson@13395
   475
text{*Re-expresses eclose using "iterates"*}
paulson@13395
   476
lemma eclose_eq_Union:
paulson@13395
   477
     "eclose(A) = (\<Union>n\<in>nat. Union^n (A))"
paulson@13395
   478
apply (simp add: eclose_def) 
paulson@13395
   479
apply (rule UN_cong) 
paulson@13395
   480
apply (rule refl)
paulson@13395
   481
apply (induct_tac n)
paulson@13395
   482
apply (simp add: nat_rec_0)  
paulson@13395
   483
apply (simp add: nat_rec_succ) 
paulson@13395
   484
done
paulson@13395
   485
paulson@13395
   486
constdefs
paulson@13395
   487
  is_eclose_n :: "[i=>o,i,i,i] => o"
paulson@13395
   488
    "is_eclose_n(M,A,n,Z) == 
paulson@13395
   489
      \<exists>sn[M]. \<exists>msn[M]. 
paulson@13395
   490
       successor(M,n,sn) & membership(M,sn,msn) &
paulson@13395
   491
       is_wfrec(M, iterates_MH(M, big_union(M), A), msn, n, Z)"
paulson@13395
   492
  
paulson@13395
   493
  mem_eclose :: "[i=>o,i,i] => o"
paulson@13395
   494
    "mem_eclose(M,A,l) == 
paulson@13395
   495
      \<exists>n[M]. \<exists>eclosen[M]. 
paulson@13395
   496
       finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen"
paulson@13395
   497
paulson@13395
   498
  is_eclose :: "[i=>o,i,i] => o"
paulson@13395
   499
    "is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z <-> mem_eclose(M,A,u)"
paulson@13395
   500
paulson@13395
   501
paulson@13395
   502
locale (open) M_eclose = M_wfrank +
paulson@13395
   503
 assumes eclose_replacement1: 
paulson@13395
   504
   "M(A) ==> iterates_replacement(M, big_union(M), A)"
paulson@13395
   505
  and eclose_replacement2: 
paulson@13395
   506
   "M(A) ==> strong_replacement(M, 
paulson@13395
   507
         \<lambda>n y. n\<in>nat & 
paulson@13395
   508
               (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
paulson@13395
   509
               is_wfrec(M, iterates_MH(M,big_union(M), A), 
paulson@13395
   510
                        msn, n, y)))"
paulson@13395
   511
paulson@13395
   512
lemma (in M_eclose) eclose_replacement2': 
paulson@13395
   513
  "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))"
paulson@13395
   514
apply (insert eclose_replacement2 [of A]) 
paulson@13395
   515
apply (rule strong_replacement_cong [THEN iffD1])  
paulson@13395
   516
apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]]) 
paulson@13395
   517
apply (simp_all add: eclose_replacement1 relativize1_def) 
paulson@13395
   518
done
paulson@13395
   519
paulson@13395
   520
lemma (in M_eclose) eclose_closed [intro,simp]:
paulson@13395
   521
     "M(A) ==> M(eclose(A))"
paulson@13395
   522
apply (insert eclose_replacement1)
paulson@13395
   523
by  (simp add: RepFun_closed2 eclose_eq_Union 
paulson@13395
   524
               eclose_replacement2' relativize1_def
paulson@13395
   525
               iterates_closed [of "big_union(M)"])
paulson@13395
   526
paulson@13395
   527
lemma (in M_eclose) is_eclose_n_abs [simp]:
paulson@13395
   528
     "[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) <-> Z = Union^n (A)"
paulson@13395
   529
apply (insert eclose_replacement1)
paulson@13395
   530
apply (simp add: is_eclose_n_def relativize1_def nat_into_M
paulson@13395
   531
                 iterates_abs [of "big_union(M)" _ "Union"])
paulson@13395
   532
done
paulson@13395
   533
paulson@13395
   534
lemma (in M_eclose) mem_eclose_abs [simp]:
paulson@13395
   535
     "M(A) ==> mem_eclose(M,A,l) <-> l \<in> eclose(A)"
paulson@13395
   536
apply (insert eclose_replacement1)
paulson@13395
   537
apply (simp add: mem_eclose_def relativize1_def eclose_eq_Union
paulson@13395
   538
                 iterates_closed [of "big_union(M)"]) 
paulson@13395
   539
done
paulson@13395
   540
paulson@13395
   541
lemma (in M_eclose) eclose_abs [simp]:
paulson@13395
   542
     "[|M(A); M(Z)|] ==> is_eclose(M,A,Z) <-> Z = eclose(A)"
paulson@13395
   543
apply (simp add: is_eclose_def, safe)
paulson@13395
   544
apply (rule M_equalityI, simp_all)
paulson@13395
   545
done
paulson@13395
   546
paulson@13395
   547
paulson@13395
   548
paulson@13395
   549
paulson@13395
   550
subsection {*Absoluteness for @{term transrec}*}
paulson@13395
   551
paulson@13395
   552
paulson@13395
   553
text{* @{term "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"} *}
paulson@13395
   554
constdefs
paulson@13395
   555
paulson@13395
   556
  is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o"
paulson@13395
   557
   "is_transrec(M,MH,a,z) == 
paulson@13395
   558
      \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. 
paulson@13395
   559
       upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
paulson@13395
   560
       is_wfrec(M,MH,mesa,a,z)"
paulson@13395
   561
paulson@13395
   562
  transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o"
paulson@13395
   563
   "transrec_replacement(M,MH,a) ==
paulson@13395
   564
      \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. 
paulson@13395
   565
       upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
paulson@13395
   566
       wfrec_replacement(M,MH,mesa)"
paulson@13395
   567
paulson@13395
   568
text{*The condition @{term "Ord(i)"} lets us use the simpler 
paulson@13395
   569
  @{text "trans_wfrec_abs"} rather than @{text "trans_wfrec_abs"},
paulson@13395
   570
  which I haven't even proved yet. *}
paulson@13395
   571
theorem (in M_eclose) transrec_abs:
paulson@13395
   572
  "[|Ord(i);  M(i);  M(z);
paulson@13395
   573
     transrec_replacement(M,MH,i);  relativize2(M,MH,H);
paulson@13395
   574
     \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
paulson@13395
   575
   ==> is_transrec(M,MH,i,z) <-> z = transrec(i,H)" 
paulson@13395
   576
by (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def
paulson@13395
   577
       transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
paulson@13395
   578
paulson@13395
   579
paulson@13395
   580
theorem (in M_eclose) transrec_closed:
paulson@13395
   581
     "[|Ord(i);  M(i);  M(z);
paulson@13395
   582
	transrec_replacement(M,MH,i);  relativize2(M,MH,H);
paulson@13395
   583
	\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
paulson@13395
   584
      ==> M(transrec(i,H))"
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by (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def
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       transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
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subsection{*Absoluteness for the List Operator @{term length}*}
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constdefs
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  is_length :: "[i=>o,i,i,i] => o"
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    "is_length(M,A,l,n) == 
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       \<exists>sn[M]. \<exists>list_n[M]. \<exists>list_sn[M]. 
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        is_list_N(M,A,n,list_n) & l \<notin> list_n &
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        successor(M,n,sn) & is_list_N(M,A,sn,list_sn) & l \<in> list_sn"
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lemma (in M_datatypes) length_abs [simp]:
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     "[|M(A); l \<in> list(A); n \<in> nat|] ==> is_length(M,A,l,n) <-> n = length(l)"
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apply (subgoal_tac "M(l) & M(n)")
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 prefer 2 apply (blast dest: transM)  
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apply (simp add: is_length_def)
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apply (blast intro: list_imp_list_N nat_into_Ord list_N_imp_eq_length
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             dest: list_N_imp_length_lt)
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   607
done
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text{*Proof is trivial since @{term length} returns natural numbers.*}
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lemma (in M_triv_axioms) length_closed [intro,simp]:
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     "l \<in> list(A) ==> M(length(l))"
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by (simp add: nat_into_M ) 
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subsection {*Absoluteness for @{term nth}*}
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lemma nth_eq_hd_iterates_tl [rule_format]:
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     "xs \<in> list(A) ==> \<forall>n \<in> nat. nth(n,xs) = hd' (tl'^n (xs))"
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apply (induct_tac xs) 
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   620
apply (simp add: iterates_tl_Nil hd'_Nil, clarify) 
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   621
apply (erule natE)
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apply (simp add: hd'_Cons) 
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   623
apply (simp add: tl'_Cons iterates_commute) 
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done
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lemma (in M_axioms) iterates_tl'_closed:
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     "[|n \<in> nat; M(x)|] ==> M(tl'^n (x))"
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apply (induct_tac n, simp) 
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   629
apply (simp add: tl'_Cons tl'_closed) 
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done
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locale (open) M_nth = M_datatypes +
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 assumes nth_replacement1: 
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   "M(xs) ==> iterates_replacement(M, %l t. is_tl(M,l,t), xs)"
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text{*Immediate by type-checking*}
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lemma (in M_datatypes) nth_closed [intro,simp]:
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     "[|xs \<in> list(A); n \<in> nat; M(A)|] ==> M(nth(n,xs))" 
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apply (case_tac "n < length(xs)")
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   640
 apply (blast intro: nth_type transM)
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   641
apply (simp add: not_lt_iff_le nth_eq_0)
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   642
done
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constdefs
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  is_nth :: "[i=>o,i,i,i] => o"
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   646
    "is_nth(M,n,l,Z) == 
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   647
      \<exists>X[M]. \<exists>sn[M]. \<exists>msn[M]. 
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       successor(M,n,sn) & membership(M,sn,msn) &
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       is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) &
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       is_hd(M,X,Z)"
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   651
 
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lemma (in M_nth) nth_abs [simp]:
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     "[|M(A); n \<in> nat; l \<in> list(A); M(Z)|] 
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   654
      ==> is_nth(M,n,l,Z) <-> Z = nth(n,l)"
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   655
apply (subgoal_tac "M(l)") 
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   656
 prefer 2 apply (blast intro: transM)
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   657
apply (insert nth_replacement1)
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   658
apply (simp add: is_nth_def nth_eq_hd_iterates_tl nat_into_M
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   659
                 tl'_closed iterates_tl'_closed 
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   660
                 iterates_abs [OF _ relativize1_tl])
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   661
done
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   662
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   663
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   664
end