src/ZF/Constructible/Datatype_absolute.thy
author wenzelm
Thu Dec 14 11:24:26 2017 +0100 (21 months ago)
changeset 67198 694f29a5433b
parent 61798 27f3c10b0b50
child 67443 3abf6a722518
permissions -rw-r--r--
merged
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(*  Title:      ZF/Constructible/Datatype_absolute.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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*)
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section \<open>Absoluteness Properties for Recursive Datatypes\<close>
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theory Datatype_absolute imports Formula WF_absolute begin
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subsection\<open>The lfp of a continuous function can be expressed as a union\<close>
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definition
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  directed :: "i=>o" where
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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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definition
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  contin :: "(i=>i) => o" where
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   "contin(h) == (\<forall>A. directed(A) \<longrightarrow> 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) \<subseteq> 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 \<longrightarrow> 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) \<subseteq> (\<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)) \<subseteq> 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\<open>Some Standard Datatype Constructions Preserve Continuity\<close>
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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 \<open>Absoluteness for "Iterates"\<close>
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definition
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  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o" where
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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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definition
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  is_iterates :: "[i=>o, [i,i]=>o, i, i, i] => o" where
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    "is_iterates(M,isF,v,n,Z) == 
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      \<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
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                       is_wfrec(M, iterates_MH(M,isF,v), msn, n, Z)"
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definition
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  iterates_replacement :: "[i=>o, [i,i]=>o, i] => o" where
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   "iterates_replacement(M,isF,v) ==
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      \<forall>n[M]. n\<in>nat \<longrightarrow> 
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         wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))"
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lemma (in M_basic) iterates_MH_abs:
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  "[| relation1(M,isF,F); M(n); M(g); M(z) |] 
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   ==> iterates_MH(M,isF,v,n,g,z) \<longleftrightarrow> 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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              relation1_def iterates_MH_def)  
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lemma (in M_basic) iterates_imp_wfrec_replacement:
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  "[|relation1(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); relation1(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_iterates(M,isF,v,n,z) \<longleftrightarrow> 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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                 is_iterates_def relation2_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_trancl) iterates_closed [intro,simp]:
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  "[| iterates_replacement(M,isF,v); relation1(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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                 relation2_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 \<open>lists without univ\<close>
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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\<open>Re-expresses lists using sum and product\<close>
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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\<open>Opposite inclusion\<close>
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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\<open>Re-expresses lists using "iterates", no univ.\<close>
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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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definition
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  is_list_functor :: "[i=>o,i,i,i] => o" where
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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_basic) list_functor_abs [simp]: 
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     "[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) \<longleftrightarrow> (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 \<open>formulas without univ\<close>
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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))"
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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))"
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by (intro sum_contin prod_contin id_contin const_contin) 
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text\<open>Re-expresses formulas using sum and product\<close>
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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))"
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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 Nand_def Forall_def)
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 apply blast 
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txt\<open>Opposite inclusion\<close>
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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 Nand_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\<open>Re-expresses formulas using "iterates", no univ.\<close>
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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)) ^ 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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definition
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  is_formula_functor :: "[i=>o,i,i] => o" where
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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]. 
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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) & 
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          is_sum(M,natnatsum,X3,Z)"
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lemma (in M_basic) formula_functor_abs [simp]: 
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     "[| M(X); M(Z) |] 
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      ==> is_formula_functor(M,X,Z) \<longleftrightarrow> 
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          Z = ((nat*nat) + (nat*nat)) + (X*X + X)"
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by (simp add: is_formula_functor_def) 
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subsection\<open>@{term M} Contains the List and Formula Datatypes\<close>
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definition
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  list_N :: "[i,i] => i" where
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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)) \<longleftrightarrow> 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\<open>These two aren't simprules because they reveal the underlying
wenzelm@60770
   294
list representation.\<close>
paulson@13397
   295
lemma list_N_0: "list_N(A,0) = 0"
paulson@13397
   296
by (simp add: list_N_def)
paulson@13397
   297
paulson@13397
   298
lemma list_N_succ: "list_N(A,succ(n)) = {0} + A * (list_N(A,n))"
paulson@13397
   299
by (simp add: list_N_def)
paulson@13397
   300
paulson@13397
   301
lemma list_N_imp_list:
paulson@13397
   302
  "[| l \<in> list_N(A,n); n \<in> nat |] ==> l \<in> list(A)"
paulson@13397
   303
by (force simp add: list_eq_Union list_N_def)
paulson@13397
   304
paulson@13397
   305
lemma list_N_imp_length_lt [rule_format]:
paulson@13397
   306
     "n \<in> nat ==> \<forall>l \<in> list_N(A,n). length(l) < n"
paulson@13397
   307
apply (induct_tac n)  
paulson@13397
   308
apply (auto simp add: list_N_0 list_N_succ 
paulson@13397
   309
                      Nil_def [symmetric] Cons_def [symmetric]) 
paulson@13397
   310
done
paulson@13397
   311
paulson@13397
   312
lemma list_imp_list_N [rule_format]:
paulson@46823
   313
     "l \<in> list(A) ==> \<forall>n\<in>nat. length(l) < n \<longrightarrow> l \<in> list_N(A, n)"
paulson@13397
   314
apply (induct_tac l)
paulson@13397
   315
apply (force elim: natE)+
paulson@13397
   316
done
paulson@13397
   317
paulson@13397
   318
lemma list_N_imp_eq_length:
paulson@13397
   319
      "[|n \<in> nat; l \<notin> list_N(A, n); l \<in> list_N(A, succ(n))|] 
paulson@13397
   320
       ==> n = length(l)"
paulson@13397
   321
apply (rule le_anti_sym) 
paulson@13397
   322
 prefer 2 apply (simp add: list_N_imp_length_lt) 
paulson@13397
   323
apply (frule list_N_imp_list, simp)
paulson@13397
   324
apply (simp add: not_lt_iff_le [symmetric]) 
paulson@13397
   325
apply (blast intro: list_imp_list_N) 
paulson@13397
   326
done
paulson@13397
   327
  
wenzelm@60770
   328
text\<open>Express @{term list_rec} without using @{term rank} or @{term Vset},
wenzelm@60770
   329
neither of which is absolute.\<close>
paulson@13564
   330
lemma (in M_trivial) list_rec_eq:
paulson@13397
   331
  "l \<in> list(A) ==>
paulson@13397
   332
   list_rec(a,g,l) = 
paulson@13397
   333
   transrec (succ(length(l)),
paulson@13409
   334
      \<lambda>x h. Lambda (list(A),
paulson@13409
   335
                    list_case' (a, 
paulson@13409
   336
                           \<lambda>a l. g(a, l, h ` succ(length(l)) ` l)))) ` l"
paulson@13397
   337
apply (induct_tac l) 
paulson@13397
   338
apply (subst transrec, simp) 
paulson@13397
   339
apply (subst transrec) 
paulson@13397
   340
apply (simp add: list_imp_list_N) 
paulson@13397
   341
done
paulson@13397
   342
wenzelm@21233
   343
definition
wenzelm@21404
   344
  is_list_N :: "[i=>o,i,i,i] => o" where
paulson@13397
   345
    "is_list_N(M,A,n,Z) == 
paulson@13655
   346
      \<exists>zero[M]. empty(M,zero) & 
paulson@13655
   347
                is_iterates(M, is_list_functor(M,A), zero, n, Z)"
wenzelm@21404
   348
wenzelm@21404
   349
definition  
wenzelm@21404
   350
  mem_list :: "[i=>o,i,i] => o" where
paulson@13395
   351
    "mem_list(M,A,l) == 
paulson@13395
   352
      \<exists>n[M]. \<exists>listn[M]. 
paulson@13397
   353
       finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l \<in> listn"
paulson@13395
   354
wenzelm@21404
   355
definition
wenzelm@21404
   356
  is_list :: "[i=>o,i,i] => o" where
paulson@46823
   357
    "is_list(M,A,Z) == \<forall>l[M]. l \<in> Z \<longleftrightarrow> mem_list(M,A,l)"
paulson@13395
   358
wenzelm@60770
   359
subsubsection\<open>Towards Absoluteness of @{term formula_rec}\<close>
paulson@13493
   360
paulson@13493
   361
consts   depth :: "i=>i"
paulson@13493
   362
primrec
paulson@13493
   363
  "depth(Member(x,y)) = 0"
paulson@13493
   364
  "depth(Equal(x,y))  = 0"
paulson@13493
   365
  "depth(Nand(p,q)) = succ(depth(p) \<union> depth(q))"
paulson@13493
   366
  "depth(Forall(p)) = succ(depth(p))"
paulson@13493
   367
paulson@13493
   368
lemma depth_type [TC]: "p \<in> formula ==> depth(p) \<in> nat"
paulson@13493
   369
by (induct_tac p, simp_all) 
paulson@13493
   370
paulson@13493
   371
wenzelm@21233
   372
definition
wenzelm@21404
   373
  formula_N :: "i => i" where
paulson@13493
   374
    "formula_N(n) == (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)) ^ n (0)"
paulson@13493
   375
paulson@13493
   376
lemma Member_in_formula_N [simp]:
paulson@46823
   377
     "Member(x,y) \<in> formula_N(succ(n)) \<longleftrightarrow> x \<in> nat & y \<in> nat"
paulson@13493
   378
by (simp add: formula_N_def Member_def) 
paulson@13493
   379
paulson@13493
   380
lemma Equal_in_formula_N [simp]:
paulson@46823
   381
     "Equal(x,y) \<in> formula_N(succ(n)) \<longleftrightarrow> x \<in> nat & y \<in> nat"
paulson@13493
   382
by (simp add: formula_N_def Equal_def) 
paulson@13493
   383
paulson@13493
   384
lemma Nand_in_formula_N [simp]:
paulson@46823
   385
     "Nand(x,y) \<in> formula_N(succ(n)) \<longleftrightarrow> x \<in> formula_N(n) & y \<in> formula_N(n)"
paulson@13493
   386
by (simp add: formula_N_def Nand_def) 
paulson@13493
   387
paulson@13493
   388
lemma Forall_in_formula_N [simp]:
paulson@46823
   389
     "Forall(x) \<in> formula_N(succ(n)) \<longleftrightarrow> x \<in> formula_N(n)"
paulson@13493
   390
by (simp add: formula_N_def Forall_def) 
paulson@13493
   391
wenzelm@60770
   392
text\<open>These two aren't simprules because they reveal the underlying
wenzelm@60770
   393
formula representation.\<close>
paulson@13493
   394
lemma formula_N_0: "formula_N(0) = 0"
paulson@13493
   395
by (simp add: formula_N_def)
paulson@13493
   396
paulson@13493
   397
lemma formula_N_succ:
paulson@13493
   398
     "formula_N(succ(n)) = 
paulson@13493
   399
      ((nat*nat) + (nat*nat)) + (formula_N(n) * formula_N(n) + formula_N(n))"
paulson@13493
   400
by (simp add: formula_N_def)
paulson@13493
   401
paulson@13493
   402
lemma formula_N_imp_formula:
paulson@13493
   403
  "[| p \<in> formula_N(n); n \<in> nat |] ==> p \<in> formula"
paulson@13493
   404
by (force simp add: formula_eq_Union formula_N_def)
paulson@13493
   405
paulson@13493
   406
lemma formula_N_imp_depth_lt [rule_format]:
paulson@13493
   407
     "n \<in> nat ==> \<forall>p \<in> formula_N(n). depth(p) < n"
paulson@13493
   408
apply (induct_tac n)  
paulson@13493
   409
apply (auto simp add: formula_N_0 formula_N_succ 
paulson@13493
   410
                      depth_type formula_N_imp_formula Un_least_lt_iff
paulson@13493
   411
                      Member_def [symmetric] Equal_def [symmetric]
paulson@13493
   412
                      Nand_def [symmetric] Forall_def [symmetric]) 
paulson@13493
   413
done
paulson@13493
   414
paulson@13493
   415
lemma formula_imp_formula_N [rule_format]:
paulson@46823
   416
     "p \<in> formula ==> \<forall>n\<in>nat. depth(p) < n \<longrightarrow> p \<in> formula_N(n)"
paulson@13493
   417
apply (induct_tac p)
paulson@13493
   418
apply (simp_all add: succ_Un_distrib Un_least_lt_iff) 
paulson@13493
   419
apply (force elim: natE)+
paulson@13493
   420
done
paulson@13493
   421
paulson@13493
   422
lemma formula_N_imp_eq_depth:
paulson@13493
   423
      "[|n \<in> nat; p \<notin> formula_N(n); p \<in> formula_N(succ(n))|] 
paulson@13493
   424
       ==> n = depth(p)"
paulson@13493
   425
apply (rule le_anti_sym) 
paulson@13493
   426
 prefer 2 apply (simp add: formula_N_imp_depth_lt) 
paulson@13493
   427
apply (frule formula_N_imp_formula, simp)
paulson@13493
   428
apply (simp add: not_lt_iff_le [symmetric]) 
paulson@13493
   429
apply (blast intro: formula_imp_formula_N) 
paulson@13493
   430
done
paulson@13493
   431
paulson@13493
   432
wenzelm@60770
   433
text\<open>This result and the next are unused.\<close>
paulson@13493
   434
lemma formula_N_mono [rule_format]:
paulson@46823
   435
  "[| m \<in> nat; n \<in> nat |] ==> m\<le>n \<longrightarrow> formula_N(m) \<subseteq> formula_N(n)"
paulson@13493
   436
apply (rule_tac m = m and n = n in diff_induct)
paulson@13493
   437
apply (simp_all add: formula_N_0 formula_N_succ, blast) 
paulson@13493
   438
done
paulson@13493
   439
paulson@13493
   440
lemma formula_N_distrib:
paulson@13493
   441
  "[| m \<in> nat; n \<in> nat |] ==> formula_N(m \<union> n) = formula_N(m) \<union> formula_N(n)"
paulson@13493
   442
apply (rule_tac i = m and j = n in Ord_linear_le, auto) 
paulson@13493
   443
apply (simp_all add: subset_Un_iff [THEN iffD1] subset_Un_iff2 [THEN iffD1] 
paulson@13493
   444
                     le_imp_subset formula_N_mono)
paulson@13493
   445
done
paulson@13493
   446
wenzelm@21233
   447
definition
wenzelm@21404
   448
  is_formula_N :: "[i=>o,i,i] => o" where
paulson@13493
   449
    "is_formula_N(M,n,Z) == 
paulson@13655
   450
      \<exists>zero[M]. empty(M,zero) & 
paulson@13655
   451
                is_iterates(M, is_formula_functor(M), zero, n, Z)"
paulson@13655
   452
paulson@13493
   453
wenzelm@21404
   454
definition  
wenzelm@21404
   455
  mem_formula :: "[i=>o,i] => o" where
paulson@13395
   456
    "mem_formula(M,p) == 
paulson@13395
   457
      \<exists>n[M]. \<exists>formn[M]. 
paulson@13493
   458
       finite_ordinal(M,n) & is_formula_N(M,n,formn) & p \<in> formn"
paulson@13395
   459
wenzelm@21404
   460
definition
wenzelm@21404
   461
  is_formula :: "[i=>o,i] => o" where
paulson@46823
   462
    "is_formula(M,Z) == \<forall>p[M]. p \<in> Z \<longleftrightarrow> mem_formula(M,p)"
paulson@13395
   463
paulson@13634
   464
locale M_datatypes = M_trancl +
paulson@13655
   465
 assumes list_replacement1:
paulson@13363
   466
   "M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)"
paulson@13655
   467
  and list_replacement2:
paulson@13655
   468
   "M(A) ==> strong_replacement(M,
paulson@13655
   469
         \<lambda>n y. n\<in>nat & is_iterates(M, is_list_functor(M,A), 0, n, y))"
paulson@13655
   470
  and formula_replacement1:
paulson@13386
   471
   "iterates_replacement(M, is_formula_functor(M), 0)"
paulson@13655
   472
  and formula_replacement2:
paulson@13655
   473
   "strong_replacement(M,
paulson@13655
   474
         \<lambda>n y. n\<in>nat & is_iterates(M, is_formula_functor(M), 0, n, y))"
paulson@13422
   475
  and nth_replacement:
paulson@13422
   476
   "M(l) ==> iterates_replacement(M, %l t. is_tl(M,l,t), l)"
paulson@13655
   477
paulson@13395
   478
wenzelm@60770
   479
subsubsection\<open>Absoluteness of the List Construction\<close>
paulson@13395
   480
paulson@13655
   481
lemma (in M_datatypes) list_replacement2':
paulson@13353
   482
  "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"
paulson@13655
   483
apply (insert list_replacement2 [of A])
paulson@13655
   484
apply (rule strong_replacement_cong [THEN iffD1])
paulson@13655
   485
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]])
paulson@13655
   486
apply (simp_all add: list_replacement1 relation1_def)
paulson@13353
   487
done
paulson@13268
   488
paulson@13268
   489
lemma (in M_datatypes) list_closed [intro,simp]:
paulson@13268
   490
     "M(A) ==> M(list(A))"
paulson@13353
   491
apply (insert list_replacement1)
paulson@13655
   492
by  (simp add: RepFun_closed2 list_eq_Union
paulson@13634
   493
               list_replacement2' relation1_def
paulson@13353
   494
               iterates_closed [of "is_list_functor(M,A)"])
paulson@13397
   495
wenzelm@61798
   496
text\<open>WARNING: use only with \<open>dest:\<close> or with variables fixed!\<close>
paulson@13423
   497
lemmas (in M_datatypes) list_into_M = transM [OF _ list_closed]
paulson@13423
   498
paulson@13397
   499
lemma (in M_datatypes) list_N_abs [simp]:
paulson@13655
   500
     "[|M(A); n\<in>nat; M(Z)|]
paulson@46823
   501
      ==> is_list_N(M,A,n,Z) \<longleftrightarrow> Z = list_N(A,n)"
paulson@13395
   502
apply (insert list_replacement1)
paulson@13634
   503
apply (simp add: is_list_N_def list_N_def relation1_def nat_into_M
paulson@13395
   504
                 iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"])
paulson@13395
   505
done
paulson@13268
   506
paulson@13397
   507
lemma (in M_datatypes) list_N_closed [intro,simp]:
paulson@13397
   508
     "[|M(A); n\<in>nat|] ==> M(list_N(A,n))"
paulson@13397
   509
apply (insert list_replacement1)
paulson@13634
   510
apply (simp add: is_list_N_def list_N_def relation1_def nat_into_M
paulson@13397
   511
                 iterates_closed [of "is_list_functor(M,A)"])
paulson@13397
   512
done
paulson@13397
   513
paulson@13395
   514
lemma (in M_datatypes) mem_list_abs [simp]:
paulson@46823
   515
     "M(A) ==> mem_list(M,A,l) \<longleftrightarrow> l \<in> list(A)"
paulson@13395
   516
apply (insert list_replacement1)
paulson@13634
   517
apply (simp add: mem_list_def list_N_def relation1_def list_eq_Union
paulson@13655
   518
                 iterates_closed [of "is_list_functor(M,A)"])
paulson@13395
   519
done
paulson@13395
   520
paulson@13395
   521
lemma (in M_datatypes) list_abs [simp]:
paulson@46823
   522
     "[|M(A); M(Z)|] ==> is_list(M,A,Z) \<longleftrightarrow> Z = list(A)"
paulson@13395
   523
apply (simp add: is_list_def, safe)
paulson@13395
   524
apply (rule M_equalityI, simp_all)
paulson@13395
   525
done
paulson@13395
   526
wenzelm@60770
   527
subsubsection\<open>Absoluteness of Formulas\<close>
paulson@13293
   528
paulson@13655
   529
lemma (in M_datatypes) formula_replacement2':
paulson@13398
   530
  "strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))^n (0))"
paulson@13655
   531
apply (insert formula_replacement2)
paulson@13655
   532
apply (rule strong_replacement_cong [THEN iffD1])
paulson@13655
   533
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]])
paulson@13655
   534
apply (simp_all add: formula_replacement1 relation1_def)
paulson@13386
   535
done
paulson@13386
   536
paulson@13386
   537
lemma (in M_datatypes) formula_closed [intro,simp]:
paulson@13386
   538
     "M(formula)"
paulson@13386
   539
apply (insert formula_replacement1)
paulson@13655
   540
apply  (simp add: RepFun_closed2 formula_eq_Union
paulson@13634
   541
                  formula_replacement2' relation1_def
paulson@13386
   542
                  iterates_closed [of "is_formula_functor(M)"])
paulson@13386
   543
done
paulson@13386
   544
paulson@13423
   545
lemmas (in M_datatypes) formula_into_M = transM [OF _ formula_closed]
paulson@13423
   546
paulson@13493
   547
lemma (in M_datatypes) formula_N_abs [simp]:
paulson@13655
   548
     "[|n\<in>nat; M(Z)|]
paulson@46823
   549
      ==> is_formula_N(M,n,Z) \<longleftrightarrow> Z = formula_N(n)"
paulson@13395
   550
apply (insert formula_replacement1)
paulson@13634
   551
apply (simp add: is_formula_N_def formula_N_def relation1_def nat_into_M
paulson@13655
   552
                 iterates_abs [of "is_formula_functor(M)" _
paulson@13493
   553
                                  "\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)"])
paulson@13493
   554
done
paulson@13493
   555
paulson@13493
   556
lemma (in M_datatypes) formula_N_closed [intro,simp]:
paulson@13493
   557
     "n\<in>nat ==> M(formula_N(n))"
paulson@13493
   558
apply (insert formula_replacement1)
paulson@13634
   559
apply (simp add: is_formula_N_def formula_N_def relation1_def nat_into_M
paulson@13493
   560
                 iterates_closed [of "is_formula_functor(M)"])
paulson@13395
   561
done
paulson@13395
   562
paulson@13395
   563
lemma (in M_datatypes) mem_formula_abs [simp]:
paulson@46823
   564
     "mem_formula(M,l) \<longleftrightarrow> l \<in> formula"
paulson@13395
   565
apply (insert formula_replacement1)
paulson@13634
   566
apply (simp add: mem_formula_def relation1_def formula_eq_Union formula_N_def
paulson@13655
   567
                 iterates_closed [of "is_formula_functor(M)"])
paulson@13395
   568
done
paulson@13395
   569
paulson@13395
   570
lemma (in M_datatypes) formula_abs [simp]:
paulson@46823
   571
     "[|M(Z)|] ==> is_formula(M,Z) \<longleftrightarrow> Z = formula"
paulson@13395
   572
apply (simp add: is_formula_def, safe)
paulson@13395
   573
apply (rule M_equalityI, simp_all)
paulson@13395
   574
done
paulson@13395
   575
paulson@13395
   576
wenzelm@61798
   577
subsection\<open>Absoluteness for \<open>\<epsilon>\<close>-Closure: the @{term eclose} Operator\<close>
paulson@13395
   578
wenzelm@60770
   579
text\<open>Re-expresses eclose using "iterates"\<close>
paulson@13395
   580
lemma eclose_eq_Union:
paulson@13395
   581
     "eclose(A) = (\<Union>n\<in>nat. Union^n (A))"
paulson@13655
   582
apply (simp add: eclose_def)
paulson@13655
   583
apply (rule UN_cong)
paulson@13395
   584
apply (rule refl)
paulson@13395
   585
apply (induct_tac n)
paulson@13655
   586
apply (simp add: nat_rec_0)
paulson@13655
   587
apply (simp add: nat_rec_succ)
paulson@13395
   588
done
paulson@13395
   589
wenzelm@21233
   590
definition
wenzelm@21404
   591
  is_eclose_n :: "[i=>o,i,i,i] => o" where
paulson@13655
   592
    "is_eclose_n(M,A,n,Z) == is_iterates(M, big_union(M), A, n, Z)"
paulson@13655
   593
wenzelm@21404
   594
definition
wenzelm@21404
   595
  mem_eclose :: "[i=>o,i,i] => o" where
paulson@13655
   596
    "mem_eclose(M,A,l) ==
paulson@13655
   597
      \<exists>n[M]. \<exists>eclosen[M].
paulson@13395
   598
       finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen"
paulson@13395
   599
wenzelm@21404
   600
definition
wenzelm@21404
   601
  is_eclose :: "[i=>o,i,i] => o" where
paulson@46823
   602
    "is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z \<longleftrightarrow> mem_eclose(M,A,u)"
paulson@13395
   603
paulson@13395
   604
wenzelm@13428
   605
locale M_eclose = M_datatypes +
paulson@13655
   606
 assumes eclose_replacement1:
paulson@13395
   607
   "M(A) ==> iterates_replacement(M, big_union(M), A)"
paulson@13655
   608
  and eclose_replacement2:
paulson@13655
   609
   "M(A) ==> strong_replacement(M,
paulson@13655
   610
         \<lambda>n y. n\<in>nat & is_iterates(M, big_union(M), A, n, y))"
paulson@13395
   611
paulson@13655
   612
lemma (in M_eclose) eclose_replacement2':
paulson@13395
   613
  "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))"
paulson@13655
   614
apply (insert eclose_replacement2 [of A])
paulson@13655
   615
apply (rule strong_replacement_cong [THEN iffD1])
paulson@13655
   616
apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]])
paulson@13655
   617
apply (simp_all add: eclose_replacement1 relation1_def)
paulson@13395
   618
done
paulson@13395
   619
paulson@13395
   620
lemma (in M_eclose) eclose_closed [intro,simp]:
paulson@13395
   621
     "M(A) ==> M(eclose(A))"
paulson@13395
   622
apply (insert eclose_replacement1)
paulson@13655
   623
by  (simp add: RepFun_closed2 eclose_eq_Union
paulson@13634
   624
               eclose_replacement2' relation1_def
paulson@13395
   625
               iterates_closed [of "big_union(M)"])
paulson@13395
   626
paulson@13395
   627
lemma (in M_eclose) is_eclose_n_abs [simp]:
paulson@46823
   628
     "[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) \<longleftrightarrow> Z = Union^n (A)"
paulson@13395
   629
apply (insert eclose_replacement1)
paulson@13634
   630
apply (simp add: is_eclose_n_def relation1_def nat_into_M
paulson@13395
   631
                 iterates_abs [of "big_union(M)" _ "Union"])
paulson@13395
   632
done
paulson@13395
   633
paulson@13395
   634
lemma (in M_eclose) mem_eclose_abs [simp]:
paulson@46823
   635
     "M(A) ==> mem_eclose(M,A,l) \<longleftrightarrow> l \<in> eclose(A)"
paulson@13395
   636
apply (insert eclose_replacement1)
paulson@13634
   637
apply (simp add: mem_eclose_def relation1_def eclose_eq_Union
paulson@13655
   638
                 iterates_closed [of "big_union(M)"])
paulson@13395
   639
done
paulson@13395
   640
paulson@13395
   641
lemma (in M_eclose) eclose_abs [simp]:
paulson@46823
   642
     "[|M(A); M(Z)|] ==> is_eclose(M,A,Z) \<longleftrightarrow> Z = eclose(A)"
paulson@13395
   643
apply (simp add: is_eclose_def, safe)
paulson@13395
   644
apply (rule M_equalityI, simp_all)
paulson@13395
   645
done
paulson@13395
   646
paulson@13395
   647
wenzelm@60770
   648
subsection \<open>Absoluteness for @{term transrec}\<close>
paulson@13395
   649
wenzelm@60770
   650
text\<open>@{prop "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"}\<close>
wenzelm@21404
   651
wenzelm@21233
   652
definition
wenzelm@21404
   653
  is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o" where
paulson@13655
   654
   "is_transrec(M,MH,a,z) ==
paulson@13655
   655
      \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
paulson@13395
   656
       upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
paulson@13395
   657
       is_wfrec(M,MH,mesa,a,z)"
paulson@13395
   658
wenzelm@21404
   659
definition
wenzelm@21404
   660
  transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o" where
paulson@13395
   661
   "transrec_replacement(M,MH,a) ==
paulson@13655
   662
      \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
paulson@13395
   663
       upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
paulson@13395
   664
       wfrec_replacement(M,MH,mesa)"
paulson@13395
   665
wenzelm@60770
   666
text\<open>The condition @{term "Ord(i)"} lets us use the simpler
wenzelm@61798
   667
  \<open>trans_wfrec_abs\<close> rather than \<open>trans_wfrec_abs\<close>,
wenzelm@60770
   668
  which I haven't even proved yet.\<close>
paulson@13395
   669
theorem (in M_eclose) transrec_abs:
paulson@13634
   670
  "[|transrec_replacement(M,MH,i);  relation2(M,MH,H);
paulson@13418
   671
     Ord(i);  M(i);  M(z);
paulson@46823
   672
     \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|]
paulson@46823
   673
   ==> is_transrec(M,MH,i,z) \<longleftrightarrow> z = transrec(i,H)"
paulson@13615
   674
by (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def
paulson@13395
   675
       transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
paulson@13395
   676
paulson@13395
   677
paulson@13395
   678
theorem (in M_eclose) transrec_closed:
paulson@13634
   679
     "[|transrec_replacement(M,MH,i);  relation2(M,MH,H);
wenzelm@32960
   680
        Ord(i);  M(i);
paulson@46823
   681
        \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|]
paulson@13395
   682
      ==> M(transrec(i,H))"
paulson@13615
   683
by (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def
paulson@13615
   684
        transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
paulson@13615
   685
paulson@13395
   686
wenzelm@60770
   687
text\<open>Helps to prove instances of @{term transrec_replacement}\<close>
paulson@13655
   688
lemma (in M_eclose) transrec_replacementI:
paulson@13440
   689
   "[|M(a);
paulson@13655
   690
      strong_replacement (M,
paulson@13655
   691
                  \<lambda>x z. \<exists>y[M]. pair(M, x, y, z) &
paulson@13655
   692
                               is_wfrec(M,MH,Memrel(eclose({a})),x,y))|]
paulson@13440
   693
    ==> transrec_replacement(M,MH,a)"
paulson@13655
   694
by (simp add: transrec_replacement_def wfrec_replacement_def)
paulson@13440
   695
paulson@13395
   696
wenzelm@60770
   697
subsection\<open>Absoluteness for the List Operator @{term length}\<close>
wenzelm@60770
   698
text\<open>But it is never used.\<close>
paulson@13647
   699
wenzelm@21233
   700
definition
wenzelm@21404
   701
  is_length :: "[i=>o,i,i,i] => o" where
paulson@13655
   702
    "is_length(M,A,l,n) ==
paulson@13655
   703
       \<exists>sn[M]. \<exists>list_n[M]. \<exists>list_sn[M].
paulson@13397
   704
        is_list_N(M,A,n,list_n) & l \<notin> list_n &
paulson@13397
   705
        successor(M,n,sn) & is_list_N(M,A,sn,list_sn) & l \<in> list_sn"
paulson@13397
   706
paulson@13397
   707
paulson@13397
   708
lemma (in M_datatypes) length_abs [simp]:
paulson@46823
   709
     "[|M(A); l \<in> list(A); n \<in> nat|] ==> is_length(M,A,l,n) \<longleftrightarrow> n = length(l)"
paulson@13397
   710
apply (subgoal_tac "M(l) & M(n)")
paulson@13655
   711
 prefer 2 apply (blast dest: transM)
paulson@13397
   712
apply (simp add: is_length_def)
paulson@13397
   713
apply (blast intro: list_imp_list_N nat_into_Ord list_N_imp_eq_length
paulson@13397
   714
             dest: list_N_imp_length_lt)
paulson@13397
   715
done
paulson@13397
   716
wenzelm@60770
   717
text\<open>Proof is trivial since @{term length} returns natural numbers.\<close>
paulson@13564
   718
lemma (in M_trivial) length_closed [intro,simp]:
paulson@13397
   719
     "l \<in> list(A) ==> M(length(l))"
paulson@13655
   720
by (simp add: nat_into_M)
paulson@13397
   721
paulson@13397
   722
wenzelm@60770
   723
subsection \<open>Absoluteness for the List Operator @{term nth}\<close>
paulson@13397
   724
paulson@13397
   725
lemma nth_eq_hd_iterates_tl [rule_format]:
paulson@13397
   726
     "xs \<in> list(A) ==> \<forall>n \<in> nat. nth(n,xs) = hd' (tl'^n (xs))"
paulson@13655
   727
apply (induct_tac xs)
paulson@13655
   728
apply (simp add: iterates_tl_Nil hd'_Nil, clarify)
paulson@13397
   729
apply (erule natE)
paulson@13655
   730
apply (simp add: hd'_Cons)
paulson@13655
   731
apply (simp add: tl'_Cons iterates_commute)
paulson@13397
   732
done
paulson@13397
   733
paulson@13564
   734
lemma (in M_basic) iterates_tl'_closed:
paulson@13397
   735
     "[|n \<in> nat; M(x)|] ==> M(tl'^n (x))"
paulson@13655
   736
apply (induct_tac n, simp)
paulson@13655
   737
apply (simp add: tl'_Cons tl'_closed)
paulson@13397
   738
done
paulson@13397
   739
wenzelm@60770
   740
text\<open>Immediate by type-checking\<close>
paulson@13397
   741
lemma (in M_datatypes) nth_closed [intro,simp]:
paulson@13655
   742
     "[|xs \<in> list(A); n \<in> nat; M(A)|] ==> M(nth(n,xs))"
paulson@13397
   743
apply (case_tac "n < length(xs)")
paulson@13397
   744
 apply (blast intro: nth_type transM)
paulson@13397
   745
apply (simp add: not_lt_iff_le nth_eq_0)
paulson@13397
   746
done
paulson@13397
   747
wenzelm@21233
   748
definition
wenzelm@21404
   749
  is_nth :: "[i=>o,i,i,i] => o" where
paulson@13655
   750
    "is_nth(M,n,l,Z) ==
paulson@13655
   751
      \<exists>X[M]. is_iterates(M, is_tl(M), l, n, X) & is_hd(M,X,Z)"
paulson@13655
   752
paulson@13409
   753
lemma (in M_datatypes) nth_abs [simp]:
paulson@13655
   754
     "[|M(A); n \<in> nat; l \<in> list(A); M(Z)|]
paulson@46823
   755
      ==> is_nth(M,n,l,Z) \<longleftrightarrow> Z = nth(n,l)"
paulson@13655
   756
apply (subgoal_tac "M(l)")
paulson@13397
   757
 prefer 2 apply (blast intro: transM)
paulson@13397
   758
apply (simp add: is_nth_def nth_eq_hd_iterates_tl nat_into_M
paulson@13655
   759
                 tl'_closed iterates_tl'_closed
paulson@13634
   760
                 iterates_abs [OF _ relation1_tl] nth_replacement)
paulson@13397
   761
done
paulson@13397
   762
paulson@13395
   763
wenzelm@60770
   764
subsection\<open>Relativization and Absoluteness for the @{term formula} Constructors\<close>
paulson@13398
   765
wenzelm@21233
   766
definition
wenzelm@21404
   767
  is_Member :: "[i=>o,i,i,i] => o" where
wenzelm@61798
   768
     \<comment>\<open>because @{term "Member(x,y) \<equiv> Inl(Inl(\<langle>x,y\<rangle>))"}\<close>
paulson@13398
   769
    "is_Member(M,x,y,Z) ==
wenzelm@32960
   770
        \<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)"
paulson@13398
   771
paulson@13564
   772
lemma (in M_trivial) Member_abs [simp]:
paulson@46823
   773
     "[|M(x); M(y); M(Z)|] ==> is_Member(M,x,y,Z) \<longleftrightarrow> (Z = Member(x,y))"
paulson@13398
   774
by (simp add: is_Member_def Member_def)
paulson@13398
   775
paulson@13564
   776
lemma (in M_trivial) Member_in_M_iff [iff]:
paulson@46823
   777
     "M(Member(x,y)) \<longleftrightarrow> M(x) & M(y)"
paulson@13655
   778
by (simp add: Member_def)
paulson@13398
   779
wenzelm@21233
   780
definition
wenzelm@21404
   781
  is_Equal :: "[i=>o,i,i,i] => o" where
wenzelm@61798
   782
     \<comment>\<open>because @{term "Equal(x,y) \<equiv> Inl(Inr(\<langle>x,y\<rangle>))"}\<close>
paulson@13398
   783
    "is_Equal(M,x,y,Z) ==
wenzelm@32960
   784
        \<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)"
paulson@13398
   785
paulson@13564
   786
lemma (in M_trivial) Equal_abs [simp]:
paulson@46823
   787
     "[|M(x); M(y); M(Z)|] ==> is_Equal(M,x,y,Z) \<longleftrightarrow> (Z = Equal(x,y))"
paulson@13398
   788
by (simp add: is_Equal_def Equal_def)
paulson@13398
   789
paulson@46823
   790
lemma (in M_trivial) Equal_in_M_iff [iff]: "M(Equal(x,y)) \<longleftrightarrow> M(x) & M(y)"
paulson@13655
   791
by (simp add: Equal_def)
paulson@13398
   792
wenzelm@21233
   793
definition
wenzelm@21404
   794
  is_Nand :: "[i=>o,i,i,i] => o" where
wenzelm@61798
   795
     \<comment>\<open>because @{term "Nand(x,y) \<equiv> Inr(Inl(\<langle>x,y\<rangle>))"}\<close>
paulson@13398
   796
    "is_Nand(M,x,y,Z) ==
wenzelm@32960
   797
        \<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)"
paulson@13398
   798
paulson@13564
   799
lemma (in M_trivial) Nand_abs [simp]:
paulson@46823
   800
     "[|M(x); M(y); M(Z)|] ==> is_Nand(M,x,y,Z) \<longleftrightarrow> (Z = Nand(x,y))"
paulson@13398
   801
by (simp add: is_Nand_def Nand_def)
paulson@13398
   802
paulson@46823
   803
lemma (in M_trivial) Nand_in_M_iff [iff]: "M(Nand(x,y)) \<longleftrightarrow> M(x) & M(y)"
paulson@13655
   804
by (simp add: Nand_def)
paulson@13398
   805
wenzelm@21233
   806
definition
wenzelm@21404
   807
  is_Forall :: "[i=>o,i,i] => o" where
wenzelm@61798
   808
     \<comment>\<open>because @{term "Forall(x) \<equiv> Inr(Inr(p))"}\<close>
paulson@13398
   809
    "is_Forall(M,p,Z) == \<exists>u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)"
paulson@13398
   810
paulson@13564
   811
lemma (in M_trivial) Forall_abs [simp]:
paulson@46823
   812
     "[|M(x); M(Z)|] ==> is_Forall(M,x,Z) \<longleftrightarrow> (Z = Forall(x))"
paulson@13398
   813
by (simp add: is_Forall_def Forall_def)
paulson@13398
   814
paulson@46823
   815
lemma (in M_trivial) Forall_in_M_iff [iff]: "M(Forall(x)) \<longleftrightarrow> M(x)"
paulson@13398
   816
by (simp add: Forall_def)
paulson@13398
   817
paulson@13398
   818
paulson@13647
   819
wenzelm@60770
   820
subsection \<open>Absoluteness for @{term formula_rec}\<close>
paulson@13398
   821
wenzelm@21233
   822
definition
wenzelm@21404
   823
  formula_rec_case :: "[[i,i]=>i, [i,i]=>i, [i,i,i,i]=>i, [i,i]=>i, i, i] => i" where
wenzelm@61798
   824
    \<comment>\<open>the instance of @{term formula_case} in @{term formula_rec}\<close>
paulson@13647
   825
   "formula_rec_case(a,b,c,d,h) ==
paulson@13647
   826
        formula_case (a, b,
paulson@13655
   827
                \<lambda>u v. c(u, v, h ` succ(depth(u)) ` u,
paulson@13647
   828
                              h ` succ(depth(v)) ` v),
paulson@13647
   829
                \<lambda>u. d(u, h ` succ(depth(u)) ` u))"
paulson@13647
   830
wenzelm@60770
   831
text\<open>Unfold @{term formula_rec} to @{term formula_rec_case}.
paulson@13647
   832
     Express @{term formula_rec} without using @{term rank} or @{term Vset},
wenzelm@60770
   833
neither of which is absolute.\<close>
paulson@13647
   834
lemma (in M_trivial) formula_rec_eq:
paulson@13647
   835
  "p \<in> formula ==>
paulson@13655
   836
   formula_rec(a,b,c,d,p) =
paulson@13647
   837
   transrec (succ(depth(p)),
paulson@13647
   838
             \<lambda>x h. Lambda (formula, formula_rec_case(a,b,c,d,h))) ` p"
paulson@13647
   839
apply (simp add: formula_rec_case_def)
paulson@13647
   840
apply (induct_tac p)
wenzelm@60770
   841
   txt\<open>Base case for @{term Member}\<close>
paulson@13655
   842
   apply (subst transrec, simp add: formula.intros)
wenzelm@60770
   843
  txt\<open>Base case for @{term Equal}\<close>
paulson@13647
   844
  apply (subst transrec, simp add: formula.intros)
wenzelm@60770
   845
 txt\<open>Inductive step for @{term Nand}\<close>
paulson@13655
   846
 apply (subst transrec)
paulson@13647
   847
 apply (simp add: succ_Un_distrib formula.intros)
wenzelm@60770
   848
txt\<open>Inductive step for @{term Forall}\<close>
paulson@13655
   849
apply (subst transrec)
paulson@13655
   850
apply (simp add: formula_imp_formula_N formula.intros)
paulson@13647
   851
done
paulson@13647
   852
paulson@13647
   853
wenzelm@60770
   854
subsubsection\<open>Absoluteness for the Formula Operator @{term depth}\<close>
wenzelm@21404
   855
wenzelm@21233
   856
definition
wenzelm@21404
   857
  is_depth :: "[i=>o,i,i] => o" where
paulson@13655
   858
    "is_depth(M,p,n) ==
paulson@13655
   859
       \<exists>sn[M]. \<exists>formula_n[M]. \<exists>formula_sn[M].
paulson@13647
   860
        is_formula_N(M,n,formula_n) & p \<notin> formula_n &
paulson@13647
   861
        successor(M,n,sn) & is_formula_N(M,sn,formula_sn) & p \<in> formula_sn"
paulson@13647
   862
paulson@13647
   863
paulson@13647
   864
lemma (in M_datatypes) depth_abs [simp]:
paulson@46823
   865
     "[|p \<in> formula; n \<in> nat|] ==> is_depth(M,p,n) \<longleftrightarrow> n = depth(p)"
paulson@13647
   866
apply (subgoal_tac "M(p) & M(n)")
paulson@13655
   867
 prefer 2 apply (blast dest: transM)
paulson@13647
   868
apply (simp add: is_depth_def)
paulson@13647
   869
apply (blast intro: formula_imp_formula_N nat_into_Ord formula_N_imp_eq_depth
paulson@13647
   870
             dest: formula_N_imp_depth_lt)
paulson@13647
   871
done
paulson@13647
   872
wenzelm@60770
   873
text\<open>Proof is trivial since @{term depth} returns natural numbers.\<close>
paulson@13647
   874
lemma (in M_trivial) depth_closed [intro,simp]:
paulson@13647
   875
     "p \<in> formula ==> M(depth(p))"
paulson@13655
   876
by (simp add: nat_into_M)
paulson@13647
   877
paulson@13647
   878
wenzelm@60770
   879
subsubsection\<open>@{term is_formula_case}: relativization of @{term formula_case}\<close>
paulson@13423
   880
wenzelm@21233
   881
definition
paulson@13655
   882
 is_formula_case ::
wenzelm@21404
   883
    "[i=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i]=>o, i, i] => o" where
wenzelm@61798
   884
  \<comment>\<open>no constraint on non-formulas\<close>
paulson@13655
   885
  "is_formula_case(M, is_a, is_b, is_c, is_d, p, z) ==
paulson@46823
   886
      (\<forall>x[M]. \<forall>y[M]. finite_ordinal(M,x) \<longrightarrow> finite_ordinal(M,y) \<longrightarrow>
paulson@46823
   887
                      is_Member(M,x,y,p) \<longrightarrow> is_a(x,y,z)) &
paulson@46823
   888
      (\<forall>x[M]. \<forall>y[M]. finite_ordinal(M,x) \<longrightarrow> finite_ordinal(M,y) \<longrightarrow>
paulson@46823
   889
                      is_Equal(M,x,y,p) \<longrightarrow> is_b(x,y,z)) &
paulson@46823
   890
      (\<forall>x[M]. \<forall>y[M]. mem_formula(M,x) \<longrightarrow> mem_formula(M,y) \<longrightarrow>
paulson@46823
   891
                     is_Nand(M,x,y,p) \<longrightarrow> is_c(x,y,z)) &
paulson@46823
   892
      (\<forall>x[M]. mem_formula(M,x) \<longrightarrow> is_Forall(M,x,p) \<longrightarrow> is_d(x,z))"
paulson@13423
   893
paulson@13655
   894
lemma (in M_datatypes) formula_case_abs [simp]:
paulson@13655
   895
     "[| Relation2(M,nat,nat,is_a,a); Relation2(M,nat,nat,is_b,b);
paulson@13655
   896
         Relation2(M,formula,formula,is_c,c); Relation1(M,formula,is_d,d);
paulson@13655
   897
         p \<in> formula; M(z) |]
paulson@46823
   898
      ==> is_formula_case(M,is_a,is_b,is_c,is_d,p,z) \<longleftrightarrow>
paulson@13423
   899
          z = formula_case(a,b,c,d,p)"
paulson@13423
   900
apply (simp add: formula_into_M is_formula_case_def)
paulson@13655
   901
apply (erule formula.cases)
paulson@13655
   902
   apply (simp_all add: Relation1_def Relation2_def)
paulson@13423
   903
done
paulson@13423
   904
paulson@13418
   905
lemma (in M_datatypes) formula_case_closed [intro,simp]:
paulson@13655
   906
  "[|p \<in> formula;
paulson@46823
   907
     \<forall>x[M]. \<forall>y[M]. x\<in>nat \<longrightarrow> y\<in>nat \<longrightarrow> M(a(x,y));
paulson@46823
   908
     \<forall>x[M]. \<forall>y[M]. x\<in>nat \<longrightarrow> y\<in>nat \<longrightarrow> M(b(x,y));
paulson@46823
   909
     \<forall>x[M]. \<forall>y[M]. x\<in>formula \<longrightarrow> y\<in>formula \<longrightarrow> M(c(x,y));
paulson@46823
   910
     \<forall>x[M]. x\<in>formula \<longrightarrow> M(d(x))|] ==> M(formula_case(a,b,c,d,p))"
paulson@13655
   911
by (erule formula.cases, simp_all)
paulson@13418
   912
paulson@13398
   913
wenzelm@60770
   914
subsubsection \<open>Absoluteness for @{term formula_rec}: Final Results\<close>
paulson@13557
   915
wenzelm@21233
   916
definition
wenzelm@21404
   917
  is_formula_rec :: "[i=>o, [i,i,i]=>o, i, i] => o" where
wenzelm@61798
   918
    \<comment>\<open>predicate to relativize the functional @{term formula_rec}\<close>
paulson@13557
   919
   "is_formula_rec(M,MH,p,z)  ==
paulson@13655
   920
      \<exists>dp[M]. \<exists>i[M]. \<exists>f[M]. finite_ordinal(M,dp) & is_depth(M,p,dp) &
paulson@13557
   921
             successor(M,dp,i) & fun_apply(M,f,p,z) & is_transrec(M,MH,i,f)"
paulson@13557
   922
paulson@13557
   923
wenzelm@60770
   924
text\<open>Sufficient conditions to relativize the instance of @{term formula_case}
wenzelm@60770
   925
      in @{term formula_rec}\<close>
paulson@13634
   926
lemma (in M_datatypes) Relation1_formula_rec_case:
paulson@13634
   927
     "[|Relation2(M, nat, nat, is_a, a);
paulson@13634
   928
        Relation2(M, nat, nat, is_b, b);
paulson@13655
   929
        Relation2 (M, formula, formula,
paulson@13557
   930
           is_c, \<lambda>u v. c(u, v, h`succ(depth(u))`u, h`succ(depth(v))`v));
paulson@13655
   931
        Relation1(M, formula,
paulson@13557
   932
           is_d, \<lambda>u. d(u, h ` succ(depth(u)) ` u));
wenzelm@32960
   933
        M(h) |]
paulson@13634
   934
      ==> Relation1(M, formula,
paulson@13557
   935
                         is_formula_case (M, is_a, is_b, is_c, is_d),
paulson@13557
   936
                         formula_rec_case(a, b, c, d, h))"
paulson@13655
   937
apply (simp (no_asm) add: formula_rec_case_def Relation1_def)
paulson@13655
   938
apply (simp add: formula_case_abs)
paulson@13557
   939
done
paulson@13557
   940
paulson@13557
   941
wenzelm@60770
   942
text\<open>This locale packages the premises of the following theorems,
paulson@13557
   943
      which is the normal purpose of locales.  It doesn't accumulate
wenzelm@60770
   944
      constraints on the class @{term M}, as in most of this deveopment.\<close>
paulson@13557
   945
locale Formula_Rec = M_eclose +
paulson@13557
   946
  fixes a and is_a and b and is_b and c and is_c and d and is_d and MH
paulson@13557
   947
  defines
paulson@13557
   948
      "MH(u::i,f,z) ==
paulson@46823
   949
        \<forall>fml[M]. is_formula(M,fml) \<longrightarrow>
paulson@13557
   950
             is_lambda
wenzelm@32960
   951
         (M, fml, is_formula_case (M, is_a, is_b, is_c(f), is_d(f)), z)"
paulson@13557
   952
paulson@13557
   953
  assumes a_closed: "[|x\<in>nat; y\<in>nat|] ==> M(a(x,y))"
paulson@13634
   954
      and a_rel:    "Relation2(M, nat, nat, is_a, a)"
paulson@13557
   955
      and b_closed: "[|x\<in>nat; y\<in>nat|] ==> M(b(x,y))"
paulson@13634
   956
      and b_rel:    "Relation2(M, nat, nat, is_b, b)"
paulson@13557
   957
      and c_closed: "[|x \<in> formula; y \<in> formula; M(gx); M(gy)|]
paulson@13557
   958
                     ==> M(c(x, y, gx, gy))"
paulson@13557
   959
      and c_rel:
paulson@13655
   960
         "M(f) ==>
paulson@13634
   961
          Relation2 (M, formula, formula, is_c(f),
paulson@13557
   962
             \<lambda>u v. c(u, v, f ` succ(depth(u)) ` u, f ` succ(depth(v)) ` v))"
paulson@13557
   963
      and d_closed: "[|x \<in> formula; M(gx)|] ==> M(d(x, gx))"
paulson@13557
   964
      and d_rel:
paulson@13655
   965
         "M(f) ==>
paulson@13634
   966
          Relation1(M, formula, is_d(f), \<lambda>u. d(u, f ` succ(depth(u)) ` u))"
paulson@13557
   967
      and fr_replace: "n \<in> nat ==> transrec_replacement(M,MH,n)"
paulson@13557
   968
      and fr_lam_replace:
paulson@13557
   969
           "M(g) ==>
paulson@13557
   970
            strong_replacement
wenzelm@32960
   971
              (M, \<lambda>x y. x \<in> formula &
wenzelm@58860
   972
                  y = \<langle>x, formula_rec_case(a,b,c,d,g,x)\<rangle>)"
paulson@13557
   973
paulson@13557
   974
lemma (in Formula_Rec) formula_rec_case_closed:
paulson@13557
   975
    "[|M(g); p \<in> formula|] ==> M(formula_rec_case(a, b, c, d, g, p))"
paulson@13655
   976
by (simp add: formula_rec_case_def a_closed b_closed c_closed d_closed)
paulson@13557
   977
paulson@13557
   978
lemma (in Formula_Rec) formula_rec_lam_closed:
paulson@13557
   979
    "M(g) ==> M(Lambda (formula, formula_rec_case(a,b,c,d,g)))"
paulson@13557
   980
by (simp add: lam_closed2 fr_lam_replace formula_rec_case_closed)
paulson@13557
   981
paulson@13557
   982
lemma (in Formula_Rec) MH_rel2:
paulson@13634
   983
     "relation2 (M, MH,
paulson@13557
   984
             \<lambda>x h. Lambda (formula, formula_rec_case(a,b,c,d,h)))"
paulson@13655
   985
apply (simp add: relation2_def MH_def, clarify)
paulson@13655
   986
apply (rule lambda_abs2)
paulson@13655
   987
apply (rule Relation1_formula_rec_case)
paulson@13655
   988
apply (simp_all add: a_rel b_rel c_rel d_rel formula_rec_case_closed)
paulson@13557
   989
done
paulson@13557
   990
paulson@13557
   991
lemma (in Formula_Rec) fr_transrec_closed:
paulson@13557
   992
    "n \<in> nat
paulson@13557
   993
     ==> M(transrec
paulson@13557
   994
          (n, \<lambda>x h. Lambda(formula, formula_rec_case(a, b, c, d, h))))"
paulson@13655
   995
by (simp add: transrec_closed [OF fr_replace MH_rel2]
paulson@13655
   996
              nat_into_M formula_rec_lam_closed)
paulson@13557
   997
wenzelm@60770
   998
text\<open>The main two results: @{term formula_rec} is absolute for @{term M}.\<close>
paulson@13557
   999
theorem (in Formula_Rec) formula_rec_closed:
paulson@13557
  1000
    "p \<in> formula ==> M(formula_rec(a,b,c,d,p))"
paulson@13655
  1001
by (simp add: formula_rec_eq fr_transrec_closed
paulson@13557
  1002
              transM [OF _ formula_closed])
paulson@13557
  1003
paulson@13557
  1004
theorem (in Formula_Rec) formula_rec_abs:
paulson@13655
  1005
  "[| p \<in> formula; M(z)|]
paulson@46823
  1006
   ==> is_formula_rec(M,MH,p,z) \<longleftrightarrow> z = formula_rec(a,b,c,d,p)"
paulson@13557
  1007
by (simp add: is_formula_rec_def formula_rec_eq transM [OF _ formula_closed]
paulson@13557
  1008
              transrec_abs [OF fr_replace MH_rel2] depth_type
paulson@13557
  1009
              fr_transrec_closed formula_rec_lam_closed eq_commute)
paulson@13557
  1010
paulson@13557
  1011
paulson@13268
  1012
end