author | wenzelm |
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permissions | -rw-r--r-- |
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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: |
207 |
"list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))" |
|
208 |
by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin) |
|
209 |
||
210 |
||
21233 | 211 |
definition |
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212 |
is_list_functor :: "[i=>o,i,i,i] => o" where |
13350 | 213 |
"is_list_functor(M,A,X,Z) == |
214 |
\<exists>n1[M]. \<exists>AX[M]. |
|
215 |
number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" |
|
216 |
||
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217 |
lemma (in M_basic) list_functor_abs [simp]: |
46823 | 218 |
"[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) \<longleftrightarrow> (Z = {0} + A*X)" |
13350 | 219 |
by (simp add: is_list_functor_def singleton_0 nat_into_M) |
220 |
||
221 |
||
60770 | 222 |
subsection \<open>formulas without univ\<close> |
13386 | 223 |
|
224 |
lemma formula_fun_bnd_mono: |
|
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225 |
"bnd_mono(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))" |
13386 | 226 |
apply (rule bnd_monoI) |
227 |
apply (intro subset_refl zero_subset_univ A_subset_univ |
|
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sum_subset_univ Sigma_subset_univ nat_subset_univ) |
13386 | 229 |
apply (rule subset_refl sum_mono Sigma_mono | assumption)+ |
230 |
done |
|
231 |
||
232 |
lemma formula_fun_contin: |
|
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"contin(\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))" |
13386 | 234 |
by (intro sum_contin prod_contin id_contin const_contin) |
235 |
||
236 |
||
60770 | 237 |
text\<open>Re-expresses formulas using sum and product\<close> |
13386 | 238 |
lemma formula_eq_lfp2: |
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239 |
"formula = lfp(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))" |
13386 | 240 |
apply (simp add: formula_def) |
241 |
apply (rule equalityI) |
|
242 |
apply (rule lfp_lowerbound) |
|
243 |
prefer 2 apply (rule lfp_subset) |
|
244 |
apply (clarify, subst lfp_unfold [OF formula_fun_bnd_mono]) |
|
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245 |
apply (simp add: Member_def Equal_def Nand_def Forall_def) |
13386 | 246 |
apply blast |
60770 | 247 |
txt\<open>Opposite inclusion\<close> |
13386 | 248 |
apply (rule lfp_lowerbound) |
249 |
prefer 2 apply (rule lfp_subset, clarify) |
|
250 |
apply (subst lfp_unfold [OF formula.bnd_mono, simplified]) |
|
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|
251 |
apply (simp add: Member_def Equal_def Nand_def Forall_def) |
13386 | 252 |
apply (elim sumE SigmaE, simp_all) |
253 |
apply (blast intro: datatype_univs dest: lfp_subset [THEN subsetD])+ |
|
254 |
done |
|
255 |
||
60770 | 256 |
text\<open>Re-expresses formulas using "iterates", no univ.\<close> |
13386 | 257 |
lemma formula_eq_Union: |
258 |
"formula = |
|
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|
259 |
(\<Union>n\<in>nat. (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)) ^ n (0))" |
13386 | 260 |
by (simp add: formula_eq_lfp2 lfp_eq_Union formula_fun_bnd_mono |
261 |
formula_fun_contin) |
|
262 |
||
263 |
||
21233 | 264 |
definition |
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|
265 |
is_formula_functor :: "[i=>o,i,i] => o" where |
13386 | 266 |
"is_formula_functor(M,X,Z) == |
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|
267 |
\<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M]. |
13386 | 268 |
omega(M,nat') & cartprod(M,nat',nat',natnat) & |
269 |
is_sum(M,natnat,natnat,natnatsum) & |
|
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|
270 |
cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & |
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|
271 |
is_sum(M,natnatsum,X3,Z)" |
13386 | 272 |
|
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|
273 |
lemma (in M_basic) formula_functor_abs [simp]: |
13386 | 274 |
"[| M(X); M(Z) |] |
46823 | 275 |
==> is_formula_functor(M,X,Z) \<longleftrightarrow> |
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|
276 |
Z = ((nat*nat) + (nat*nat)) + (X*X + X)" |
13386 | 277 |
by (simp add: is_formula_functor_def) |
278 |
||
279 |
||
60770 | 280 |
subsection\<open>@{term M} Contains the List and Formula Datatypes\<close> |
13395 | 281 |
|
21233 | 282 |
definition |
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|
283 |
list_N :: "[i,i] => i" where |
13397 | 284 |
"list_N(A,n) == (\<lambda>X. {0} + A * X)^n (0)" |
285 |
||
286 |
lemma Nil_in_list_N [simp]: "[] \<in> list_N(A,succ(n))" |
|
287 |
by (simp add: list_N_def Nil_def) |
|
288 |
||
289 |
lemma Cons_in_list_N [simp]: |
|
46823 | 290 |
"Cons(a,l) \<in> list_N(A,succ(n)) \<longleftrightarrow> a\<in>A & l \<in> list_N(A,n)" |
13397 | 291 |
by (simp add: list_N_def Cons_def) |
292 |
||
60770 | 293 |
text\<open>These two aren't simprules because they reveal the underlying |
294 |
list representation.\<close> |
|
13397 | 295 |
lemma list_N_0: "list_N(A,0) = 0" |
296 |
by (simp add: list_N_def) |
|
297 |
||
298 |
lemma list_N_succ: "list_N(A,succ(n)) = {0} + A * (list_N(A,n))" |
|
299 |
by (simp add: list_N_def) |
|
300 |
||
301 |
lemma list_N_imp_list: |
|
302 |
"[| l \<in> list_N(A,n); n \<in> nat |] ==> l \<in> list(A)" |
|
303 |
by (force simp add: list_eq_Union list_N_def) |
|
304 |
||
305 |
lemma list_N_imp_length_lt [rule_format]: |
|
306 |
"n \<in> nat ==> \<forall>l \<in> list_N(A,n). length(l) < n" |
|
307 |
apply (induct_tac n) |
|
308 |
apply (auto simp add: list_N_0 list_N_succ |
|
309 |
Nil_def [symmetric] Cons_def [symmetric]) |
|
310 |
done |
|
311 |
||
312 |
lemma list_imp_list_N [rule_format]: |
|
46823 | 313 |
"l \<in> list(A) ==> \<forall>n\<in>nat. length(l) < n \<longrightarrow> l \<in> list_N(A, n)" |
13397 | 314 |
apply (induct_tac l) |
315 |
apply (force elim: natE)+ |
|
316 |
done |
|
317 |
||
318 |
lemma list_N_imp_eq_length: |
|
319 |
"[|n \<in> nat; l \<notin> list_N(A, n); l \<in> list_N(A, succ(n))|] |
|
320 |
==> n = length(l)" |
|
321 |
apply (rule le_anti_sym) |
|
322 |
prefer 2 apply (simp add: list_N_imp_length_lt) |
|
323 |
apply (frule list_N_imp_list, simp) |
|
324 |
apply (simp add: not_lt_iff_le [symmetric]) |
|
325 |
apply (blast intro: list_imp_list_N) |
|
326 |
done |
|
327 |
||
60770 | 328 |
text\<open>Express @{term list_rec} without using @{term rank} or @{term Vset}, |
329 |
neither of which is absolute.\<close> |
|
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|
330 |
lemma (in M_trivial) list_rec_eq: |
13397 | 331 |
"l \<in> list(A) ==> |
332 |
list_rec(a,g,l) = |
|
333 |
transrec (succ(length(l)), |
|
13409
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Relativization and Separation for the function "nth"
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13398
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|
334 |
\<lambda>x h. Lambda (list(A), |
d4ea094c650e
Relativization and Separation for the function "nth"
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|
335 |
list_case' (a, |
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Relativization and Separation for the function "nth"
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|
336 |
\<lambda>a l. g(a, l, h ` succ(length(l)) ` l)))) ` l" |
13397 | 337 |
apply (induct_tac l) |
338 |
apply (subst transrec, simp) |
|
339 |
apply (subst transrec) |
|
340 |
apply (simp add: list_imp_list_N) |
|
341 |
done |
|
342 |
||
21233 | 343 |
definition |
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|
344 |
is_list_N :: "[i=>o,i,i,i] => o" where |
13397 | 345 |
"is_list_N(M,A,n,Z) == |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
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13647
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|
346 |
\<exists>zero[M]. empty(M,zero) & |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
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13647
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|
347 |
is_iterates(M, is_list_functor(M,A), zero, n, Z)" |
21404
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|
348 |
|
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|
349 |
definition |
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|
350 |
mem_list :: "[i=>o,i,i] => o" where |
13395 | 351 |
"mem_list(M,A,l) == |
352 |
\<exists>n[M]. \<exists>listn[M]. |
|
13397 | 353 |
finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l \<in> listn" |
13395 | 354 |
|
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|
355 |
definition |
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wenzelm
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21233
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changeset
|
356 |
is_list :: "[i=>o,i,i] => o" where |
46823 | 357 |
"is_list(M,A,Z) == \<forall>l[M]. l \<in> Z \<longleftrightarrow> mem_list(M,A,l)" |
13395 | 358 |
|
60770 | 359 |
subsubsection\<open>Towards Absoluteness of @{term formula_rec}\<close> |
13493
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|
360 |
|
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|
361 |
consts depth :: "i=>i" |
5aa68c051725
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|
362 |
primrec |
5aa68c051725
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|
363 |
"depth(Member(x,y)) = 0" |
5aa68c051725
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changeset
|
364 |
"depth(Equal(x,y)) = 0" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
365 |
"depth(Nand(p,q)) = succ(depth(p) \<union> depth(q))" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
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changeset
|
366 |
"depth(Forall(p)) = succ(depth(p))" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
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changeset
|
367 |
|
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Lots of new results concerning recursive datatypes, towards absoluteness of
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parents:
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diff
changeset
|
368 |
lemma depth_type [TC]: "p \<in> formula ==> depth(p) \<in> nat" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
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changeset
|
369 |
by (induct_tac p, simp_all) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
370 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
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diff
changeset
|
371 |
|
21233 | 372 |
definition |
21404
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wenzelm
parents:
21233
diff
changeset
|
373 |
formula_N :: "i => i" where |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
374 |
"formula_N(n) == (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)) ^ n (0)" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
375 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
376 |
lemma Member_in_formula_N [simp]: |
46823 | 377 |
"Member(x,y) \<in> formula_N(succ(n)) \<longleftrightarrow> x \<in> nat & y \<in> nat" |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
378 |
by (simp add: formula_N_def Member_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
379 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
380 |
lemma Equal_in_formula_N [simp]: |
46823 | 381 |
"Equal(x,y) \<in> formula_N(succ(n)) \<longleftrightarrow> x \<in> nat & y \<in> nat" |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
382 |
by (simp add: formula_N_def Equal_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
383 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
384 |
lemma Nand_in_formula_N [simp]: |
46823 | 385 |
"Nand(x,y) \<in> formula_N(succ(n)) \<longleftrightarrow> x \<in> formula_N(n) & y \<in> formula_N(n)" |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
386 |
by (simp add: formula_N_def Nand_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
387 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
388 |
lemma Forall_in_formula_N [simp]: |
46823 | 389 |
"Forall(x) \<in> formula_N(succ(n)) \<longleftrightarrow> x \<in> formula_N(n)" |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
390 |
by (simp add: formula_N_def Forall_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
391 |
|
60770 | 392 |
text\<open>These two aren't simprules because they reveal the underlying |
393 |
formula representation.\<close> |
|
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
394 |
lemma formula_N_0: "formula_N(0) = 0" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
395 |
by (simp add: formula_N_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
396 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
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diff
changeset
|
397 |
lemma formula_N_succ: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
398 |
"formula_N(succ(n)) = |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
399 |
((nat*nat) + (nat*nat)) + (formula_N(n) * formula_N(n) + formula_N(n))" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
400 |
by (simp add: formula_N_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
401 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
402 |
lemma formula_N_imp_formula: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
403 |
"[| p \<in> formula_N(n); n \<in> nat |] ==> p \<in> formula" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
404 |
by (force simp add: formula_eq_Union formula_N_def) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
405 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
406 |
lemma formula_N_imp_depth_lt [rule_format]: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
407 |
"n \<in> nat ==> \<forall>p \<in> formula_N(n). depth(p) < n" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
408 |
apply (induct_tac n) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
409 |
apply (auto simp add: formula_N_0 formula_N_succ |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
410 |
depth_type formula_N_imp_formula Un_least_lt_iff |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
411 |
Member_def [symmetric] Equal_def [symmetric] |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
412 |
Nand_def [symmetric] Forall_def [symmetric]) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
413 |
done |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
414 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
415 |
lemma formula_imp_formula_N [rule_format]: |
46823 | 416 |
"p \<in> formula ==> \<forall>n\<in>nat. depth(p) < n \<longrightarrow> p \<in> formula_N(n)" |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
417 |
apply (induct_tac p) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
418 |
apply (simp_all add: succ_Un_distrib Un_least_lt_iff) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
419 |
apply (force elim: natE)+ |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
420 |
done |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
421 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
422 |
lemma formula_N_imp_eq_depth: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
423 |
"[|n \<in> nat; p \<notin> formula_N(n); p \<in> formula_N(succ(n))|] |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
424 |
==> n = depth(p)" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
425 |
apply (rule le_anti_sym) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
426 |
prefer 2 apply (simp add: formula_N_imp_depth_lt) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
427 |
apply (frule formula_N_imp_formula, simp) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
428 |
apply (simp add: not_lt_iff_le [symmetric]) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
429 |
apply (blast intro: formula_imp_formula_N) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
430 |
done |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
431 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
432 |
|
60770 | 433 |
text\<open>This result and the next are unused.\<close> |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
434 |
lemma formula_N_mono [rule_format]: |
46823 | 435 |
"[| m \<in> nat; n \<in> nat |] ==> m\<le>n \<longrightarrow> formula_N(m) \<subseteq> formula_N(n)" |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
436 |
apply (rule_tac m = m and n = n in diff_induct) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
437 |
apply (simp_all add: formula_N_0 formula_N_succ, blast) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
438 |
done |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
439 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
440 |
lemma formula_N_distrib: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
441 |
"[| m \<in> nat; n \<in> nat |] ==> formula_N(m \<union> n) = formula_N(m) \<union> formula_N(n)" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
442 |
apply (rule_tac i = m and j = n in Ord_linear_le, auto) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
443 |
apply (simp_all add: subset_Un_iff [THEN iffD1] subset_Un_iff2 [THEN iffD1] |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
444 |
le_imp_subset formula_N_mono) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
445 |
done |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
446 |
|
21233 | 447 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
448 |
is_formula_N :: "[i=>o,i,i] => o" where |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
449 |
"is_formula_N(M,n,Z) == |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
450 |
\<exists>zero[M]. empty(M,zero) & |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
451 |
is_iterates(M, is_formula_functor(M), zero, n, Z)" |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
452 |
|
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
453 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
454 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
455 |
mem_formula :: "[i=>o,i] => o" where |
13395 | 456 |
"mem_formula(M,p) == |
457 |
\<exists>n[M]. \<exists>formn[M]. |
|
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
458 |
finite_ordinal(M,n) & is_formula_N(M,n,formn) & p \<in> formn" |
13395 | 459 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
460 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
461 |
is_formula :: "[i=>o,i] => o" where |
46823 | 462 |
"is_formula(M,Z) == \<forall>p[M]. p \<in> Z \<longleftrightarrow> mem_formula(M,p)" |
13395 | 463 |
|
13634 | 464 |
locale M_datatypes = M_trancl + |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
465 |
assumes list_replacement1: |
13363 | 466 |
"M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
467 |
and list_replacement2: |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
468 |
"M(A) ==> strong_replacement(M, |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
469 |
\<lambda>n y. n\<in>nat & is_iterates(M, is_list_functor(M,A), 0, n, y))" |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
470 |
and formula_replacement1: |
13386 | 471 |
"iterates_replacement(M, is_formula_functor(M), 0)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
472 |
and formula_replacement2: |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
473 |
"strong_replacement(M, |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
474 |
\<lambda>n y. n\<in>nat & is_iterates(M, is_formula_functor(M), 0, n, y))" |
13422
af9bc8d87a75
Added the assumption nth_replacement to locale M_datatypes.
paulson
parents:
13418
diff
changeset
|
475 |
and nth_replacement: |
af9bc8d87a75
Added the assumption nth_replacement to locale M_datatypes.
paulson
parents:
13418
diff
changeset
|
476 |
"M(l) ==> iterates_replacement(M, %l t. is_tl(M,l,t), l)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
477 |
|
13395 | 478 |
|
60770 | 479 |
subsubsection\<open>Absoluteness of the List Construction\<close> |
13395 | 480 |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
481 |
lemma (in M_datatypes) list_replacement2': |
13353 | 482 |
"M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
483 |
apply (insert list_replacement2 [of A]) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
484 |
apply (rule strong_replacement_cong [THEN iffD1]) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
485 |
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]]) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
486 |
apply (simp_all add: list_replacement1 relation1_def) |
13353 | 487 |
done |
13268 | 488 |
|
489 |
lemma (in M_datatypes) list_closed [intro,simp]: |
|
490 |
"M(A) ==> M(list(A))" |
|
13353 | 491 |
apply (insert list_replacement1) |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
492 |
by (simp add: RepFun_closed2 list_eq_Union |
13634 | 493 |
list_replacement2' relation1_def |
13353 | 494 |
iterates_closed [of "is_list_functor(M,A)"]) |
13397 | 495 |
|
61798 | 496 |
text\<open>WARNING: use only with \<open>dest:\<close> or with variables fixed!\<close> |
13423
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
497 |
lemmas (in M_datatypes) list_into_M = transM [OF _ list_closed] |
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
498 |
|
13397 | 499 |
lemma (in M_datatypes) list_N_abs [simp]: |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
500 |
"[|M(A); n\<in>nat; M(Z)|] |
46823 | 501 |
==> is_list_N(M,A,n,Z) \<longleftrightarrow> Z = list_N(A,n)" |
13395 | 502 |
apply (insert list_replacement1) |
13634 | 503 |
apply (simp add: is_list_N_def list_N_def relation1_def nat_into_M |
13395 | 504 |
iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"]) |
505 |
done |
|
13268 | 506 |
|
13397 | 507 |
lemma (in M_datatypes) list_N_closed [intro,simp]: |
508 |
"[|M(A); n\<in>nat|] ==> M(list_N(A,n))" |
|
509 |
apply (insert list_replacement1) |
|
13634 | 510 |
apply (simp add: is_list_N_def list_N_def relation1_def nat_into_M |
13397 | 511 |
iterates_closed [of "is_list_functor(M,A)"]) |
512 |
done |
|
513 |
||
13395 | 514 |
lemma (in M_datatypes) mem_list_abs [simp]: |
46823 | 515 |
"M(A) ==> mem_list(M,A,l) \<longleftrightarrow> l \<in> list(A)" |
13395 | 516 |
apply (insert list_replacement1) |
13634 | 517 |
apply (simp add: mem_list_def list_N_def relation1_def list_eq_Union |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
518 |
iterates_closed [of "is_list_functor(M,A)"]) |
13395 | 519 |
done |
520 |
||
521 |
lemma (in M_datatypes) list_abs [simp]: |
|
46823 | 522 |
"[|M(A); M(Z)|] ==> is_list(M,A,Z) \<longleftrightarrow> Z = list(A)" |
13395 | 523 |
apply (simp add: is_list_def, safe) |
524 |
apply (rule M_equalityI, simp_all) |
|
525 |
done |
|
526 |
||
60770 | 527 |
subsubsection\<open>Absoluteness of Formulas\<close> |
13293 | 528 |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
529 |
lemma (in M_datatypes) formula_replacement2': |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
530 |
"strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))^n (0))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
531 |
apply (insert formula_replacement2) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
532 |
apply (rule strong_replacement_cong [THEN iffD1]) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
533 |
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]]) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
534 |
apply (simp_all add: formula_replacement1 relation1_def) |
13386 | 535 |
done |
536 |
||
537 |
lemma (in M_datatypes) formula_closed [intro,simp]: |
|
538 |
"M(formula)" |
|
539 |
apply (insert formula_replacement1) |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
540 |
apply (simp add: RepFun_closed2 formula_eq_Union |
13634 | 541 |
formula_replacement2' relation1_def |
13386 | 542 |
iterates_closed [of "is_formula_functor(M)"]) |
543 |
done |
|
544 |
||
13423
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
545 |
lemmas (in M_datatypes) formula_into_M = transM [OF _ formula_closed] |
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
546 |
|
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
547 |
lemma (in M_datatypes) formula_N_abs [simp]: |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
548 |
"[|n\<in>nat; M(Z)|] |
46823 | 549 |
==> is_formula_N(M,n,Z) \<longleftrightarrow> Z = formula_N(n)" |
13395 | 550 |
apply (insert formula_replacement1) |
13634 | 551 |
apply (simp add: is_formula_N_def formula_N_def relation1_def nat_into_M |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
552 |
iterates_abs [of "is_formula_functor(M)" _ |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
553 |
"\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)"]) |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
554 |
done |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
555 |
|
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
556 |
lemma (in M_datatypes) formula_N_closed [intro,simp]: |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
557 |
"n\<in>nat ==> M(formula_N(n))" |
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
558 |
apply (insert formula_replacement1) |
13634 | 559 |
apply (simp add: is_formula_N_def formula_N_def relation1_def nat_into_M |
13493
5aa68c051725
Lots of new results concerning recursive datatypes, towards absoluteness of
paulson
parents:
13440
diff
changeset
|
560 |
iterates_closed [of "is_formula_functor(M)"]) |
13395 | 561 |
done |
562 |
||
563 |
lemma (in M_datatypes) mem_formula_abs [simp]: |
|
46823 | 564 |
"mem_formula(M,l) \<longleftrightarrow> l \<in> formula" |
13395 | 565 |
apply (insert formula_replacement1) |
13634 | 566 |
apply (simp add: mem_formula_def relation1_def formula_eq_Union formula_N_def |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
567 |
iterates_closed [of "is_formula_functor(M)"]) |
13395 | 568 |
done |
569 |
||
570 |
lemma (in M_datatypes) formula_abs [simp]: |
|
46823 | 571 |
"[|M(Z)|] ==> is_formula(M,Z) \<longleftrightarrow> Z = formula" |
13395 | 572 |
apply (simp add: is_formula_def, safe) |
573 |
apply (rule M_equalityI, simp_all) |
|
574 |
done |
|
575 |
||
576 |
||
61798 | 577 |
subsection\<open>Absoluteness for \<open>\<epsilon>\<close>-Closure: the @{term eclose} Operator\<close> |
13395 | 578 |
|
60770 | 579 |
text\<open>Re-expresses eclose using "iterates"\<close> |
13395 | 580 |
lemma eclose_eq_Union: |
581 |
"eclose(A) = (\<Union>n\<in>nat. Union^n (A))" |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
582 |
apply (simp add: eclose_def) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
583 |
apply (rule UN_cong) |
13395 | 584 |
apply (rule refl) |
585 |
apply (induct_tac n) |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
586 |
apply (simp add: nat_rec_0) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
587 |
apply (simp add: nat_rec_succ) |
13395 | 588 |
done |
589 |
||
21233 | 590 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
591 |
is_eclose_n :: "[i=>o,i,i,i] => o" where |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
592 |
"is_eclose_n(M,A,n,Z) == is_iterates(M, big_union(M), A, n, Z)" |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
593 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
594 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
595 |
mem_eclose :: "[i=>o,i,i] => o" where |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
596 |
"mem_eclose(M,A,l) == |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
597 |
\<exists>n[M]. \<exists>eclosen[M]. |
13395 | 598 |
finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen" |
599 |
||
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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diff
changeset
|
600 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
601 |
is_eclose :: "[i=>o,i,i] => o" where |
46823 | 602 |
"is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z \<longleftrightarrow> mem_eclose(M,A,u)" |
13395 | 603 |
|
604 |
||
13428 | 605 |
locale M_eclose = M_datatypes + |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
606 |
assumes eclose_replacement1: |
13395 | 607 |
"M(A) ==> iterates_replacement(M, big_union(M), A)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
608 |
and eclose_replacement2: |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
609 |
"M(A) ==> strong_replacement(M, |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
610 |
\<lambda>n y. n\<in>nat & is_iterates(M, big_union(M), A, n, y))" |
13395 | 611 |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
612 |
lemma (in M_eclose) eclose_replacement2': |
13395 | 613 |
"M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
614 |
apply (insert eclose_replacement2 [of A]) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
615 |
apply (rule strong_replacement_cong [THEN iffD1]) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
616 |
apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]]) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
617 |
apply (simp_all add: eclose_replacement1 relation1_def) |
13395 | 618 |
done |
619 |
||
620 |
lemma (in M_eclose) eclose_closed [intro,simp]: |
|
621 |
"M(A) ==> M(eclose(A))" |
|
622 |
apply (insert eclose_replacement1) |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
623 |
by (simp add: RepFun_closed2 eclose_eq_Union |
13634 | 624 |
eclose_replacement2' relation1_def |
13395 | 625 |
iterates_closed [of "big_union(M)"]) |
626 |
||
627 |
lemma (in M_eclose) is_eclose_n_abs [simp]: |
|
46823 | 628 |
"[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) \<longleftrightarrow> Z = Union^n (A)" |
13395 | 629 |
apply (insert eclose_replacement1) |
13634 | 630 |
apply (simp add: is_eclose_n_def relation1_def nat_into_M |
13395 | 631 |
iterates_abs [of "big_union(M)" _ "Union"]) |
632 |
done |
|
633 |
||
634 |
lemma (in M_eclose) mem_eclose_abs [simp]: |
|
46823 | 635 |
"M(A) ==> mem_eclose(M,A,l) \<longleftrightarrow> l \<in> eclose(A)" |
13395 | 636 |
apply (insert eclose_replacement1) |
13634 | 637 |
apply (simp add: mem_eclose_def relation1_def eclose_eq_Union |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
638 |
iterates_closed [of "big_union(M)"]) |
13395 | 639 |
done |
640 |
||
641 |
lemma (in M_eclose) eclose_abs [simp]: |
|
46823 | 642 |
"[|M(A); M(Z)|] ==> is_eclose(M,A,Z) \<longleftrightarrow> Z = eclose(A)" |
13395 | 643 |
apply (simp add: is_eclose_def, safe) |
644 |
apply (rule M_equalityI, simp_all) |
|
645 |
done |
|
646 |
||
647 |
||
60770 | 648 |
subsection \<open>Absoluteness for @{term transrec}\<close> |
13395 | 649 |
|
60770 | 650 |
text\<open>@{prop "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"}\<close> |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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diff
changeset
|
651 |
|
21233 | 652 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
653 |
is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o" where |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
654 |
"is_transrec(M,MH,a,z) == |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
655 |
\<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. |
13395 | 656 |
upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) & |
657 |
is_wfrec(M,MH,mesa,a,z)" |
|
658 |
||
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
659 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
660 |
transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o" where |
13395 | 661 |
"transrec_replacement(M,MH,a) == |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
662 |
\<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. |
13395 | 663 |
upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) & |
664 |
wfrec_replacement(M,MH,mesa)" |
|
665 |
||
60770 | 666 |
text\<open>The condition @{term "Ord(i)"} lets us use the simpler |
61798 | 667 |
\<open>trans_wfrec_abs\<close> rather than \<open>trans_wfrec_abs\<close>, |
60770 | 668 |
which I haven't even proved yet.\<close> |
13395 | 669 |
theorem (in M_eclose) transrec_abs: |
13634 | 670 |
"[|transrec_replacement(M,MH,i); relation2(M,MH,H); |
13418
7c0ba9dba978
tweaks, aiming towards relativization of "satisfies"
paulson
parents:
13409
diff
changeset
|
671 |
Ord(i); M(i); M(z); |
46823 | 672 |
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|] |
673 |
==> is_transrec(M,MH,i,z) \<longleftrightarrow> z = transrec(i,H)" |
|
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13564
diff
changeset
|
674 |
by (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def |
13395 | 675 |
transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel) |
676 |
||
677 |
||
678 |
theorem (in M_eclose) transrec_closed: |
|
13634 | 679 |
"[|transrec_replacement(M,MH,i); relation2(M,MH,H); |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
22710
diff
changeset
|
680 |
Ord(i); M(i); |
46823 | 681 |
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|] |
13395 | 682 |
==> M(transrec(i,H))" |
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13564
diff
changeset
|
683 |
by (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def |
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13564
diff
changeset
|
684 |
transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel) |
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13564
diff
changeset
|
685 |
|
13395 | 686 |
|
60770 | 687 |
text\<open>Helps to prove instances of @{term transrec_replacement}\<close> |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
688 |
lemma (in M_eclose) transrec_replacementI: |
13440 | 689 |
"[|M(a); |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
690 |
strong_replacement (M, |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
691 |
\<lambda>x z. \<exists>y[M]. pair(M, x, y, z) & |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
692 |
is_wfrec(M,MH,Memrel(eclose({a})),x,y))|] |
13440 | 693 |
==> transrec_replacement(M,MH,a)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
694 |
by (simp add: transrec_replacement_def wfrec_replacement_def) |
13440 | 695 |
|
13395 | 696 |
|
60770 | 697 |
subsection\<open>Absoluteness for the List Operator @{term length}\<close> |
698 |
text\<open>But it is never used.\<close> |
|
13647 | 699 |
|
21233 | 700 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
701 |
is_length :: "[i=>o,i,i,i] => o" where |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
702 |
"is_length(M,A,l,n) == |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
703 |
\<exists>sn[M]. \<exists>list_n[M]. \<exists>list_sn[M]. |
13397 | 704 |
is_list_N(M,A,n,list_n) & l \<notin> list_n & |
705 |
successor(M,n,sn) & is_list_N(M,A,sn,list_sn) & l \<in> list_sn" |
|
706 |
||
707 |
||
708 |
lemma (in M_datatypes) length_abs [simp]: |
|
46823 | 709 |
"[|M(A); l \<in> list(A); n \<in> nat|] ==> is_length(M,A,l,n) \<longleftrightarrow> n = length(l)" |
13397 | 710 |
apply (subgoal_tac "M(l) & M(n)") |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
711 |
prefer 2 apply (blast dest: transM) |
13397 | 712 |
apply (simp add: is_length_def) |
713 |
apply (blast intro: list_imp_list_N nat_into_Ord list_N_imp_eq_length |
|
714 |
dest: list_N_imp_length_lt) |
|
715 |
done |
|
716 |
||
60770 | 717 |
text\<open>Proof is trivial since @{term length} returns natural numbers.\<close> |
13564
1500a2e48d44
renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents:
13557
diff
changeset
|
718 |
lemma (in M_trivial) length_closed [intro,simp]: |
13397 | 719 |
"l \<in> list(A) ==> M(length(l))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
720 |
by (simp add: nat_into_M) |
13397 | 721 |
|
722 |
||
60770 | 723 |
subsection \<open>Absoluteness for the List Operator @{term nth}\<close> |
13397 | 724 |
|
725 |
lemma nth_eq_hd_iterates_tl [rule_format]: |
|
726 |
"xs \<in> list(A) ==> \<forall>n \<in> nat. nth(n,xs) = hd' (tl'^n (xs))" |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
727 |
apply (induct_tac xs) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
728 |
apply (simp add: iterates_tl_Nil hd'_Nil, clarify) |
13397 | 729 |
apply (erule natE) |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
730 |
apply (simp add: hd'_Cons) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
731 |
apply (simp add: tl'_Cons iterates_commute) |
13397 | 732 |
done |
733 |
||
13564
1500a2e48d44
renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents:
13557
diff
changeset
|
734 |
lemma (in M_basic) iterates_tl'_closed: |
13397 | 735 |
"[|n \<in> nat; M(x)|] ==> M(tl'^n (x))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
736 |
apply (induct_tac n, simp) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
737 |
apply (simp add: tl'_Cons tl'_closed) |
13397 | 738 |
done |
739 |
||
60770 | 740 |
text\<open>Immediate by type-checking\<close> |
13397 | 741 |
lemma (in M_datatypes) nth_closed [intro,simp]: |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
742 |
"[|xs \<in> list(A); n \<in> nat; M(A)|] ==> M(nth(n,xs))" |
13397 | 743 |
apply (case_tac "n < length(xs)") |
744 |
apply (blast intro: nth_type transM) |
|
745 |
apply (simp add: not_lt_iff_le nth_eq_0) |
|
746 |
done |
|
747 |
||
21233 | 748 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
749 |
is_nth :: "[i=>o,i,i,i] => o" where |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
750 |
"is_nth(M,n,l,Z) == |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
751 |
\<exists>X[M]. is_iterates(M, is_tl(M), l, n, X) & is_hd(M,X,Z)" |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
752 |
|
13409
d4ea094c650e
Relativization and Separation for the function "nth"
paulson
parents:
13398
diff
changeset
|
753 |
lemma (in M_datatypes) nth_abs [simp]: |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
754 |
"[|M(A); n \<in> nat; l \<in> list(A); M(Z)|] |
46823 | 755 |
==> is_nth(M,n,l,Z) \<longleftrightarrow> Z = nth(n,l)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
756 |
apply (subgoal_tac "M(l)") |
13397 | 757 |
prefer 2 apply (blast intro: transM) |
758 |
apply (simp add: is_nth_def nth_eq_hd_iterates_tl nat_into_M |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
759 |
tl'_closed iterates_tl'_closed |
13634 | 760 |
iterates_abs [OF _ relation1_tl] nth_replacement) |
13397 | 761 |
done |
762 |
||
13395 | 763 |
|
60770 | 764 |
subsection\<open>Relativization and Absoluteness for the @{term formula} Constructors\<close> |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
765 |
|
21233 | 766 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
767 |
is_Member :: "[i=>o,i,i,i] => o" where |
61798 | 768 |
\<comment>\<open>because @{term "Member(x,y) \<equiv> Inl(Inl(\<langle>x,y\<rangle>))"}\<close> |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
769 |
"is_Member(M,x,y,Z) == |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
22710
diff
changeset
|
770 |
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)" |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
771 |
|
13564
1500a2e48d44
renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents:
13557
diff
changeset
|
772 |
lemma (in M_trivial) Member_abs [simp]: |
46823 | 773 |
"[|M(x); M(y); M(Z)|] ==> is_Member(M,x,y,Z) \<longleftrightarrow> (Z = Member(x,y))" |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
774 |
by (simp add: is_Member_def Member_def) |
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
775 |
|
13564
1500a2e48d44
renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents:
13557
diff
changeset
|
776 |
lemma (in M_trivial) Member_in_M_iff [iff]: |
46823 | 777 |
"M(Member(x,y)) \<longleftrightarrow> M(x) & M(y)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
778 |
by (simp add: Member_def) |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
779 |
|
21233 | 780 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
781 |
is_Equal :: "[i=>o,i,i,i] => o" where |
61798 | 782 |
\<comment>\<open>because @{term "Equal(x,y) \<equiv> Inl(Inr(\<langle>x,y\<rangle>))"}\<close> |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
783 |
"is_Equal(M,x,y,Z) == |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
22710
diff
changeset
|
784 |
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)" |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
785 |
|
13564
1500a2e48d44
renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents:
13557
diff
changeset
|
786 |
lemma (in M_trivial) Equal_abs [simp]: |
46823 | 787 |
"[|M(x); M(y); M(Z)|] ==> is_Equal(M,x,y,Z) \<longleftrightarrow> (Z = Equal(x,y))" |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
788 |
by (simp add: is_Equal_def Equal_def) |
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
789 |
|
46823 | 790 |
lemma (in M_trivial) Equal_in_M_iff [iff]: "M(Equal(x,y)) \<longleftrightarrow> M(x) & M(y)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
791 |
by (simp add: Equal_def) |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
792 |
|
21233 | 793 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
794 |
is_Nand :: "[i=>o,i,i,i] => o" where |
61798 | 795 |
\<comment>\<open>because @{term "Nand(x,y) \<equiv> Inr(Inl(\<langle>x,y\<rangle>))"}\<close> |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
796 |
"is_Nand(M,x,y,Z) == |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
22710
diff
changeset
|
797 |
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)" |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
798 |
|
13564
1500a2e48d44
renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents:
13557
diff
changeset
|
799 |
lemma (in M_trivial) Nand_abs [simp]: |
46823 | 800 |
"[|M(x); M(y); M(Z)|] ==> is_Nand(M,x,y,Z) \<longleftrightarrow> (Z = Nand(x,y))" |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
801 |
by (simp add: is_Nand_def Nand_def) |
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
802 |
|
46823 | 803 |
lemma (in M_trivial) Nand_in_M_iff [iff]: "M(Nand(x,y)) \<longleftrightarrow> M(x) & M(y)" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
804 |
by (simp add: Nand_def) |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
805 |
|
21233 | 806 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
807 |
is_Forall :: "[i=>o,i,i] => o" where |
61798 | 808 |
\<comment>\<open>because @{term "Forall(x) \<equiv> Inr(Inr(p))"}\<close> |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
809 |
"is_Forall(M,p,Z) == \<exists>u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)" |
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
810 |
|
13564
1500a2e48d44
renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents:
13557
diff
changeset
|
811 |
lemma (in M_trivial) Forall_abs [simp]: |
46823 | 812 |
"[|M(x); M(Z)|] ==> is_Forall(M,x,Z) \<longleftrightarrow> (Z = Forall(x))" |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
813 |
by (simp add: is_Forall_def Forall_def) |
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
814 |
|
46823 | 815 |
lemma (in M_trivial) Forall_in_M_iff [iff]: "M(Forall(x)) \<longleftrightarrow> M(x)" |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
816 |
by (simp add: Forall_def) |
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
817 |
|
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
818 |
|
13647 | 819 |
|
60770 | 820 |
subsection \<open>Absoluteness for @{term formula_rec}\<close> |
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
821 |
|
21233 | 822 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
823 |
formula_rec_case :: "[[i,i]=>i, [i,i]=>i, [i,i,i,i]=>i, [i,i]=>i, i, i] => i" where |
61798 | 824 |
\<comment>\<open>the instance of @{term formula_case} in @{term formula_rec}\<close> |
13647 | 825 |
"formula_rec_case(a,b,c,d,h) == |
826 |
formula_case (a, b, |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
827 |
\<lambda>u v. c(u, v, h ` succ(depth(u)) ` u, |
13647 | 828 |
h ` succ(depth(v)) ` v), |
829 |
\<lambda>u. d(u, h ` succ(depth(u)) ` u))" |
|
830 |
||
60770 | 831 |
text\<open>Unfold @{term formula_rec} to @{term formula_rec_case}. |
13647 | 832 |
Express @{term formula_rec} without using @{term rank} or @{term Vset}, |
60770 | 833 |
neither of which is absolute.\<close> |
13647 | 834 |
lemma (in M_trivial) formula_rec_eq: |
835 |
"p \<in> formula ==> |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
836 |
formula_rec(a,b,c,d,p) = |
13647 | 837 |
transrec (succ(depth(p)), |
838 |
\<lambda>x h. Lambda (formula, formula_rec_case(a,b,c,d,h))) ` p" |
|
839 |
apply (simp add: formula_rec_case_def) |
|
840 |
apply (induct_tac p) |
|
60770 | 841 |
txt\<open>Base case for @{term Member}\<close> |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
842 |
apply (subst transrec, simp add: formula.intros) |
60770 | 843 |
txt\<open>Base case for @{term Equal}\<close> |
13647 | 844 |
apply (subst transrec, simp add: formula.intros) |
60770 | 845 |
txt\<open>Inductive step for @{term Nand}\<close> |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
846 |
apply (subst transrec) |
13647 | 847 |
apply (simp add: succ_Un_distrib formula.intros) |
60770 | 848 |
txt\<open>Inductive step for @{term Forall}\<close> |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
849 |
apply (subst transrec) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
850 |
apply (simp add: formula_imp_formula_N formula.intros) |
13647 | 851 |
done |
852 |
||
853 |
||
60770 | 854 |
subsubsection\<open>Absoluteness for the Formula Operator @{term depth}\<close> |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
855 |
|
21233 | 856 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
857 |
is_depth :: "[i=>o,i,i] => o" where |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
858 |
"is_depth(M,p,n) == |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
859 |
\<exists>sn[M]. \<exists>formula_n[M]. \<exists>formula_sn[M]. |
13647 | 860 |
is_formula_N(M,n,formula_n) & p \<notin> formula_n & |
861 |
successor(M,n,sn) & is_formula_N(M,sn,formula_sn) & p \<in> formula_sn" |
|
862 |
||
863 |
||
864 |
lemma (in M_datatypes) depth_abs [simp]: |
|
46823 | 865 |
"[|p \<in> formula; n \<in> nat|] ==> is_depth(M,p,n) \<longleftrightarrow> n = depth(p)" |
13647 | 866 |
apply (subgoal_tac "M(p) & M(n)") |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
867 |
prefer 2 apply (blast dest: transM) |
13647 | 868 |
apply (simp add: is_depth_def) |
869 |
apply (blast intro: formula_imp_formula_N nat_into_Ord formula_N_imp_eq_depth |
|
870 |
dest: formula_N_imp_depth_lt) |
|
871 |
done |
|
872 |
||
60770 | 873 |
text\<open>Proof is trivial since @{term depth} returns natural numbers.\<close> |
13647 | 874 |
lemma (in M_trivial) depth_closed [intro,simp]: |
875 |
"p \<in> formula ==> M(depth(p))" |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
876 |
by (simp add: nat_into_M) |
13647 | 877 |
|
878 |
||
60770 | 879 |
subsubsection\<open>@{term is_formula_case}: relativization of @{term formula_case}\<close> |
13423
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
880 |
|
21233 | 881 |
definition |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
882 |
is_formula_case :: |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
883 |
"[i=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i]=>o, i, i] => o" where |
61798 | 884 |
\<comment>\<open>no constraint on non-formulas\<close> |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
885 |
"is_formula_case(M, is_a, is_b, is_c, is_d, p, z) == |
46823 | 886 |
(\<forall>x[M]. \<forall>y[M]. finite_ordinal(M,x) \<longrightarrow> finite_ordinal(M,y) \<longrightarrow> |
887 |
is_Member(M,x,y,p) \<longrightarrow> is_a(x,y,z)) & |
|
888 |
(\<forall>x[M]. \<forall>y[M]. finite_ordinal(M,x) \<longrightarrow> finite_ordinal(M,y) \<longrightarrow> |
|
889 |
is_Equal(M,x,y,p) \<longrightarrow> is_b(x,y,z)) & |
|
890 |
(\<forall>x[M]. \<forall>y[M]. mem_formula(M,x) \<longrightarrow> mem_formula(M,y) \<longrightarrow> |
|
891 |
is_Nand(M,x,y,p) \<longrightarrow> is_c(x,y,z)) & |
|
892 |
(\<forall>x[M]. mem_formula(M,x) \<longrightarrow> is_Forall(M,x,p) \<longrightarrow> is_d(x,z))" |
|
13423
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
893 |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
894 |
lemma (in M_datatypes) formula_case_abs [simp]: |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
895 |
"[| Relation2(M,nat,nat,is_a,a); Relation2(M,nat,nat,is_b,b); |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
896 |
Relation2(M,formula,formula,is_c,c); Relation1(M,formula,is_d,d); |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
897 |
p \<in> formula; M(z) |] |
46823 | 898 |
==> is_formula_case(M,is_a,is_b,is_c,is_d,p,z) \<longleftrightarrow> |
13423
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
899 |
z = formula_case(a,b,c,d,p)" |
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
900 |
apply (simp add: formula_into_M is_formula_case_def) |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
901 |
apply (erule formula.cases) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
902 |
apply (simp_all add: Relation1_def Relation2_def) |
13423
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
903 |
done |
7ec771711c09
More lemmas, working towards relativization of "satisfies"
paulson
parents:
13422
diff
changeset
|
904 |
|
13418
7c0ba9dba978
tweaks, aiming towards relativization of "satisfies"
paulson
parents:
13409
diff
changeset
|
905 |
lemma (in M_datatypes) formula_case_closed [intro,simp]: |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
906 |
"[|p \<in> formula; |
46823 | 907 |
\<forall>x[M]. \<forall>y[M]. x\<in>nat \<longrightarrow> y\<in>nat \<longrightarrow> M(a(x,y)); |
908 |
\<forall>x[M]. \<forall>y[M]. x\<in>nat \<longrightarrow> y\<in>nat \<longrightarrow> M(b(x,y)); |
|
909 |
\<forall>x[M]. \<forall>y[M]. x\<in>formula \<longrightarrow> y\<in>formula \<longrightarrow> M(c(x,y)); |
|
910 |
\<forall>x[M]. x\<in>formula \<longrightarrow> M(d(x))|] ==> M(formula_case(a,b,c,d,p))" |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
911 |
by (erule formula.cases, simp_all) |
13418
7c0ba9dba978
tweaks, aiming towards relativization of "satisfies"
paulson
parents:
13409
diff
changeset
|
912 |
|
13398
1cadd412da48
Towards relativization and absoluteness of formula_rec
paulson
parents:
13397
diff
changeset
|
913 |
|
60770 | 914 |
subsubsection \<open>Absoluteness for @{term formula_rec}: Final Results\<close> |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
915 |
|
21233 | 916 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
917 |
is_formula_rec :: "[i=>o, [i,i,i]=>o, i, i] => o" where |
61798 | 918 |
\<comment>\<open>predicate to relativize the functional @{term formula_rec}\<close> |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
919 |
"is_formula_rec(M,MH,p,z) == |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
920 |
\<exists>dp[M]. \<exists>i[M]. \<exists>f[M]. finite_ordinal(M,dp) & is_depth(M,p,dp) & |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
921 |
successor(M,dp,i) & fun_apply(M,f,p,z) & is_transrec(M,MH,i,f)" |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
922 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
923 |
|
60770 | 924 |
text\<open>Sufficient conditions to relativize the instance of @{term formula_case} |
925 |
in @{term formula_rec}\<close> |
|
13634 | 926 |
lemma (in M_datatypes) Relation1_formula_rec_case: |
927 |
"[|Relation2(M, nat, nat, is_a, a); |
|
928 |
Relation2(M, nat, nat, is_b, b); |
|
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
929 |
Relation2 (M, formula, formula, |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
930 |
is_c, \<lambda>u v. c(u, v, h`succ(depth(u))`u, h`succ(depth(v))`v)); |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
931 |
Relation1(M, formula, |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
932 |
is_d, \<lambda>u. d(u, h ` succ(depth(u)) ` u)); |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
22710
diff
changeset
|
933 |
M(h) |] |
13634 | 934 |
==> Relation1(M, formula, |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
935 |
is_formula_case (M, is_a, is_b, is_c, is_d), |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
936 |
formula_rec_case(a, b, c, d, h))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
937 |
apply (simp (no_asm) add: formula_rec_case_def Relation1_def) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
938 |
apply (simp add: formula_case_abs) |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
939 |
done |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
940 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
941 |
|
60770 | 942 |
text\<open>This locale packages the premises of the following theorems, |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
943 |
which is the normal purpose of locales. It doesn't accumulate |
60770 | 944 |
constraints on the class @{term M}, as in most of this deveopment.\<close> |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
945 |
locale Formula_Rec = M_eclose + |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
946 |
fixes a and is_a and b and is_b and c and is_c and d and is_d and MH |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
947 |
defines |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
948 |
"MH(u::i,f,z) == |
46823 | 949 |
\<forall>fml[M]. is_formula(M,fml) \<longrightarrow> |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
950 |
is_lambda |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
22710
diff
changeset
|
951 |
(M, fml, is_formula_case (M, is_a, is_b, is_c(f), is_d(f)), z)" |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
952 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
953 |
assumes a_closed: "[|x\<in>nat; y\<in>nat|] ==> M(a(x,y))" |
13634 | 954 |
and a_rel: "Relation2(M, nat, nat, is_a, a)" |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
955 |
and b_closed: "[|x\<in>nat; y\<in>nat|] ==> M(b(x,y))" |
13634 | 956 |
and b_rel: "Relation2(M, nat, nat, is_b, b)" |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
957 |
and c_closed: "[|x \<in> formula; y \<in> formula; M(gx); M(gy)|] |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
958 |
==> M(c(x, y, gx, gy))" |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
959 |
and c_rel: |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
960 |
"M(f) ==> |
13634 | 961 |
Relation2 (M, formula, formula, is_c(f), |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
962 |
\<lambda>u v. c(u, v, f ` succ(depth(u)) ` u, f ` succ(depth(v)) ` v))" |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
963 |
and d_closed: "[|x \<in> formula; M(gx)|] ==> M(d(x, gx))" |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
964 |
and d_rel: |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
965 |
"M(f) ==> |
13634 | 966 |
Relation1(M, formula, is_d(f), \<lambda>u. d(u, f ` succ(depth(u)) ` u))" |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
967 |
and fr_replace: "n \<in> nat ==> transrec_replacement(M,MH,n)" |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
968 |
and fr_lam_replace: |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
969 |
"M(g) ==> |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
970 |
strong_replacement |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
22710
diff
changeset
|
971 |
(M, \<lambda>x y. x \<in> formula & |
58860 | 972 |
y = \<langle>x, formula_rec_case(a,b,c,d,g,x)\<rangle>)" |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
973 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
974 |
lemma (in Formula_Rec) formula_rec_case_closed: |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
975 |
"[|M(g); p \<in> formula|] ==> M(formula_rec_case(a, b, c, d, g, p))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
976 |
by (simp add: formula_rec_case_def a_closed b_closed c_closed d_closed) |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
977 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
978 |
lemma (in Formula_Rec) formula_rec_lam_closed: |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
979 |
"M(g) ==> M(Lambda (formula, formula_rec_case(a,b,c,d,g)))" |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
980 |
by (simp add: lam_closed2 fr_lam_replace formula_rec_case_closed) |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
981 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
982 |
lemma (in Formula_Rec) MH_rel2: |
13634 | 983 |
"relation2 (M, MH, |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
984 |
\<lambda>x h. Lambda (formula, formula_rec_case(a,b,c,d,h)))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
985 |
apply (simp add: relation2_def MH_def, clarify) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
986 |
apply (rule lambda_abs2) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
987 |
apply (rule Relation1_formula_rec_case) |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
988 |
apply (simp_all add: a_rel b_rel c_rel d_rel formula_rec_case_closed) |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
989 |
done |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
990 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
991 |
lemma (in Formula_Rec) fr_transrec_closed: |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
992 |
"n \<in> nat |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
993 |
==> M(transrec |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
994 |
(n, \<lambda>x h. Lambda(formula, formula_rec_case(a, b, c, d, h))))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
995 |
by (simp add: transrec_closed [OF fr_replace MH_rel2] |
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
996 |
nat_into_M formula_rec_lam_closed) |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
997 |
|
60770 | 998 |
text\<open>The main two results: @{term formula_rec} is absolute for @{term M}.\<close> |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
999 |
theorem (in Formula_Rec) formula_rec_closed: |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1000 |
"p \<in> formula ==> M(formula_rec(a,b,c,d,p))" |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
1001 |
by (simp add: formula_rec_eq fr_transrec_closed |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1002 |
transM [OF _ formula_closed]) |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1003 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1004 |
theorem (in Formula_Rec) formula_rec_abs: |
13655
95b95cdb4704
Tidying up. New primitives is_iterates and is_iterates_fm.
paulson
parents:
13647
diff
changeset
|
1005 |
"[| p \<in> formula; M(z)|] |
46823 | 1006 |
==> is_formula_rec(M,MH,p,z) \<longleftrightarrow> z = formula_rec(a,b,c,d,p)" |
13557
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1007 |
by (simp add: is_formula_rec_def formula_rec_eq transM [OF _ formula_closed] |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1008 |
transrec_abs [OF fr_replace MH_rel2] depth_type |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1009 |
fr_transrec_closed formula_rec_lam_closed eq_commute) |
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1010 |
|
6061d0045409
deleted redundant material (quasiformula, ...) and rationalized
paulson
parents:
13505
diff
changeset
|
1011 |
|
13268 | 1012 |
end |