author | paulson |
Fri, 19 Jul 2002 13:29:22 +0200 | |
changeset 13397 | 6e5f4d911435 |
parent 13395 | 4eb948d1eb4e |
child 13398 | 1cadd412da48 |
permissions | -rw-r--r-- |
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header {*Absoluteness Properties for Recursive Datatypes*} |
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theory Datatype_absolute = Formula + WF_absolute: |
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subsection{*The lfp of a continuous function can be expressed as a union*} |
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constdefs |
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directed :: "i=>o" |
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"directed(A) == A\<noteq>0 & (\<forall>x\<in>A. \<forall>y\<in>A. x \<union> y \<in> A)" |
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contin :: "(i=>i) => o" |
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"contin(h) == (\<forall>A. directed(A) --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))" |
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lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D" |
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apply (induct_tac n) |
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apply (simp_all add: bnd_mono_def, blast) |
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done |
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lemma bnd_mono_increasing [rule_format]: |
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"[|i \<in> nat; j \<in> nat; bnd_mono(D,h)|] ==> i \<le> j --> h^i(0) \<subseteq> h^j(0)" |
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apply (rule_tac m=i and n=j in diff_induct, simp_all) |
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apply (blast del: subsetI |
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intro: bnd_mono_iterates_subset bnd_monoD2 [of concl: h] ) |
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done |
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lemma directed_iterates: "bnd_mono(D,h) ==> directed({h^n (0). n\<in>nat})" |
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apply (simp add: directed_def, clarify) |
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apply (rename_tac i j) |
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apply (rule_tac x="i \<union> j" in bexI) |
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apply (rule_tac i = i and j = j in Ord_linear_le) |
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apply (simp_all add: subset_Un_iff [THEN iffD1] le_imp_subset |
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subset_Un_iff2 [THEN iffD1]) |
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apply (simp_all add: subset_Un_iff [THEN iff_sym] bnd_mono_increasing |
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subset_Un_iff2 [THEN iff_sym]) |
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done |
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lemma contin_iterates_eq: |
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"[|bnd_mono(D, h); contin(h)|] |
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==> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))" |
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apply (simp add: contin_def directed_iterates) |
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apply (rule trans) |
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apply (rule equalityI) |
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apply (simp_all add: UN_subset_iff) |
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apply safe |
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apply (erule_tac [2] natE) |
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apply (rule_tac a="succ(x)" in UN_I) |
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apply simp_all |
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apply blast |
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done |
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lemma lfp_subset_Union: |
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"[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) <= (\<Union>n\<in>nat. h^n(0))" |
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apply (rule lfp_lowerbound) |
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apply (simp add: contin_iterates_eq) |
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apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff) |
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done |
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lemma Union_subset_lfp: |
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"bnd_mono(D,h) ==> (\<Union>n\<in>nat. h^n(0)) <= lfp(D,h)" |
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apply (simp add: UN_subset_iff) |
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apply (rule ballI) |
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apply (induct_tac n, simp_all) |
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apply (rule subset_trans [of _ "h(lfp(D,h))"]) |
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apply (blast dest: bnd_monoD2 [OF _ _ lfp_subset] ) |
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apply (erule lfp_lemma2) |
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done |
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lemma lfp_eq_Union: |
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"[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) = (\<Union>n\<in>nat. h^n(0))" |
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by (blast del: subsetI |
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intro: lfp_subset_Union Union_subset_lfp) |
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subsubsection{*Some Standard Datatype Constructions Preserve Continuity*} |
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lemma contin_imp_mono: "[|X\<subseteq>Y; contin(F)|] ==> F(X) \<subseteq> F(Y)" |
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apply (simp add: contin_def) |
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apply (drule_tac x="{X,Y}" in spec) |
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apply (simp add: directed_def subset_Un_iff2 Un_commute) |
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done |
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lemma sum_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) + G(X))" |
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by (simp add: contin_def, blast) |
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lemma prod_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) * G(X))" |
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apply (subgoal_tac "\<forall>B C. F(B) \<subseteq> F(B \<union> C)") |
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prefer 2 apply (simp add: Un_upper1 contin_imp_mono) |
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apply (subgoal_tac "\<forall>B C. G(C) \<subseteq> G(B \<union> C)") |
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prefer 2 apply (simp add: Un_upper2 contin_imp_mono) |
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apply (simp add: contin_def, clarify) |
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apply (rule equalityI) |
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prefer 2 apply blast |
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apply clarify |
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apply (rename_tac B C) |
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apply (rule_tac a="B \<union> C" in UN_I) |
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apply (simp add: directed_def, blast) |
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done |
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lemma const_contin: "contin(\<lambda>X. A)" |
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by (simp add: contin_def directed_def) |
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lemma id_contin: "contin(\<lambda>X. X)" |
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by (simp add: contin_def) |
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subsection {*Absoluteness for "Iterates"*} |
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constdefs |
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iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o" |
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"iterates_MH(M,isF,v,n,g,z) == |
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is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u), |
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n, z)" |
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iterates_replacement :: "[i=>o, [i,i]=>o, i] => o" |
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"iterates_replacement(M,isF,v) == |
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\<forall>n[M]. n\<in>nat --> |
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wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))" |
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lemma (in M_axioms) iterates_MH_abs: |
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"[| relativize1(M,isF,F); M(n); M(g); M(z) |] |
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==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, \<lambda>m. F(g`m), n)" |
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by (simp add: nat_case_abs [of _ "\<lambda>m. F(g ` m)"] |
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relativize1_def iterates_MH_def) |
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lemma (in M_axioms) iterates_imp_wfrec_replacement: |
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"[|relativize1(M,isF,F); n \<in> nat; iterates_replacement(M,isF,v)|] |
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==> wfrec_replacement(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n), |
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Memrel(succ(n)))" |
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by (simp add: iterates_replacement_def iterates_MH_abs) |
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theorem (in M_trancl) iterates_abs: |
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"[| iterates_replacement(M,isF,v); relativize1(M,isF,F); |
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n \<in> nat; M(v); M(z); \<forall>x[M]. M(F(x)) |] |
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==> is_wfrec(M, iterates_MH(M,isF,v), Memrel(succ(n)), n, z) <-> |
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z = iterates(F,n,v)" |
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apply (frule iterates_imp_wfrec_replacement, assumption+) |
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apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M |
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relativize2_def iterates_MH_abs |
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iterates_nat_def recursor_def transrec_def |
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eclose_sing_Ord_eq nat_into_M |
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trans_wfrec_abs [of _ _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"]) |
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done |
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lemma (in M_wfrank) iterates_closed [intro,simp]: |
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"[| iterates_replacement(M,isF,v); relativize1(M,isF,F); |
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n \<in> nat; M(v); \<forall>x[M]. M(F(x)) |] |
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==> M(iterates(F,n,v))" |
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apply (frule iterates_imp_wfrec_replacement, assumption+) |
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apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M |
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relativize2_def iterates_MH_abs |
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iterates_nat_def recursor_def transrec_def |
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eclose_sing_Ord_eq nat_into_M |
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trans_wfrec_closed [of _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"]) |
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done |
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subsection {*lists without univ*} |
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lemmas datatype_univs = Inl_in_univ Inr_in_univ |
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Pair_in_univ nat_into_univ A_into_univ |
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lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)" |
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apply (rule bnd_monoI) |
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apply (intro subset_refl zero_subset_univ A_subset_univ |
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sum_subset_univ Sigma_subset_univ) |
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apply (rule subset_refl sum_mono Sigma_mono | assumption)+ |
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done |
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lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)" |
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by (intro sum_contin prod_contin id_contin const_contin) |
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text{*Re-expresses lists using sum and product*} |
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lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)" |
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apply (simp add: list_def) |
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apply (rule equalityI) |
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apply (rule lfp_lowerbound) |
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prefer 2 apply (rule lfp_subset) |
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apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono]) |
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apply (simp add: Nil_def Cons_def) |
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apply blast |
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txt{*Opposite inclusion*} |
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apply (rule lfp_lowerbound) |
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prefer 2 apply (rule lfp_subset) |
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apply (clarify, subst lfp_unfold [OF list.bnd_mono]) |
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apply (simp add: Nil_def Cons_def) |
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apply (blast intro: datatype_univs |
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dest: lfp_subset [THEN subsetD]) |
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done |
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text{*Re-expresses lists using "iterates", no univ.*} |
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lemma list_eq_Union: |
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"list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))" |
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by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin) |
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constdefs |
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is_list_functor :: "[i=>o,i,i,i] => o" |
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"is_list_functor(M,A,X,Z) == |
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\<exists>n1[M]. \<exists>AX[M]. |
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number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" |
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lemma (in M_axioms) list_functor_abs [simp]: |
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"[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) <-> (Z = {0} + A*X)" |
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by (simp add: is_list_functor_def singleton_0 nat_into_M) |
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subsection {*formulas without univ*} |
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lemma formula_fun_bnd_mono: |
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"bnd_mono(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))" |
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apply (rule bnd_monoI) |
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apply (intro subset_refl zero_subset_univ A_subset_univ |
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sum_subset_univ Sigma_subset_univ nat_subset_univ) |
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apply (rule subset_refl sum_mono Sigma_mono | assumption)+ |
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done |
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lemma formula_fun_contin: |
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"contin(\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))" |
|
224 |
by (intro sum_contin prod_contin id_contin const_contin) |
|
225 |
||
226 |
||
227 |
text{*Re-expresses formulas using sum and product*} |
|
228 |
lemma formula_eq_lfp2: |
|
229 |
"formula = lfp(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))" |
|
230 |
apply (simp add: formula_def) |
|
231 |
apply (rule equalityI) |
|
232 |
apply (rule lfp_lowerbound) |
|
233 |
prefer 2 apply (rule lfp_subset) |
|
234 |
apply (clarify, subst lfp_unfold [OF formula_fun_bnd_mono]) |
|
235 |
apply (simp add: Member_def Equal_def Neg_def And_def Forall_def) |
|
236 |
apply blast |
|
237 |
txt{*Opposite inclusion*} |
|
238 |
apply (rule lfp_lowerbound) |
|
239 |
prefer 2 apply (rule lfp_subset, clarify) |
|
240 |
apply (subst lfp_unfold [OF formula.bnd_mono, simplified]) |
|
241 |
apply (simp add: Member_def Equal_def Neg_def And_def Forall_def) |
|
242 |
apply (elim sumE SigmaE, simp_all) |
|
243 |
apply (blast intro: datatype_univs dest: lfp_subset [THEN subsetD])+ |
|
244 |
done |
|
245 |
||
246 |
text{*Re-expresses formulas using "iterates", no univ.*} |
|
247 |
lemma formula_eq_Union: |
|
248 |
"formula = |
|
249 |
(\<Union>n\<in>nat. (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))) ^ n (0))" |
|
250 |
by (simp add: formula_eq_lfp2 lfp_eq_Union formula_fun_bnd_mono |
|
251 |
formula_fun_contin) |
|
252 |
||
253 |
||
254 |
constdefs |
|
255 |
is_formula_functor :: "[i=>o,i,i] => o" |
|
256 |
"is_formula_functor(M,X,Z) == |
|
257 |
\<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M]. \<exists>X4[M]. |
|
258 |
omega(M,nat') & cartprod(M,nat',nat',natnat) & |
|
259 |
is_sum(M,natnat,natnat,natnatsum) & |
|
260 |
cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & is_sum(M,X,X3,X4) & |
|
261 |
is_sum(M,natnatsum,X4,Z)" |
|
262 |
||
263 |
lemma (in M_axioms) formula_functor_abs [simp]: |
|
264 |
"[| M(X); M(Z) |] |
|
265 |
==> is_formula_functor(M,X,Z) <-> |
|
266 |
Z = ((nat*nat) + (nat*nat)) + (X + (X*X + X))" |
|
267 |
by (simp add: is_formula_functor_def) |
|
268 |
||
269 |
||
270 |
subsection{*@{term M} Contains the List and Formula Datatypes*} |
|
13395 | 271 |
|
272 |
constdefs |
|
13397 | 273 |
list_N :: "[i,i] => i" |
274 |
"list_N(A,n) == (\<lambda>X. {0} + A * X)^n (0)" |
|
275 |
||
276 |
lemma Nil_in_list_N [simp]: "[] \<in> list_N(A,succ(n))" |
|
277 |
by (simp add: list_N_def Nil_def) |
|
278 |
||
279 |
lemma Cons_in_list_N [simp]: |
|
280 |
"Cons(a,l) \<in> list_N(A,succ(n)) <-> a\<in>A & l \<in> list_N(A,n)" |
|
281 |
by (simp add: list_N_def Cons_def) |
|
282 |
||
283 |
text{*These two aren't simprules because they reveal the underlying |
|
284 |
list representation.*} |
|
285 |
lemma list_N_0: "list_N(A,0) = 0" |
|
286 |
by (simp add: list_N_def) |
|
287 |
||
288 |
lemma list_N_succ: "list_N(A,succ(n)) = {0} + A * (list_N(A,n))" |
|
289 |
by (simp add: list_N_def) |
|
290 |
||
291 |
lemma list_N_imp_list: |
|
292 |
"[| l \<in> list_N(A,n); n \<in> nat |] ==> l \<in> list(A)" |
|
293 |
by (force simp add: list_eq_Union list_N_def) |
|
294 |
||
295 |
lemma list_N_imp_length_lt [rule_format]: |
|
296 |
"n \<in> nat ==> \<forall>l \<in> list_N(A,n). length(l) < n" |
|
297 |
apply (induct_tac n) |
|
298 |
apply (auto simp add: list_N_0 list_N_succ |
|
299 |
Nil_def [symmetric] Cons_def [symmetric]) |
|
300 |
done |
|
301 |
||
302 |
lemma list_imp_list_N [rule_format]: |
|
303 |
"l \<in> list(A) ==> \<forall>n\<in>nat. length(l) < n --> l \<in> list_N(A, n)" |
|
304 |
apply (induct_tac l) |
|
305 |
apply (force elim: natE)+ |
|
306 |
done |
|
307 |
||
308 |
lemma list_N_imp_eq_length: |
|
309 |
"[|n \<in> nat; l \<notin> list_N(A, n); l \<in> list_N(A, succ(n))|] |
|
310 |
==> n = length(l)" |
|
311 |
apply (rule le_anti_sym) |
|
312 |
prefer 2 apply (simp add: list_N_imp_length_lt) |
|
313 |
apply (frule list_N_imp_list, simp) |
|
314 |
apply (simp add: not_lt_iff_le [symmetric]) |
|
315 |
apply (blast intro: list_imp_list_N) |
|
316 |
done |
|
317 |
||
318 |
text{*Express @{term list_rec} without using @{term rank} or @{term Vset}, |
|
319 |
neither of which is absolute.*} |
|
320 |
lemma (in M_triv_axioms) list_rec_eq: |
|
321 |
"l \<in> list(A) ==> |
|
322 |
list_rec(a,g,l) = |
|
323 |
transrec (succ(length(l)), |
|
324 |
\<lambda>x h. Lambda (list_N(A,x), |
|
325 |
list_case' (a, |
|
326 |
\<lambda>a l. g(a, l, h ` succ(length(l)) ` l)))) ` l" |
|
327 |
apply (induct_tac l) |
|
328 |
apply (subst transrec, simp) |
|
329 |
apply (subst transrec) |
|
330 |
apply (simp add: list_imp_list_N) |
|
331 |
done |
|
332 |
||
333 |
constdefs |
|
334 |
is_list_N :: "[i=>o,i,i,i] => o" |
|
335 |
"is_list_N(M,A,n,Z) == |
|
13395 | 336 |
\<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M]. |
337 |
empty(M,zero) & |
|
338 |
successor(M,n,sn) & membership(M,sn,msn) & |
|
339 |
is_wfrec(M, iterates_MH(M, is_list_functor(M,A),zero), msn, n, Z)" |
|
340 |
||
341 |
mem_list :: "[i=>o,i,i] => o" |
|
342 |
"mem_list(M,A,l) == |
|
343 |
\<exists>n[M]. \<exists>listn[M]. |
|
13397 | 344 |
finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l \<in> listn" |
13395 | 345 |
|
346 |
is_list :: "[i=>o,i,i] => o" |
|
347 |
"is_list(M,A,Z) == \<forall>l[M]. l \<in> Z <-> mem_list(M,A,l)" |
|
348 |
||
349 |
constdefs |
|
350 |
is_formula_n :: "[i=>o,i,i] => o" |
|
351 |
"is_formula_n(M,n,Z) == |
|
352 |
\<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M]. |
|
353 |
empty(M,zero) & |
|
354 |
successor(M,n,sn) & membership(M,sn,msn) & |
|
355 |
is_wfrec(M, iterates_MH(M, is_formula_functor(M),zero), msn, n, Z)" |
|
356 |
||
357 |
mem_formula :: "[i=>o,i] => o" |
|
358 |
"mem_formula(M,p) == |
|
359 |
\<exists>n[M]. \<exists>formn[M]. |
|
360 |
finite_ordinal(M,n) & is_formula_n(M,n,formn) & p \<in> formn" |
|
361 |
||
362 |
is_formula :: "[i=>o,i] => o" |
|
363 |
"is_formula(M,Z) == \<forall>p[M]. p \<in> Z <-> mem_formula(M,p)" |
|
364 |
||
13382 | 365 |
locale (open) M_datatypes = M_wfrank + |
13353 | 366 |
assumes list_replacement1: |
13363 | 367 |
"M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)" |
13353 | 368 |
and list_replacement2: |
13363 | 369 |
"M(A) ==> strong_replacement(M, |
13353 | 370 |
\<lambda>n y. n\<in>nat & |
371 |
(\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) & |
|
13363 | 372 |
is_wfrec(M, iterates_MH(M,is_list_functor(M,A), 0), |
13353 | 373 |
msn, n, y)))" |
13386 | 374 |
and formula_replacement1: |
375 |
"iterates_replacement(M, is_formula_functor(M), 0)" |
|
376 |
and formula_replacement2: |
|
377 |
"strong_replacement(M, |
|
378 |
\<lambda>n y. n\<in>nat & |
|
379 |
(\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) & |
|
380 |
is_wfrec(M, iterates_MH(M,is_formula_functor(M), 0), |
|
381 |
msn, n, y)))" |
|
13350 | 382 |
|
13395 | 383 |
|
384 |
subsubsection{*Absoluteness of the List Construction*} |
|
385 |
||
13348 | 386 |
lemma (in M_datatypes) list_replacement2': |
13353 | 387 |
"M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))" |
388 |
apply (insert list_replacement2 [of A]) |
|
389 |
apply (rule strong_replacement_cong [THEN iffD1]) |
|
390 |
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]]) |
|
13363 | 391 |
apply (simp_all add: list_replacement1 relativize1_def) |
13353 | 392 |
done |
13268 | 393 |
|
394 |
lemma (in M_datatypes) list_closed [intro,simp]: |
|
395 |
"M(A) ==> M(list(A))" |
|
13353 | 396 |
apply (insert list_replacement1) |
397 |
by (simp add: RepFun_closed2 list_eq_Union |
|
398 |
list_replacement2' relativize1_def |
|
399 |
iterates_closed [of "is_list_functor(M,A)"]) |
|
13397 | 400 |
|
401 |
lemma (in M_datatypes) list_N_abs [simp]: |
|
13395 | 402 |
"[|M(A); n\<in>nat; M(Z)|] |
13397 | 403 |
==> is_list_N(M,A,n,Z) <-> Z = list_N(A,n)" |
13395 | 404 |
apply (insert list_replacement1) |
13397 | 405 |
apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M |
13395 | 406 |
iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"]) |
407 |
done |
|
13268 | 408 |
|
13397 | 409 |
lemma (in M_datatypes) list_N_closed [intro,simp]: |
410 |
"[|M(A); n\<in>nat|] ==> M(list_N(A,n))" |
|
411 |
apply (insert list_replacement1) |
|
412 |
apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M |
|
413 |
iterates_closed [of "is_list_functor(M,A)"]) |
|
414 |
done |
|
415 |
||
13395 | 416 |
lemma (in M_datatypes) mem_list_abs [simp]: |
417 |
"M(A) ==> mem_list(M,A,l) <-> l \<in> list(A)" |
|
418 |
apply (insert list_replacement1) |
|
13397 | 419 |
apply (simp add: mem_list_def list_N_def relativize1_def list_eq_Union |
13395 | 420 |
iterates_closed [of "is_list_functor(M,A)"]) |
421 |
done |
|
422 |
||
423 |
lemma (in M_datatypes) list_abs [simp]: |
|
424 |
"[|M(A); M(Z)|] ==> is_list(M,A,Z) <-> Z = list(A)" |
|
425 |
apply (simp add: is_list_def, safe) |
|
426 |
apply (rule M_equalityI, simp_all) |
|
427 |
done |
|
428 |
||
429 |
subsubsection{*Absoluteness of Formulas*} |
|
13293 | 430 |
|
13386 | 431 |
lemma (in M_datatypes) formula_replacement2': |
432 |
"strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0))" |
|
433 |
apply (insert formula_replacement2) |
|
434 |
apply (rule strong_replacement_cong [THEN iffD1]) |
|
435 |
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]]) |
|
436 |
apply (simp_all add: formula_replacement1 relativize1_def) |
|
437 |
done |
|
438 |
||
439 |
lemma (in M_datatypes) formula_closed [intro,simp]: |
|
440 |
"M(formula)" |
|
441 |
apply (insert formula_replacement1) |
|
442 |
apply (simp add: RepFun_closed2 formula_eq_Union |
|
443 |
formula_replacement2' relativize1_def |
|
444 |
iterates_closed [of "is_formula_functor(M)"]) |
|
445 |
done |
|
446 |
||
13395 | 447 |
lemma (in M_datatypes) is_formula_n_abs [simp]: |
448 |
"[|n\<in>nat; M(Z)|] |
|
449 |
==> is_formula_n(M,n,Z) <-> |
|
450 |
Z = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0)" |
|
451 |
apply (insert formula_replacement1) |
|
452 |
apply (simp add: is_formula_n_def relativize1_def nat_into_M |
|
453 |
iterates_abs [of "is_formula_functor(M)" _ |
|
454 |
"\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))"]) |
|
455 |
done |
|
456 |
||
457 |
lemma (in M_datatypes) mem_formula_abs [simp]: |
|
458 |
"mem_formula(M,l) <-> l \<in> formula" |
|
459 |
apply (insert formula_replacement1) |
|
460 |
apply (simp add: mem_formula_def relativize1_def formula_eq_Union |
|
461 |
iterates_closed [of "is_formula_functor(M)"]) |
|
462 |
done |
|
463 |
||
464 |
lemma (in M_datatypes) formula_abs [simp]: |
|
465 |
"[|M(Z)|] ==> is_formula(M,Z) <-> Z = formula" |
|
466 |
apply (simp add: is_formula_def, safe) |
|
467 |
apply (rule M_equalityI, simp_all) |
|
468 |
done |
|
469 |
||
470 |
||
13397 | 471 |
subsection{*Absoluteness for Some List Operators*} |
472 |
||
13395 | 473 |
subsection{*Absoluteness for @{text \<epsilon>}-Closure: the @{term eclose} Operator*} |
474 |
||
475 |
text{*Re-expresses eclose using "iterates"*} |
|
476 |
lemma eclose_eq_Union: |
|
477 |
"eclose(A) = (\<Union>n\<in>nat. Union^n (A))" |
|
478 |
apply (simp add: eclose_def) |
|
479 |
apply (rule UN_cong) |
|
480 |
apply (rule refl) |
|
481 |
apply (induct_tac n) |
|
482 |
apply (simp add: nat_rec_0) |
|
483 |
apply (simp add: nat_rec_succ) |
|
484 |
done |
|
485 |
||
486 |
constdefs |
|
487 |
is_eclose_n :: "[i=>o,i,i,i] => o" |
|
488 |
"is_eclose_n(M,A,n,Z) == |
|
489 |
\<exists>sn[M]. \<exists>msn[M]. |
|
490 |
successor(M,n,sn) & membership(M,sn,msn) & |
|
491 |
is_wfrec(M, iterates_MH(M, big_union(M), A), msn, n, Z)" |
|
492 |
||
493 |
mem_eclose :: "[i=>o,i,i] => o" |
|
494 |
"mem_eclose(M,A,l) == |
|
495 |
\<exists>n[M]. \<exists>eclosen[M]. |
|
496 |
finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen" |
|
497 |
||
498 |
is_eclose :: "[i=>o,i,i] => o" |
|
499 |
"is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z <-> mem_eclose(M,A,u)" |
|
500 |
||
501 |
||
502 |
locale (open) M_eclose = M_wfrank + |
|
503 |
assumes eclose_replacement1: |
|
504 |
"M(A) ==> iterates_replacement(M, big_union(M), A)" |
|
505 |
and eclose_replacement2: |
|
506 |
"M(A) ==> strong_replacement(M, |
|
507 |
\<lambda>n y. n\<in>nat & |
|
508 |
(\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) & |
|
509 |
is_wfrec(M, iterates_MH(M,big_union(M), A), |
|
510 |
msn, n, y)))" |
|
511 |
||
512 |
lemma (in M_eclose) eclose_replacement2': |
|
513 |
"M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))" |
|
514 |
apply (insert eclose_replacement2 [of A]) |
|
515 |
apply (rule strong_replacement_cong [THEN iffD1]) |
|
516 |
apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]]) |
|
517 |
apply (simp_all add: eclose_replacement1 relativize1_def) |
|
518 |
done |
|
519 |
||
520 |
lemma (in M_eclose) eclose_closed [intro,simp]: |
|
521 |
"M(A) ==> M(eclose(A))" |
|
522 |
apply (insert eclose_replacement1) |
|
523 |
by (simp add: RepFun_closed2 eclose_eq_Union |
|
524 |
eclose_replacement2' relativize1_def |
|
525 |
iterates_closed [of "big_union(M)"]) |
|
526 |
||
527 |
lemma (in M_eclose) is_eclose_n_abs [simp]: |
|
528 |
"[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) <-> Z = Union^n (A)" |
|
529 |
apply (insert eclose_replacement1) |
|
530 |
apply (simp add: is_eclose_n_def relativize1_def nat_into_M |
|
531 |
iterates_abs [of "big_union(M)" _ "Union"]) |
|
532 |
done |
|
533 |
||
534 |
lemma (in M_eclose) mem_eclose_abs [simp]: |
|
535 |
"M(A) ==> mem_eclose(M,A,l) <-> l \<in> eclose(A)" |
|
536 |
apply (insert eclose_replacement1) |
|
537 |
apply (simp add: mem_eclose_def relativize1_def eclose_eq_Union |
|
538 |
iterates_closed [of "big_union(M)"]) |
|
539 |
done |
|
540 |
||
541 |
lemma (in M_eclose) eclose_abs [simp]: |
|
542 |
"[|M(A); M(Z)|] ==> is_eclose(M,A,Z) <-> Z = eclose(A)" |
|
543 |
apply (simp add: is_eclose_def, safe) |
|
544 |
apply (rule M_equalityI, simp_all) |
|
545 |
done |
|
546 |
||
547 |
||
548 |
||
549 |
||
550 |
subsection {*Absoluteness for @{term transrec}*} |
|
551 |
||
552 |
||
553 |
text{* @{term "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"} *} |
|
554 |
constdefs |
|
555 |
||
556 |
is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o" |
|
557 |
"is_transrec(M,MH,a,z) == |
|
558 |
\<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. |
|
559 |
upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) & |
|
560 |
is_wfrec(M,MH,mesa,a,z)" |
|
561 |
||
562 |
transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o" |
|
563 |
"transrec_replacement(M,MH,a) == |
|
564 |
\<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. |
|
565 |
upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) & |
|
566 |
wfrec_replacement(M,MH,mesa)" |
|
567 |
||
568 |
text{*The condition @{term "Ord(i)"} lets us use the simpler |
|
569 |
@{text "trans_wfrec_abs"} rather than @{text "trans_wfrec_abs"}, |
|
570 |
which I haven't even proved yet. *} |
|
571 |
theorem (in M_eclose) transrec_abs: |
|
572 |
"[|Ord(i); M(i); M(z); |
|
573 |
transrec_replacement(M,MH,i); relativize2(M,MH,H); |
|
574 |
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] |
|
575 |
==> is_transrec(M,MH,i,z) <-> z = transrec(i,H)" |
|
576 |
by (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def |
|
577 |
transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel) |
|
578 |
||
579 |
||
580 |
theorem (in M_eclose) transrec_closed: |
|
581 |
"[|Ord(i); M(i); M(z); |
|
582 |
transrec_replacement(M,MH,i); relativize2(M,MH,H); |
|
583 |
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] |
|
584 |
==> M(transrec(i,H))" |
|
585 |
by (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def |
|
586 |
transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel) |
|
587 |
||
588 |
||
589 |
||
13397 | 590 |
subsection{*Absoluteness for the List Operator @{term length}*} |
591 |
constdefs |
|
592 |
||
593 |
is_length :: "[i=>o,i,i,i] => o" |
|
594 |
"is_length(M,A,l,n) == |
|
595 |
\<exists>sn[M]. \<exists>list_n[M]. \<exists>list_sn[M]. |
|
596 |
is_list_N(M,A,n,list_n) & l \<notin> list_n & |
|
597 |
successor(M,n,sn) & is_list_N(M,A,sn,list_sn) & l \<in> list_sn" |
|
598 |
||
599 |
||
600 |
lemma (in M_datatypes) length_abs [simp]: |
|
601 |
"[|M(A); l \<in> list(A); n \<in> nat|] ==> is_length(M,A,l,n) <-> n = length(l)" |
|
602 |
apply (subgoal_tac "M(l) & M(n)") |
|
603 |
prefer 2 apply (blast dest: transM) |
|
604 |
apply (simp add: is_length_def) |
|
605 |
apply (blast intro: list_imp_list_N nat_into_Ord list_N_imp_eq_length |
|
606 |
dest: list_N_imp_length_lt) |
|
607 |
done |
|
608 |
||
609 |
text{*Proof is trivial since @{term length} returns natural numbers.*} |
|
610 |
lemma (in M_triv_axioms) length_closed [intro,simp]: |
|
611 |
"l \<in> list(A) ==> M(length(l))" |
|
612 |
by (simp add: nat_into_M ) |
|
613 |
||
614 |
||
615 |
subsection {*Absoluteness for @{term nth}*} |
|
616 |
||
617 |
lemma nth_eq_hd_iterates_tl [rule_format]: |
|
618 |
"xs \<in> list(A) ==> \<forall>n \<in> nat. nth(n,xs) = hd' (tl'^n (xs))" |
|
619 |
apply (induct_tac xs) |
|
620 |
apply (simp add: iterates_tl_Nil hd'_Nil, clarify) |
|
621 |
apply (erule natE) |
|
622 |
apply (simp add: hd'_Cons) |
|
623 |
apply (simp add: tl'_Cons iterates_commute) |
|
624 |
done |
|
625 |
||
626 |
lemma (in M_axioms) iterates_tl'_closed: |
|
627 |
"[|n \<in> nat; M(x)|] ==> M(tl'^n (x))" |
|
628 |
apply (induct_tac n, simp) |
|
629 |
apply (simp add: tl'_Cons tl'_closed) |
|
630 |
done |
|
631 |
||
632 |
locale (open) M_nth = M_datatypes + |
|
633 |
assumes nth_replacement1: |
|
634 |
"M(xs) ==> iterates_replacement(M, %l t. is_tl(M,l,t), xs)" |
|
635 |
||
636 |
text{*Immediate by type-checking*} |
|
637 |
lemma (in M_datatypes) nth_closed [intro,simp]: |
|
638 |
"[|xs \<in> list(A); n \<in> nat; M(A)|] ==> M(nth(n,xs))" |
|
639 |
apply (case_tac "n < length(xs)") |
|
640 |
apply (blast intro: nth_type transM) |
|
641 |
apply (simp add: not_lt_iff_le nth_eq_0) |
|
642 |
done |
|
643 |
||
644 |
constdefs |
|
645 |
is_nth :: "[i=>o,i,i,i] => o" |
|
646 |
"is_nth(M,n,l,Z) == |
|
647 |
\<exists>X[M]. \<exists>sn[M]. \<exists>msn[M]. |
|
648 |
successor(M,n,sn) & membership(M,sn,msn) & |
|
649 |
is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) & |
|
650 |
is_hd(M,X,Z)" |
|
651 |
||
652 |
lemma (in M_nth) nth_abs [simp]: |
|
653 |
"[|M(A); n \<in> nat; l \<in> list(A); M(Z)|] |
|
654 |
==> is_nth(M,n,l,Z) <-> Z = nth(n,l)" |
|
655 |
apply (subgoal_tac "M(l)") |
|
656 |
prefer 2 apply (blast intro: transM) |
|
657 |
apply (insert nth_replacement1) |
|
658 |
apply (simp add: is_nth_def nth_eq_hd_iterates_tl nat_into_M |
|
659 |
tl'_closed iterates_tl'_closed |
|
660 |
iterates_abs [OF _ relativize1_tl]) |
|
661 |
done |
|
662 |
||
13395 | 663 |
|
13268 | 664 |
end |