(*  Title:      ZF/Constructible/Datatype_absolute.thy

    ID: $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   2002  University of Cambridge

*)

header {*Absoluteness Properties for Recursive Datatypes*}

theory Datatype_absolute = Formula + WF_absolute:

subsection{*The lfp of a continuous function can be expressed as a union*}

constdefs

  directed :: "i=>o"

   "directed(A) == A\<noteq>0 & (\<forall>x\<in>A. \<forall>y\<in>A. x \<union> y \<in> A)"

  contin :: "(i=>i) => o"

   "contin(h) == (\<forall>A. directed(A) --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))"

lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D"

apply (induct_tac n) 

 apply (simp_all add: bnd_mono_def, blast) 

done

lemma bnd_mono_increasing [rule_format]:

     "[|i \<in> nat; j \<in> nat; bnd_mono(D,h)|] ==> i \<le> j --> h^i(0) \<subseteq> h^j(0)"

apply (rule_tac m=i and n=j in diff_induct, simp_all)

apply (blast del: subsetI

	     intro: bnd_mono_iterates_subset bnd_monoD2 [of concl: h]) 

done

lemma directed_iterates: "bnd_mono(D,h) ==> directed({h^n (0). n\<in>nat})"

apply (simp add: directed_def, clarify) 

apply (rename_tac i j)

apply (rule_tac x="i \<union> j" in bexI) 

apply (rule_tac i = i and j = j in Ord_linear_le)

apply (simp_all add: subset_Un_iff [THEN iffD1] le_imp_subset

                     subset_Un_iff2 [THEN iffD1])

apply (simp_all add: subset_Un_iff [THEN iff_sym] bnd_mono_increasing

                     subset_Un_iff2 [THEN iff_sym])

done

lemma contin_iterates_eq: 

    "[|bnd_mono(D, h); contin(h)|] 

     ==> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))"

apply (simp add: contin_def directed_iterates) 

apply (rule trans) 

apply (rule equalityI) 

 apply (simp_all add: UN_subset_iff)

 apply safe

 apply (erule_tac [2] natE) 

  apply (rule_tac a="succ(x)" in UN_I) 

   apply simp_all 

apply blast 

done

lemma lfp_subset_Union:

     "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) <= (\<Union>n\<in>nat. h^n(0))"

apply (rule lfp_lowerbound) 

 apply (simp add: contin_iterates_eq) 

apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff) 

done

lemma Union_subset_lfp:

     "bnd_mono(D,h) ==> (\<Union>n\<in>nat. h^n(0)) <= lfp(D,h)"

apply (simp add: UN_subset_iff)

apply (rule ballI)  

apply (induct_tac n, simp_all) 

apply (rule subset_trans [of _ "h(lfp(D,h))"])

 apply (blast dest: bnd_monoD2 [OF _ _ lfp_subset])  

apply (erule lfp_lemma2) 

done

lemma lfp_eq_Union:

     "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) = (\<Union>n\<in>nat. h^n(0))"

by (blast del: subsetI 

          intro: lfp_subset_Union Union_subset_lfp)

subsubsection{*Some Standard Datatype Constructions Preserve Continuity*}

lemma contin_imp_mono: "[|X\<subseteq>Y; contin(F)|] ==> F(X) \<subseteq> F(Y)"

apply (simp add: contin_def) 

apply (drule_tac x="{X,Y}" in spec) 

apply (simp add: directed_def subset_Un_iff2 Un_commute) 

done

lemma sum_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) + G(X))"

by (simp add: contin_def, blast)

lemma prod_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) * G(X))" 

apply (subgoal_tac "\<forall>B C. F(B) \<subseteq> F(B \<union> C)")

 prefer 2 apply (simp add: Un_upper1 contin_imp_mono) 

apply (subgoal_tac "\<forall>B C. G(C) \<subseteq> G(B \<union> C)")

 prefer 2 apply (simp add: Un_upper2 contin_imp_mono) 

apply (simp add: contin_def, clarify) 

apply (rule equalityI) 

 prefer 2 apply blast 

apply clarify 

apply (rename_tac B C) 

apply (rule_tac a="B \<union> C" in UN_I) 

 apply (simp add: directed_def, blast)  

done

lemma const_contin: "contin(\<lambda>X. A)"

by (simp add: contin_def directed_def)

lemma id_contin: "contin(\<lambda>X. X)"

by (simp add: contin_def)

subsection {*Absoluteness for "Iterates"*}

constdefs

  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"

   "iterates_MH(M,isF,v,n,g,z) ==

        is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),

                    n, z)"

  iterates_replacement :: "[i=>o, [i,i]=>o, i] => o"

   "iterates_replacement(M,isF,v) ==

      \<forall>n[M]. n\<in>nat --> 

         wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))"

lemma (in M_basic) iterates_MH_abs:

  "[| relativize1(M,isF,F); M(n); M(g); M(z) |] 

   ==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, \<lambda>m. F(g`m), n)"

by (simp add: nat_case_abs [of _ "\<lambda>m. F(g ` m)"]

              relativize1_def iterates_MH_def)  

lemma (in M_basic) iterates_imp_wfrec_replacement:

  "[|relativize1(M,isF,F); n \<in> nat; iterates_replacement(M,isF,v)|] 

   ==> wfrec_replacement(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n), 

                       Memrel(succ(n)))" 

by (simp add: iterates_replacement_def iterates_MH_abs)

theorem (in M_trancl) iterates_abs:

  "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);

      n \<in> nat; M(v); M(z); \<forall>x[M]. M(F(x)) |] 

   ==> is_wfrec(M, iterates_MH(M,isF,v), Memrel(succ(n)), n, z) <->

       z = iterates(F,n,v)" 

apply (frule iterates_imp_wfrec_replacement, assumption+)

apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M

                 relativize2_def iterates_MH_abs 

                 iterates_nat_def recursor_def transrec_def 

                 eclose_sing_Ord_eq nat_into_M

         trans_wfrec_abs [of _ _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])

done

lemma (in M_wfrank) iterates_closed [intro,simp]:

  "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);

      n \<in> nat; M(v); \<forall>x[M]. M(F(x)) |] 

   ==> M(iterates(F,n,v))"

apply (frule iterates_imp_wfrec_replacement, assumption+)

apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M

                 relativize2_def iterates_MH_abs 

                 iterates_nat_def recursor_def transrec_def 

                 eclose_sing_Ord_eq nat_into_M

         trans_wfrec_closed [of _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])

done

subsection {*lists without univ*}

lemmas datatype_univs = Inl_in_univ Inr_in_univ 

                        Pair_in_univ nat_into_univ A_into_univ 

lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"

apply (rule bnd_monoI)

 apply (intro subset_refl zero_subset_univ A_subset_univ 

	      sum_subset_univ Sigma_subset_univ) 

apply (rule subset_refl sum_mono Sigma_mono | assumption)+

done

lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"

by (intro sum_contin prod_contin id_contin const_contin) 

text{*Re-expresses lists using sum and product*}

lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"

apply (simp add: list_def) 

apply (rule equalityI) 

 apply (rule lfp_lowerbound) 

  prefer 2 apply (rule lfp_subset)

 apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])

 apply (simp add: Nil_def Cons_def)

 apply blast 

txt{*Opposite inclusion*}

apply (rule lfp_lowerbound) 

 prefer 2 apply (rule lfp_subset) 

apply (clarify, subst lfp_unfold [OF list.bnd_mono]) 

apply (simp add: Nil_def Cons_def)

apply (blast intro: datatype_univs

             dest: lfp_subset [THEN subsetD])

done

text{*Re-expresses lists using "iterates", no univ.*}

lemma list_eq_Union:

     "list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"

by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)

constdefs

  is_list_functor :: "[i=>o,i,i,i] => o"

    "is_list_functor(M,A,X,Z) == 

        \<exists>n1[M]. \<exists>AX[M]. 

         number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)"

lemma (in M_basic) list_functor_abs [simp]: 

     "[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) <-> (Z = {0} + A*X)"

by (simp add: is_list_functor_def singleton_0 nat_into_M)

subsection {*formulas without univ*}

lemma formula_fun_bnd_mono:

     "bnd_mono(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))"

apply (rule bnd_monoI)

 apply (intro subset_refl zero_subset_univ A_subset_univ 

	      sum_subset_univ Sigma_subset_univ nat_subset_univ) 

apply (rule subset_refl sum_mono Sigma_mono | assumption)+

done

lemma formula_fun_contin:

     "contin(\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))"

by (intro sum_contin prod_contin id_contin const_contin) 

text{*Re-expresses formulas using sum and product*}

lemma formula_eq_lfp2:

    "formula = lfp(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))"

apply (simp add: formula_def) 

apply (rule equalityI) 

 apply (rule lfp_lowerbound) 

  prefer 2 apply (rule lfp_subset)

 apply (clarify, subst lfp_unfold [OF formula_fun_bnd_mono])

 apply (simp add: Member_def Equal_def Nand_def Forall_def)

 apply blast 

txt{*Opposite inclusion*}

apply (rule lfp_lowerbound) 

 prefer 2 apply (rule lfp_subset, clarify) 

apply (subst lfp_unfold [OF formula.bnd_mono, simplified]) 

apply (simp add: Member_def Equal_def Nand_def Forall_def)  

apply (elim sumE SigmaE, simp_all) 

apply (blast intro: datatype_univs dest: lfp_subset [THEN subsetD])+  

done

text{*Re-expresses formulas using "iterates", no univ.*}

lemma formula_eq_Union:

     "formula = 

      (\<Union>n\<in>nat. (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)) ^ n (0))"

by (simp add: formula_eq_lfp2 lfp_eq_Union formula_fun_bnd_mono 

              formula_fun_contin)

constdefs

  is_formula_functor :: "[i=>o,i,i] => o"

    "is_formula_functor(M,X,Z) == 

        \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M]. 

          omega(M,nat') & cartprod(M,nat',nat',natnat) & 

          is_sum(M,natnat,natnat,natnatsum) &

          cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & 

          is_sum(M,natnatsum,X3,Z)"

lemma (in M_basic) formula_functor_abs [simp]: 

     "[| M(X); M(Z) |] 

      ==> is_formula_functor(M,X,Z) <-> 

          Z = ((nat*nat) + (nat*nat)) + (X*X + X)"

by (simp add: is_formula_functor_def) 

subsection{*@{term M} Contains the List and Formula Datatypes*}

constdefs

  list_N :: "[i,i] => i"

    "list_N(A,n) == (\<lambda>X. {0} + A * X)^n (0)"

lemma Nil_in_list_N [simp]: "[] \<in> list_N(A,succ(n))"

by (simp add: list_N_def Nil_def)

lemma Cons_in_list_N [simp]:

     "Cons(a,l) \<in> list_N(A,succ(n)) <-> a\<in>A & l \<in> list_N(A,n)"

by (simp add: list_N_def Cons_def) 

text{*These two aren't simprules because they reveal the underlying

list representation.*}

lemma list_N_0: "list_N(A,0) = 0"

by (simp add: list_N_def)

lemma list_N_succ: "list_N(A,succ(n)) = {0} + A * (list_N(A,n))"

by (simp add: list_N_def)

lemma list_N_imp_list:

  "[| l \<in> list_N(A,n); n \<in> nat |] ==> l \<in> list(A)"

by (force simp add: list_eq_Union list_N_def)

lemma list_N_imp_length_lt [rule_format]:

     "n \<in> nat ==> \<forall>l \<in> list_N(A,n). length(l) < n"

apply (induct_tac n)  

apply (auto simp add: list_N_0 list_N_succ 

                      Nil_def [symmetric] Cons_def [symmetric]) 

done

lemma list_imp_list_N [rule_format]:

     "l \<in> list(A) ==> \<forall>n\<in>nat. length(l) < n --> l \<in> list_N(A, n)"

apply (induct_tac l)

apply (force elim: natE)+

done

lemma list_N_imp_eq_length:

      "[|n \<in> nat; l \<notin> list_N(A, n); l \<in> list_N(A, succ(n))|] 

       ==> n = length(l)"

apply (rule le_anti_sym) 

 prefer 2 apply (simp add: list_N_imp_length_lt) 

apply (frule list_N_imp_list, simp)

apply (simp add: not_lt_iff_le [symmetric]) 

apply (blast intro: list_imp_list_N) 

done

text{*Express @{term list_rec} without using @{term rank} or @{term Vset},

neither of which is absolute.*}

lemma (in M_trivial) list_rec_eq:

  "l \<in> list(A) ==>

   list_rec(a,g,l) = 

   transrec (succ(length(l)),

      \<lambda>x h. Lambda (list(A),

                    list_case' (a, 

                           \<lambda>a l. g(a, l, h ` succ(length(l)) ` l)))) ` l"

apply (induct_tac l) 

apply (subst transrec, simp) 

apply (subst transrec) 

apply (simp add: list_imp_list_N) 

done

constdefs

  is_list_N :: "[i=>o,i,i,i] => o"

    "is_list_N(M,A,n,Z) == 

      \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M]. 

       empty(M,zero) & 

       successor(M,n,sn) & membership(M,sn,msn) &

       is_wfrec(M, iterates_MH(M, is_list_functor(M,A),zero), msn, n, Z)"

  mem_list :: "[i=>o,i,i] => o"

    "mem_list(M,A,l) == 

      \<exists>n[M]. \<exists>listn[M]. 

       finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l \<in> listn"

  is_list :: "[i=>o,i,i] => o"

    "is_list(M,A,Z) == \<forall>l[M]. l \<in> Z <-> mem_list(M,A,l)"

subsubsection{*Towards Absoluteness of @{term formula_rec}*}

consts   depth :: "i=>i"

primrec

  "depth(Member(x,y)) = 0"

  "depth(Equal(x,y))  = 0"

  "depth(Nand(p,q)) = succ(depth(p) \<union> depth(q))"

  "depth(Forall(p)) = succ(depth(p))"

lemma depth_type [TC]: "p \<in> formula ==> depth(p) \<in> nat"

by (induct_tac p, simp_all) 

constdefs

  formula_N :: "i => i"

    "formula_N(n) == (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)) ^ n (0)"

lemma Member_in_formula_N [simp]:

     "Member(x,y) \<in> formula_N(succ(n)) <-> x \<in> nat & y \<in> nat"

by (simp add: formula_N_def Member_def) 

lemma Equal_in_formula_N [simp]:

     "Equal(x,y) \<in> formula_N(succ(n)) <-> x \<in> nat & y \<in> nat"

by (simp add: formula_N_def Equal_def) 

lemma Nand_in_formula_N [simp]:

     "Nand(x,y) \<in> formula_N(succ(n)) <-> x \<in> formula_N(n) & y \<in> formula_N(n)"

by (simp add: formula_N_def Nand_def) 

lemma Forall_in_formula_N [simp]:

     "Forall(x) \<in> formula_N(succ(n)) <-> x \<in> formula_N(n)"

by (simp add: formula_N_def Forall_def) 

text{*These two aren't simprules because they reveal the underlying

formula representation.*}

lemma formula_N_0: "formula_N(0) = 0"

by (simp add: formula_N_def)

lemma formula_N_succ:

     "formula_N(succ(n)) = 

      ((nat*nat) + (nat*nat)) + (formula_N(n) * formula_N(n) + formula_N(n))"

by (simp add: formula_N_def)

lemma formula_N_imp_formula:

  "[| p \<in> formula_N(n); n \<in> nat |] ==> p \<in> formula"

by (force simp add: formula_eq_Union formula_N_def)

lemma formula_N_imp_depth_lt [rule_format]:

     "n \<in> nat ==> \<forall>p \<in> formula_N(n). depth(p) < n"

apply (induct_tac n)  

apply (auto simp add: formula_N_0 formula_N_succ 

                      depth_type formula_N_imp_formula Un_least_lt_iff

                      Member_def [symmetric] Equal_def [symmetric]

                      Nand_def [symmetric] Forall_def [symmetric]) 

done

lemma formula_imp_formula_N [rule_format]:

     "p \<in> formula ==> \<forall>n\<in>nat. depth(p) < n --> p \<in> formula_N(n)"

apply (induct_tac p)

apply (simp_all add: succ_Un_distrib Un_least_lt_iff) 

apply (force elim: natE)+

done

lemma formula_N_imp_eq_depth:

      "[|n \<in> nat; p \<notin> formula_N(n); p \<in> formula_N(succ(n))|] 

       ==> n = depth(p)"

apply (rule le_anti_sym) 

 prefer 2 apply (simp add: formula_N_imp_depth_lt) 

apply (frule formula_N_imp_formula, simp)

apply (simp add: not_lt_iff_le [symmetric]) 

apply (blast intro: formula_imp_formula_N) 

done

lemma formula_N_mono [rule_format]:

  "[| m \<in> nat; n \<in> nat |] ==> m\<le>n --> formula_N(m) \<subseteq> formula_N(n)"

apply (rule_tac m = m and n = n in diff_induct)

apply (simp_all add: formula_N_0 formula_N_succ, blast) 

done

lemma formula_N_distrib:

  "[| m \<in> nat; n \<in> nat |] ==> formula_N(m \<union> n) = formula_N(m) \<union> formula_N(n)"

apply (rule_tac i = m and j = n in Ord_linear_le, auto) 

apply (simp_all add: subset_Un_iff [THEN iffD1] subset_Un_iff2 [THEN iffD1] 

                     le_imp_subset formula_N_mono)

done

constdefs

  is_formula_N :: "[i=>o,i,i] => o"

    "is_formula_N(M,n,Z) == 

      \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M]. 

       empty(M,zero) & 

       successor(M,n,sn) & membership(M,sn,msn) &

       is_wfrec(M, iterates_MH(M, is_formula_functor(M),zero), msn, n, Z)"

constdefs

  mem_formula :: "[i=>o,i] => o"

    "mem_formula(M,p) == 

      \<exists>n[M]. \<exists>formn[M]. 

       finite_ordinal(M,n) & is_formula_N(M,n,formn) & p \<in> formn"

  is_formula :: "[i=>o,i] => o"

    "is_formula(M,Z) == \<forall>p[M]. p \<in> Z <-> mem_formula(M,p)"

locale M_datatypes = M_wfrank +

 assumes list_replacement1: 

   "M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)"

  and list_replacement2: 

   "M(A) ==> strong_replacement(M, 

         \<lambda>n y. n\<in>nat & 

               (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &

               is_wfrec(M, iterates_MH(M,is_list_functor(M,A), 0), 

                        msn, n, y)))"

  and formula_replacement1: 

   "iterates_replacement(M, is_formula_functor(M), 0)"

  and formula_replacement2: 

   "strong_replacement(M, 

         \<lambda>n y. n\<in>nat & 

               (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &

               is_wfrec(M, iterates_MH(M,is_formula_functor(M), 0), 

                        msn, n, y)))"

  and nth_replacement:

   "M(l) ==> iterates_replacement(M, %l t. is_tl(M,l,t), l)"

subsubsection{*Absoluteness of the List Construction*}

lemma (in M_datatypes) list_replacement2': 

  "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"

apply (insert list_replacement2 [of A]) 

apply (rule strong_replacement_cong [THEN iffD1])  

apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]]) 

apply (simp_all add: list_replacement1 relativize1_def) 

done

lemma (in M_datatypes) list_closed [intro,simp]:

     "M(A) ==> M(list(A))"

apply (insert list_replacement1)

by  (simp add: RepFun_closed2 list_eq_Union 

               list_replacement2' relativize1_def

               iterates_closed [of "is_list_functor(M,A)"])

text{*WARNING: use only with @{text "dest:"} or with variables fixed!*}

lemmas (in M_datatypes) list_into_M = transM [OF _ list_closed]

lemma (in M_datatypes) list_N_abs [simp]:

     "[|M(A); n\<in>nat; M(Z)|] 

      ==> is_list_N(M,A,n,Z) <-> Z = list_N(A,n)"

apply (insert list_replacement1)

apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M

                 iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"])

done

lemma (in M_datatypes) list_N_closed [intro,simp]:

     "[|M(A); n\<in>nat|] ==> M(list_N(A,n))"

apply (insert list_replacement1)

apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M

                 iterates_closed [of "is_list_functor(M,A)"])

done

lemma (in M_datatypes) mem_list_abs [simp]:

     "M(A) ==> mem_list(M,A,l) <-> l \<in> list(A)"

apply (insert list_replacement1)

apply (simp add: mem_list_def list_N_def relativize1_def list_eq_Union

                 iterates_closed [of "is_list_functor(M,A)"]) 

done

lemma (in M_datatypes) list_abs [simp]:

     "[|M(A); M(Z)|] ==> is_list(M,A,Z) <-> Z = list(A)"

apply (simp add: is_list_def, safe)

apply (rule M_equalityI, simp_all)

done

subsubsection{*Absoluteness of Formulas*}

lemma (in M_datatypes) formula_replacement2': 

  "strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))^n (0))"

apply (insert formula_replacement2) 

apply (rule strong_replacement_cong [THEN iffD1])  

apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]]) 

apply (simp_all add: formula_replacement1 relativize1_def) 

done

lemma (in M_datatypes) formula_closed [intro,simp]:

     "M(formula)"

apply (insert formula_replacement1)

apply  (simp add: RepFun_closed2 formula_eq_Union 

                  formula_replacement2' relativize1_def

                  iterates_closed [of "is_formula_functor(M)"])

done

lemmas (in M_datatypes) formula_into_M = transM [OF _ formula_closed]

lemma (in M_datatypes) formula_N_abs [simp]:

     "[|n\<in>nat; M(Z)|] 

      ==> is_formula_N(M,n,Z) <-> Z = formula_N(n)"

apply (insert formula_replacement1)

apply (simp add: is_formula_N_def formula_N_def relativize1_def nat_into_M

                 iterates_abs [of "is_formula_functor(M)" _ 

                                  "\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)"])

done

lemma (in M_datatypes) formula_N_closed [intro,simp]:

     "n\<in>nat ==> M(formula_N(n))"

apply (insert formula_replacement1)

apply (simp add: is_formula_N_def formula_N_def relativize1_def nat_into_M

                 iterates_closed [of "is_formula_functor(M)"])

done

lemma (in M_datatypes) mem_formula_abs [simp]:

     "mem_formula(M,l) <-> l \<in> formula"

apply (insert formula_replacement1)

apply (simp add: mem_formula_def relativize1_def formula_eq_Union formula_N_def

                 iterates_closed [of "is_formula_functor(M)"]) 

done

lemma (in M_datatypes) formula_abs [simp]:

     "[|M(Z)|] ==> is_formula(M,Z) <-> Z = formula"

apply (simp add: is_formula_def, safe)

apply (rule M_equalityI, simp_all)

done

subsection{*Absoluteness for Some List Operators*}

subsection{*Absoluteness for @{text \<epsilon>}-Closure: the @{term eclose} Operator*}

text{*Re-expresses eclose using "iterates"*}

lemma eclose_eq_Union:

     "eclose(A) = (\<Union>n\<in>nat. Union^n (A))"

apply (simp add: eclose_def) 

apply (rule UN_cong) 

apply (rule refl)

apply (induct_tac n)

apply (simp add: nat_rec_0)  

apply (simp add: nat_rec_succ) 

done

constdefs

  is_eclose_n :: "[i=>o,i,i,i] => o"

    "is_eclose_n(M,A,n,Z) == 

      \<exists>sn[M]. \<exists>msn[M]. 

       successor(M,n,sn) & membership(M,sn,msn) &

       is_wfrec(M, iterates_MH(M, big_union(M), A), msn, n, Z)"

  mem_eclose :: "[i=>o,i,i] => o"

    "mem_eclose(M,A,l) == 

      \<exists>n[M]. \<exists>eclosen[M]. 

       finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen"

  is_eclose :: "[i=>o,i,i] => o"

    "is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z <-> mem_eclose(M,A,u)"

locale M_eclose = M_datatypes +

 assumes eclose_replacement1: 

   "M(A) ==> iterates_replacement(M, big_union(M), A)"

  and eclose_replacement2: 

   "M(A) ==> strong_replacement(M, 

         \<lambda>n y. n\<in>nat & 

               (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &

               is_wfrec(M, iterates_MH(M,big_union(M), A), 

                        msn, n, y)))"

lemma (in M_eclose) eclose_replacement2': 

  "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))"

apply (insert eclose_replacement2 [of A]) 

apply (rule strong_replacement_cong [THEN iffD1])  

apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]]) 

apply (simp_all add: eclose_replacement1 relativize1_def) 

done

lemma (in M_eclose) eclose_closed [intro,simp]:

     "M(A) ==> M(eclose(A))"

apply (insert eclose_replacement1)

by  (simp add: RepFun_closed2 eclose_eq_Union 

               eclose_replacement2' relativize1_def

               iterates_closed [of "big_union(M)"])

lemma (in M_eclose) is_eclose_n_abs [simp]:

     "[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) <-> Z = Union^n (A)"

apply (insert eclose_replacement1)

apply (simp add: is_eclose_n_def relativize1_def nat_into_M

                 iterates_abs [of "big_union(M)" _ "Union"])

done

lemma (in M_eclose) mem_eclose_abs [simp]:

     "M(A) ==> mem_eclose(M,A,l) <-> l \<in> eclose(A)"

apply (insert eclose_replacement1)

apply (simp add: mem_eclose_def relativize1_def eclose_eq_Union

                 iterates_closed [of "big_union(M)"]) 

done

lemma (in M_eclose) eclose_abs [simp]:

     "[|M(A); M(Z)|] ==> is_eclose(M,A,Z) <-> Z = eclose(A)"

apply (simp add: is_eclose_def, safe)

apply (rule M_equalityI, simp_all)

done

subsection {*Absoluteness for @{term transrec}*}

text{* @{term "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"} *}

constdefs

  is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o"

   "is_transrec(M,MH,a,z) == 

      \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. 

       upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &

       is_wfrec(M,MH,mesa,a,z)"

  transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o"

   "transrec_replacement(M,MH,a) ==

      \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. 

       upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &

       wfrec_replacement(M,MH,mesa)"

text{*The condition @{term "Ord(i)"} lets us use the simpler 

  @{text "trans_wfrec_abs"} rather than @{text "trans_wfrec_abs"},

  which I haven't even proved yet. *}

theorem (in M_eclose) transrec_abs:

  "[|transrec_replacement(M,MH,i);  relativize2(M,MH,H);

     Ord(i);  M(i);  M(z);

     \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 

   ==> is_transrec(M,MH,i,z) <-> z = transrec(i,H)" 

apply (rotate_tac 2) 

apply (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def

       transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)

done

theorem (in M_eclose) transrec_closed:

     "[|transrec_replacement(M,MH,i);  relativize2(M,MH,H);

	Ord(i);  M(i);  

	\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 

      ==> M(transrec(i,H))"

apply (rotate_tac 2) 

apply (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def

       transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)

done

text{*Helps to prove instances of @{term transrec_replacement}*}

lemma (in M_eclose) transrec_replacementI: 

   "[|M(a);

    strong_replacement (M, 

                  \<lambda>x z. \<exists>y[M]. pair(M, x, y, z) \<and> is_wfrec(M,MH,Memrel(eclose({a})),x,y))|]

    ==> transrec_replacement(M,MH,a)"

by (simp add: transrec_replacement_def wfrec_replacement_def) 

subsection{*Absoluteness for the List Operator @{term length}*}

constdefs

  is_length :: "[i=>o,i,i,i] => o"

    "is_length(M,A,l,n) == 

       \<exists>sn[M]. \<exists>list_n[M]. \<exists>list_sn[M]. 

        is_list_N(M,A,n,list_n) & l \<notin> list_n &

        successor(M,n,sn) & is_list_N(M,A,sn,list_sn) & l \<in> list_sn"

lemma (in M_datatypes) length_abs [simp]:

     "[|M(A); l \<in> list(A); n \<in> nat|] ==> is_length(M,A,l,n) <-> n = length(l)"

apply (subgoal_tac "M(l) & M(n)")

 prefer 2 apply (blast dest: transM)  

apply (simp add: is_length_def)

apply (blast intro: list_imp_list_N nat_into_Ord list_N_imp_eq_length

             dest: list_N_imp_length_lt)

done

text{*Proof is trivial since @{term length} returns natural numbers.*}

lemma (in M_trivial) length_closed [intro,simp]:

     "l \<in> list(A) ==> M(length(l))"

by (simp add: nat_into_M) 

subsection {*Absoluteness for @{term nth}*}

lemma nth_eq_hd_iterates_tl [rule_format]:

     "xs \<in> list(A) ==> \<forall>n \<in> nat. nth(n,xs) = hd' (tl'^n (xs))"

apply (induct_tac xs) 

apply (simp add: iterates_tl_Nil hd'_Nil, clarify) 

apply (erule natE)

apply (simp add: hd'_Cons) 

apply (simp add: tl'_Cons iterates_commute) 

done

lemma (in M_basic) iterates_tl'_closed:

     "[|n \<in> nat; M(x)|] ==> M(tl'^n (x))"

apply (induct_tac n, simp) 

apply (simp add: tl'_Cons tl'_closed) 

done

text{*Immediate by type-checking*}

lemma (in M_datatypes) nth_closed [intro,simp]:

     "[|xs \<in> list(A); n \<in> nat; M(A)|] ==> M(nth(n,xs))" 

apply (case_tac "n < length(xs)")

 apply (blast intro: nth_type transM)

apply (simp add: not_lt_iff_le nth_eq_0)

done

constdefs

  is_nth :: "[i=>o,i,i,i] => o"

    "is_nth(M,n,l,Z) == 

      \<exists>X[M]. \<exists>sn[M]. \<exists>msn[M]. 

       successor(M,n,sn) & membership(M,sn,msn) &

       is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) &

       is_hd(M,X,Z)"

lemma (in M_datatypes) nth_abs [simp]:

     "[|M(A); n \<in> nat; l \<in> list(A); M(Z)|] 

      ==> is_nth(M,n,l,Z) <-> Z = nth(n,l)"

apply (subgoal_tac "M(l)") 

 prefer 2 apply (blast intro: transM)

apply (simp add: is_nth_def nth_eq_hd_iterates_tl nat_into_M

                 tl'_closed iterates_tl'_closed 

                 iterates_abs [OF _ relativize1_tl] nth_replacement)

done

subsection{*Relativization and Absoluteness for the @{term formula} Constructors*}

constdefs

  is_Member :: "[i=>o,i,i,i] => o"

     --{* because @{term "Member(x,y) \<equiv> Inl(Inl(\<langle>x,y\<rangle>))"}*}

    "is_Member(M,x,y,Z) ==

	\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)"

lemma (in M_trivial) Member_abs [simp]:

     "[|M(x); M(y); M(Z)|] ==> is_Member(M,x,y,Z) <-> (Z = Member(x,y))"

by (simp add: is_Member_def Member_def)

lemma (in M_trivial) Member_in_M_iff [iff]:

     "M(Member(x,y)) <-> M(x) & M(y)"

by (simp add: Member_def) 

constdefs

  is_Equal :: "[i=>o,i,i,i] => o"

     --{* because @{term "Equal(x,y) \<equiv> Inl(Inr(\<langle>x,y\<rangle>))"}*}

    "is_Equal(M,x,y,Z) ==

	\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)"

lemma (in M_trivial) Equal_abs [simp]:

     "[|M(x); M(y); M(Z)|] ==> is_Equal(M,x,y,Z) <-> (Z = Equal(x,y))"

by (simp add: is_Equal_def Equal_def)

lemma (in M_trivial) Equal_in_M_iff [iff]: "M(Equal(x,y)) <-> M(x) & M(y)"

by (simp add: Equal_def) 

constdefs

  is_Nand :: "[i=>o,i,i,i] => o"

     --{* because @{term "Nand(x,y) \<equiv> Inr(Inl(\<langle>x,y\<rangle>))"}*}

    "is_Nand(M,x,y,Z) ==

	\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)"

lemma (in M_trivial) Nand_abs [simp]:

     "[|M(x); M(y); M(Z)|] ==> is_Nand(M,x,y,Z) <-> (Z = Nand(x,y))"

by (simp add: is_Nand_def Nand_def)

lemma (in M_trivial) Nand_in_M_iff [iff]: "M(Nand(x,y)) <-> M(x) & M(y)"

by (simp add: Nand_def) 

constdefs

  is_Forall :: "[i=>o,i,i] => o"

     --{* because @{term "Forall(x) \<equiv> Inr(Inr(p))"}*}

    "is_Forall(M,p,Z) == \<exists>u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)"

lemma (in M_trivial) Forall_abs [simp]:

     "[|M(x); M(Z)|] ==> is_Forall(M,x,Z) <-> (Z = Forall(x))"

by (simp add: is_Forall_def Forall_def)

lemma (in M_trivial) Forall_in_M_iff [iff]: "M(Forall(x)) <-> M(x)"

by (simp add: Forall_def)

subsection {*Absoluteness for @{term formula_rec}*}

subsubsection{*@{term is_formula_case}: relativization of @{term formula_case}*}

constdefs

 is_formula_case :: 

    "[i=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i]=>o, i, i] => o"

  --{*no constraint on non-formulas*}

  "is_formula_case(M, is_a, is_b, is_c, is_d, p, z) == 

      (\<forall>x[M]. \<forall>y[M]. finite_ordinal(M,x) --> finite_ordinal(M,y) --> 

                      is_Member(M,x,y,p) --> is_a(x,y,z)) &

      (\<forall>x[M]. \<forall>y[M]. finite_ordinal(M,x) --> finite_ordinal(M,y) --> 

                      is_Equal(M,x,y,p) --> is_b(x,y,z)) &

      (\<forall>x[M]. \<forall>y[M]. mem_formula(M,x) --> mem_formula(M,y) --> 

                     is_Nand(M,x,y,p) --> is_c(x,y,z)) &

      (\<forall>x[M]. mem_formula(M,x) --> is_Forall(M,x,p) --> is_d(x,z))"

lemma (in M_datatypes) formula_case_abs [simp]: 

     "[| Relativize2(M,nat,nat,is_a,a); Relativize2(M,nat,nat,is_b,b); 

         Relativize2(M,formula,formula,is_c,c); Relativize1(M,formula,is_d,d); 

         p \<in> formula; M(z) |] 

      ==> is_formula_case(M,is_a,is_b,is_c,is_d,p,z) <-> 

          z = formula_case(a,b,c,d,p)"

apply (simp add: formula_into_M is_formula_case_def)

apply (erule formula.cases) 

   apply (simp_all add: Relativize1_def Relativize2_def) 

done

lemma (in M_datatypes) formula_case_closed [intro,simp]:

  "[|p \<in> formula; 

     \<forall>x[M]. \<forall>y[M]. x\<in>nat --> y\<in>nat --> M(a(x,y)); 

     \<forall>x[M]. \<forall>y[M]. x\<in>nat --> y\<in>nat --> M(b(x,y)); 

     \<forall>x[M]. \<forall>y[M]. x\<in>formula --> y\<in>formula --> M(c(x,y)); 

     \<forall>x[M]. x\<in>formula --> M(d(x))|] ==> M(formula_case(a,b,c,d,p))"

by (erule formula.cases, simp_all) 

subsection{*Absoluteness for the Formula Operator @{term depth}*}

constdefs

  is_depth :: "[i=>o,i,i] => o"

    "is_depth(M,p,n) == 

       \<exists>sn[M]. \<exists>formula_n[M]. \<exists>formula_sn[M]. 

        is_formula_N(M,n,formula_n) & p \<notin> formula_n &

        successor(M,n,sn) & is_formula_N(M,sn,formula_sn) & p \<in> formula_sn"

lemma (in M_datatypes) depth_abs [simp]:

     "[|p \<in> formula; n \<in> nat|] ==> is_depth(M,p,n) <-> n = depth(p)"

apply (subgoal_tac "M(p) & M(n)")

 prefer 2 apply (blast dest: transM)  

apply (simp add: is_depth_def)

apply (blast intro: formula_imp_formula_N nat_into_Ord formula_N_imp_eq_depth

             dest: formula_N_imp_depth_lt)

done

text{*Proof is trivial since @{term depth} returns natural numbers.*}

lemma (in M_trivial) depth_closed [intro,simp]:

     "p \<in> formula ==> M(depth(p))"

by (simp add: nat_into_M) 

subsection {*Absoluteness for @{term formula_rec}*}

constdefs

  formula_rec_case :: "[[i,i]=>i, [i,i]=>i, [i,i,i,i]=>i, [i,i]=>i, i, i] => i"

    --{* the instance of @{term formula_case} in @{term formula_rec}*}

   "formula_rec_case(a,b,c,d,h) ==

        formula_case (a, b,

                \<lambda>u v. c(u, v, h ` succ(depth(u)) ` u, 

                              h ` succ(depth(v)) ` v),

                \<lambda>u. d(u, h ` succ(depth(u)) ` u))"

  is_formula_rec :: "[i=>o, [i,i,i]=>o, i, i] => o"

    --{* predicate to relativize the functional @{term formula_rec}*}

   "is_formula_rec(M,MH,p,z)  ==

      \<exists>dp[M]. \<exists>i[M]. \<exists>f[M]. finite_ordinal(M,dp) & is_depth(M,p,dp) & 

             successor(M,dp,i) & fun_apply(M,f,p,z) & is_transrec(M,MH,i,f)"

text{*Unfold @{term formula_rec} to @{term formula_rec_case}.

     Express @{term formula_rec} without using @{term rank} or @{term Vset},

neither of which is absolute.*}

lemma (in M_trivial) formula_rec_eq:

  "p \<in> formula ==>

   formula_rec(a,b,c,d,p) = 

   transrec (succ(depth(p)),

             \<lambda>x h. Lambda (formula, formula_rec_case(a,b,c,d,h))) ` p"

apply (simp add: formula_rec_case_def)

apply (induct_tac p)

   txt{*Base case for @{term Member}*}

   apply (subst transrec, simp add: formula.intros) 

  txt{*Base case for @{term Equal}*}

  apply (subst transrec, simp add: formula.intros)

 txt{*Inductive step for @{term Nand}*}

 apply (subst transrec) 

 apply (simp add: succ_Un_distrib formula.intros)

txt{*Inductive step for @{term Forall}*}

apply (subst transrec) 

apply (simp add: formula_imp_formula_N formula.intros) 

done

text{*Sufficient conditions to relative the instance of @{term formula_case}

      in @{term formula_rec}*}

lemma (in M_datatypes) Relativize1_formula_rec_case:

     "[|Relativize2(M, nat, nat, is_a, a);

        Relativize2(M, nat, nat, is_b, b);

        Relativize2 (M, formula, formula, 

           is_c, \<lambda>u v. c(u, v, h`succ(depth(u))`u, h`succ(depth(v))`v));

        Relativize1(M, formula, 

           is_d, \<lambda>u. d(u, h ` succ(depth(u)) ` u));

 	M(h) |] 

      ==> Relativize1(M, formula,

                         is_formula_case (M, is_a, is_b, is_c, is_d),

                         formula_rec_case(a, b, c, d, h))"

apply (simp (no_asm) add: formula_rec_case_def Relativize1_def) 

apply (simp add: formula_case_abs) 

done

text{*This locale packages the premises of the following theorems,

      which is the normal purpose of locales.  It doesn't accumulate

      constraints on the class @{term M}, as in most of this deveopment.*}

locale Formula_Rec = M_eclose +

  fixes a and is_a and b and is_b and c and is_c and d and is_d and MH

  defines

      "MH(u::i,f,z) ==

	\<forall>fml[M]. is_formula(M,fml) -->

             is_lambda

	 (M, fml, is_formula_case (M, is_a, is_b, is_c(f), is_d(f)), z)"

  assumes a_closed: "[|x\<in>nat; y\<in>nat|] ==> M(a(x,y))"

      and a_rel:    "Relativize2(M, nat, nat, is_a, a)"

      and b_closed: "[|x\<in>nat; y\<in>nat|] ==> M(b(x,y))"

      and b_rel:    "Relativize2(M, nat, nat, is_b, b)"

      and c_closed: "[|x \<in> formula; y \<in> formula; M(gx); M(gy)|]

                     ==> M(c(x, y, gx, gy))"

      and c_rel:

         "M(f) ==> 

          Relativize2 (M, formula, formula, is_c(f),

             \<lambda>u v. c(u, v, f ` succ(depth(u)) ` u, f ` succ(depth(v)) ` v))"

      and d_closed: "[|x \<in> formula; M(gx)|] ==> M(d(x, gx))"

      and d_rel:

         "M(f) ==> 

          Relativize1(M, formula, is_d(f), \<lambda>u. d(u, f ` succ(depth(u)) ` u))"

      and fr_replace: "n \<in> nat ==> transrec_replacement(M,MH,n)"

      and fr_lam_replace:

           "M(g) ==>

            strong_replacement

	      (M, \<lambda>x y. x \<in> formula &

		  y = \<langle>x, formula_rec_case(a,b,c,d,g,x)\<rangle>)";

lemma (in Formula_Rec) formula_rec_case_closed:

    "[|M(g); p \<in> formula|] ==> M(formula_rec_case(a, b, c, d, g, p))"

by (simp add: formula_rec_case_def a_closed b_closed c_closed d_closed) 

lemma (in Formula_Rec) formula_rec_lam_closed: