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theory Datatype_absolute(* Title: ZF/Constructible/Datatype_absolute.thy ID: $Id: Datatype_absolute.thy,v 1.33 2005/06/17 14:15:10 haftmann Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory *) header {*Absoluteness Properties for Recursive Datatypes*} theory Datatype_absolute imports Formula WF_absolute begin subsection{*The lfp of a continuous function can be expressed as a union*} constdefs directed :: "i=>o" "directed(A) == A≠0 & (∀x∈A. ∀y∈A. x ∪ y ∈ A)" contin :: "(i=>i) => o" "contin(h) == (∀A. directed(A) --> h(\<Union>A) = (\<Union>X∈A. h(X)))" lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n ∈ 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 ∈ nat; j ∈ nat; bnd_mono(D,h)|] ==> i ≤ j --> h^i(0) ⊆ 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∈nat})" apply (simp add: directed_def, clarify) apply (rename_tac i j) apply (rule_tac x="i ∪ 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∈nat. h^n (0)) = (\<Union>n∈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∈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∈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∈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⊆Y; contin(F)|] ==> F(X) ⊆ 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(λX. F(X) + G(X))" by (simp add: contin_def, blast) lemma prod_contin: "[|contin(F); contin(G)|] ==> contin(λX. F(X) * G(X))" apply (subgoal_tac "∀B C. F(B) ⊆ F(B ∪ C)") prefer 2 apply (simp add: Un_upper1 contin_imp_mono) apply (subgoal_tac "∀B C. G(C) ⊆ G(B ∪ 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 ∪ C" in UN_I) apply (simp add: directed_def, blast) done lemma const_contin: "contin(λX. A)" by (simp add: contin_def directed_def) lemma id_contin: "contin(λX. 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, λm u. ∃gm[M]. fun_apply(M,g,m,gm) & isF(gm,u), n, z)" is_iterates :: "[i=>o, [i,i]=>o, i, i, i] => o" "is_iterates(M,isF,v,n,Z) == ∃sn[M]. ∃msn[M]. successor(M,n,sn) & membership(M,sn,msn) & is_wfrec(M, iterates_MH(M,isF,v), msn, n, Z)" iterates_replacement :: "[i=>o, [i,i]=>o, i] => o" "iterates_replacement(M,isF,v) == ∀n[M]. n∈nat --> wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))" lemma (in M_basic) iterates_MH_abs: "[| relation1(M,isF,F); M(n); M(g); M(z) |] ==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, λm. F(g`m), n)" by (simp add: nat_case_abs [of _ "λm. F(g ` m)"] relation1_def iterates_MH_def) lemma (in M_basic) iterates_imp_wfrec_replacement: "[|relation1(M,isF,F); n ∈ nat; iterates_replacement(M,isF,v)|] ==> wfrec_replacement(M, λn f z. z = nat_case(v, λm. 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); relation1(M,isF,F); n ∈ nat; M(v); M(z); ∀x[M]. M(F(x)) |] ==> is_iterates(M,isF,v,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 is_iterates_def relation2_def iterates_MH_abs iterates_nat_def recursor_def transrec_def eclose_sing_Ord_eq nat_into_M trans_wfrec_abs [of _ _ _ _ "λn g. nat_case(v, λm. F(g`m), n)"]) done lemma (in M_trancl) iterates_closed [intro,simp]: "[| iterates_replacement(M,isF,v); relation1(M,isF,F); n ∈ nat; M(v); ∀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 relation2_def iterates_MH_abs iterates_nat_def recursor_def transrec_def eclose_sing_Ord_eq nat_into_M trans_wfrec_closed [of _ _ _ "λn g. nat_case(v, λm. 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), λX. {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(λX. {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), λX. {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∈nat. (λX. {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) == ∃n1[M]. ∃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), λX. ((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(λX. ((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), λX. ((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∈nat. (λX. ((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) == ∃nat'[M]. ∃natnat[M]. ∃natnatsum[M]. ∃XX[M]. ∃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) == (λX. {0} + A * X)^n (0)" lemma Nil_in_list_N [simp]: "[] ∈ list_N(A,succ(n))" by (simp add: list_N_def Nil_def) lemma Cons_in_list_N [simp]: "Cons(a,l) ∈ list_N(A,succ(n)) <-> a∈A & l ∈ 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 ∈ list_N(A,n); n ∈ nat |] ==> l ∈ list(A)" by (force simp add: list_eq_Union list_N_def) lemma list_N_imp_length_lt [rule_format]: "n ∈ nat ==> ∀l ∈ 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 ∈ list(A) ==> ∀n∈nat. length(l) < n --> l ∈ list_N(A, n)" apply (induct_tac l) apply (force elim: natE)+ done lemma list_N_imp_eq_length: "[|n ∈ nat; l ∉ list_N(A, n); l ∈ 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 ∈ list(A) ==> list_rec(a,g,l) = transrec (succ(length(l)), λx h. Lambda (list(A), list_case' (a, λa 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) == ∃zero[M]. empty(M,zero) & is_iterates(M, is_list_functor(M,A), zero, n, Z)" mem_list :: "[i=>o,i,i] => o" "mem_list(M,A,l) == ∃n[M]. ∃listn[M]. finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l ∈ listn" is_list :: "[i=>o,i,i] => o" "is_list(M,A,Z) == ∀l[M]. l ∈ 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) ∪ depth(q))" "depth(Forall(p)) = succ(depth(p))" lemma depth_type [TC]: "p ∈ formula ==> depth(p) ∈ nat" by (induct_tac p, simp_all) constdefs formula_N :: "i => i" "formula_N(n) == (λX. ((nat*nat) + (nat*nat)) + (X*X + X)) ^ n (0)" lemma Member_in_formula_N [simp]: "Member(x,y) ∈ formula_N(succ(n)) <-> x ∈ nat & y ∈ nat" by (simp add: formula_N_def Member_def) lemma Equal_in_formula_N [simp]: "Equal(x,y) ∈ formula_N(succ(n)) <-> x ∈ nat & y ∈ nat" by (simp add: formula_N_def Equal_def) lemma Nand_in_formula_N [simp]: "Nand(x,y) ∈ formula_N(succ(n)) <-> x ∈ formula_N(n) & y ∈ formula_N(n)" by (simp add: formula_N_def Nand_def) lemma Forall_in_formula_N [simp]: "Forall(x) ∈ formula_N(succ(n)) <-> x ∈ 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 ∈ formula_N(n); n ∈ nat |] ==> p ∈ formula" by (force simp add: formula_eq_Union formula_N_def) lemma formula_N_imp_depth_lt [rule_format]: "n ∈ nat ==> ∀p ∈ 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 ∈ formula ==> ∀n∈nat. depth(p) < n --> p ∈ 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 ∈ nat; p ∉ formula_N(n); p ∈ 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 text{*This result and the next are unused.*} lemma formula_N_mono [rule_format]: "[| m ∈ nat; n ∈ nat |] ==> m≤n --> formula_N(m) ⊆ 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 ∈ nat; n ∈ nat |] ==> formula_N(m ∪ n) = formula_N(m) ∪ 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) == ∃zero[M]. empty(M,zero) & is_iterates(M, is_formula_functor(M), zero, n, Z)" constdefs mem_formula :: "[i=>o,i] => o" "mem_formula(M,p) == ∃n[M]. ∃formn[M]. finite_ordinal(M,n) & is_formula_N(M,n,formn) & p ∈ formn" is_formula :: "[i=>o,i] => o" "is_formula(M,Z) == ∀p[M]. p ∈ Z <-> mem_formula(M,p)" locale M_datatypes = M_trancl + assumes list_replacement1: "M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)" and list_replacement2: "M(A) ==> strong_replacement(M, λn y. n∈nat & is_iterates(M, is_list_functor(M,A), 0, n, y))" and formula_replacement1: "iterates_replacement(M, is_formula_functor(M), 0)" and formula_replacement2: "strong_replacement(M, λn y. n∈nat & is_iterates(M, is_formula_functor(M), 0, 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, λn y. n∈nat & y = (λX. {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 relation1_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' relation1_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∈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 relation1_def nat_into_M iterates_abs [of "is_list_functor(M,A)" _ "λX. {0} + A*X"]) done lemma (in M_datatypes) list_N_closed [intro,simp]: "[|M(A); n∈nat|] ==> M(list_N(A,n))" apply (insert list_replacement1) apply (simp add: is_list_N_def list_N_def relation1_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 ∈ list(A)" apply (insert list_replacement1) apply (simp add: mem_list_def list_N_def relation1_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, λn y. n∈nat & y = (λX. ((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 relation1_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' relation1_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∈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 relation1_def nat_into_M iterates_abs [of "is_formula_functor(M)" _ "λX. ((nat*nat) + (nat*nat)) + (X*X + X)"]) done lemma (in M_datatypes) formula_N_closed [intro,simp]: "n∈nat ==> M(formula_N(n))" apply (insert formula_replacement1) apply (simp add: is_formula_N_def formula_N_def relation1_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 ∈ formula" apply (insert formula_replacement1) apply (simp add: mem_formula_def relation1_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 @{text ε}-Closure: the @{term eclose} Operator*} text{*Re-expresses eclose using "iterates"*} lemma eclose_eq_Union: "eclose(A) = (\<Union>n∈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) == is_iterates(M, big_union(M), A, n, Z)" mem_eclose :: "[i=>o,i,i] => o" "mem_eclose(M,A,l) == ∃n[M]. ∃eclosen[M]. finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l ∈ eclosen" is_eclose :: "[i=>o,i,i] => o" "is_eclose(M,A,Z) == ∀u[M]. u ∈ 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, λn y. n∈nat & is_iterates(M, big_union(M), A, n, y))" lemma (in M_eclose) eclose_replacement2': "M(A) ==> strong_replacement(M, λn y. n∈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 relation1_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' relation1_def iterates_closed [of "big_union(M)"]) lemma (in M_eclose) is_eclose_n_abs [simp]: "[|M(A); n∈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 relation1_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 ∈ eclose(A)" apply (insert eclose_replacement1) apply (simp add: mem_eclose_def relation1_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) ≡ wfrec(Memrel(eclose({a})), a, H)"} *} constdefs is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o" "is_transrec(M,MH,a,z) == ∃sa[M]. ∃esa[M]. ∃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) == ∃sa[M]. ∃esa[M]. ∃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); relation2(M,MH,H); Ord(i); M(i); M(z); ∀x[M]. ∀g[M]. function(g) --> M(H(x,g))|] ==> is_transrec(M,MH,i,z) <-> z = transrec(i,H)" by (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel) theorem (in M_eclose) transrec_closed: "[|transrec_replacement(M,MH,i); relation2(M,MH,H); Ord(i); M(i); ∀x[M]. ∀g[M]. function(g) --> M(H(x,g))|] ==> M(transrec(i,H))" by (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel) text{*Helps to prove instances of @{term transrec_replacement}*} lemma (in M_eclose) transrec_replacementI: "[|M(a); strong_replacement (M, λx z. ∃y[M]. pair(M, x, y, z) & 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}*} text{*But it is never used.*} constdefs is_length :: "[i=>o,i,i,i] => o" "is_length(M,A,l,n) == ∃sn[M]. ∃list_n[M]. ∃list_sn[M]. is_list_N(M,A,n,list_n) & l ∉ list_n & successor(M,n,sn) & is_list_N(M,A,sn,list_sn) & l ∈ list_sn" lemma (in M_datatypes) length_abs [simp]: "[|M(A); l ∈ list(A); n ∈ 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 ∈ list(A) ==> M(length(l))" by (simp add: nat_into_M) subsection {*Absoluteness for the List Operator @{term nth}*} lemma nth_eq_hd_iterates_tl [rule_format]: "xs ∈ list(A) ==> ∀n ∈ 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 ∈ 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 ∈ list(A); n ∈ 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) == ∃X[M]. is_iterates(M, is_tl(M), l, n, X) & is_hd(M,X,Z)" lemma (in M_datatypes) nth_abs [simp]: "[|M(A); n ∈ nat; l ∈ 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 _ relation1_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) ≡ Inl(Inl(〈x,y〉))"}*} "is_Member(M,x,y,Z) == ∃p[M]. ∃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) ≡ Inl(Inr(〈x,y〉))"}*} "is_Equal(M,x,y,Z) == ∃p[M]. ∃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) ≡ Inr(Inl(〈x,y〉))"}*} "is_Nand(M,x,y,Z) == ∃p[M]. ∃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) ≡ Inr(Inr(p))"}*} "is_Forall(M,p,Z) == ∃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}*} 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, λu v. c(u, v, h ` succ(depth(u)) ` u, h ` succ(depth(v)) ` v), λu. d(u, h ` succ(depth(u)) ` u))" 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 ∈ formula ==> formula_rec(a,b,c,d,p) = transrec (succ(depth(p)), λx 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 subsubsection{*Absoluteness for the Formula Operator @{term depth}*} constdefs is_depth :: "[i=>o,i,i] => o" "is_depth(M,p,n) == ∃sn[M]. ∃formula_n[M]. ∃formula_sn[M]. is_formula_N(M,n,formula_n) & p ∉ formula_n & successor(M,n,sn) & is_formula_N(M,sn,formula_sn) & p ∈ formula_sn" lemma (in M_datatypes) depth_abs [simp]: "[|p ∈ formula; n ∈ 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 ∈ formula ==> M(depth(p))" by (simp add: nat_into_M) 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) == (∀x[M]. ∀y[M]. finite_ordinal(M,x) --> finite_ordinal(M,y) --> is_Member(M,x,y,p) --> is_a(x,y,z)) & (∀x[M]. ∀y[M]. finite_ordinal(M,x) --> finite_ordinal(M,y) --> is_Equal(M,x,y,p) --> is_b(x,y,z)) & (∀x[M]. ∀y[M]. mem_formula(M,x) --> mem_formula(M,y) --> is_Nand(M,x,y,p) --> is_c(x,y,z)) & (∀x[M]. mem_formula(M,x) --> is_Forall(M,x,p) --> is_d(x,z))" lemma (in M_datatypes) formula_case_abs [simp]: "[| Relation2(M,nat,nat,is_a,a); Relation2(M,nat,nat,is_b,b); Relation2(M,formula,formula,is_c,c); Relation1(M,formula,is_d,d); p ∈ 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: Relation1_def Relation2_def) done lemma (in M_datatypes) formula_case_closed [intro,simp]: "[|p ∈ formula; ∀x[M]. ∀y[M]. x∈nat --> y∈nat --> M(a(x,y)); ∀x[M]. ∀y[M]. x∈nat --> y∈nat --> M(b(x,y)); ∀x[M]. ∀y[M]. x∈formula --> y∈formula --> M(c(x,y)); ∀x[M]. x∈formula --> M(d(x))|] ==> M(formula_case(a,b,c,d,p))" by (erule formula.cases, simp_all) subsubsection {*Absoluteness for @{term formula_rec}: Final Results*} constdefs 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) == ∃dp[M]. ∃i[M]. ∃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{*Sufficient conditions to relativize the instance of @{term formula_case} in @{term formula_rec}*} lemma (in M_datatypes) Relation1_formula_rec_case: "[|Relation2(M, nat, nat, is_a, a); Relation2(M, nat, nat, is_b, b); Relation2 (M, formula, formula, is_c, λu v. c(u, v, h`succ(depth(u))`u, h`succ(depth(v))`v)); Relation1(M, formula, is_d, λu. d(u, h ` succ(depth(u)) ` u)); M(h) |] ==> Relation1(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 Relation1_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) == ∀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∈nat; y∈nat|] ==> M(a(x,y))" and a_rel: "Relation2(M, nat, nat, is_a, a)" and b_closed: "[|x∈nat; y∈nat|] ==> M(b(x,y))" and b_rel: "Relation2(M, nat, nat, is_b, b)" and c_closed: "[|x ∈ formula; y ∈ formula; M(gx); M(gy)|] ==> M(c(x, y, gx, gy))" and c_rel: "M(f) ==> Relation2 (M, formula, formula, is_c(f), λu v. c(u, v, f ` succ(depth(u)) ` u, f ` succ(depth(v)) ` v))" and d_closed: "[|x ∈ formula; M(gx)|] ==> M(d(x, gx))" and d_rel: "M(f) ==> Relation1(M, formula, is_d(f), λu. d(u, f ` succ(depth(u)) ` u))" and fr_replace: "n ∈ nat ==> transrec_replacement(M,MH,n)" and fr_lam_replace: "M(g) ==> strong_replacement (M, λx y. x ∈ formula & y = 〈x, formula_rec_case(a,b,c,d,g,x)〉)"; lemma (in Formula_Rec) formula_rec_case_closed: "[|M(g); p ∈ 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: "M(g) ==> M(Lambda (formula, formula_rec_case(a,b,c,d,g)))" by (simp add: lam_closed2 fr_lam_replace formula_rec_case_closed) lemma (in Formula_Rec) MH_rel2: "relation2 (M, MH, λx h. Lambda (formula, formula_rec_case(a,b,c,d,h)))" apply (simp add: relation2_def MH_def, clarify) apply (rule lambda_abs2) apply (rule Relation1_formula_rec_case) apply (simp_all add: a_rel b_rel c_rel d_rel formula_rec_case_closed) done lemma (in Formula_Rec) fr_transrec_closed: "n ∈ nat ==> M(transrec (n, λx h. Lambda(formula, formula_rec_case(a, b, c, d, h))))" by (simp add: transrec_closed [OF fr_replace MH_rel2] nat_into_M formula_rec_lam_closed) text{*The main two results: @{term formula_rec} is absolute for @{term M}.*} theorem (in Formula_Rec) formula_rec_closed: "p ∈ formula ==> M(formula_rec(a,b,c,d,p))" by (simp add: formula_rec_eq fr_transrec_closed transM [OF _ formula_closed]) theorem (in Formula_Rec) formula_rec_abs: "[| p ∈ formula; M(z)|] ==> is_formula_rec(M,MH,p,z) <-> z = formula_rec(a,b,c,d,p)" by (simp add: is_formula_rec_def formula_rec_eq transM [OF _ formula_closed] transrec_abs [OF fr_replace MH_rel2] depth_type fr_transrec_closed formula_rec_lam_closed eq_commute) end
lemma bnd_mono_iterates_subset:
[| bnd_mono(D, h); n ∈ nat |] ==> h^n (0) ⊆ D
lemma bnd_mono_increasing:
[| i ∈ nat; j ∈ nat; bnd_mono(D, h); i ≤ j |] ==> h^i (0) ⊆ h^j (0)
lemma directed_iterates:
bnd_mono(D, h) ==> directed({h^n (0) . n ∈ nat})
lemma contin_iterates_eq:
[| bnd_mono(D, h); contin(h) |] ==> h(\<Union>n∈nat. h^n (0)) = (\<Union>n∈nat. h^n (0))
lemma lfp_subset_Union:
[| bnd_mono(D, h); contin(h) |] ==> lfp(D, h) ⊆ (\<Union>n∈nat. h^n (0))
lemma Union_subset_lfp:
bnd_mono(D, h) ==> (\<Union>n∈nat. h^n (0)) ⊆ lfp(D, h)
lemma lfp_eq_Union:
[| bnd_mono(D, h); contin(h) |] ==> lfp(D, h) = (\<Union>n∈nat. h^n (0))
lemma contin_imp_mono:
[| X ⊆ Y; contin(F) |] ==> F(X) ⊆ F(Y)
lemma sum_contin:
[| contin(F); contin(G) |] ==> contin(%X. F(X) + G(X))
lemma prod_contin:
[| contin(F); contin(G) |] ==> contin(%X. F(X) × G(X))
lemma const_contin:
contin(%X. A)
lemma id_contin:
contin(%X. X)
lemma iterates_MH_abs:
[| PROP M_basic(M); relation1(M, isF, F); M(n); M(g); M(z) |] ==> iterates_MH(M, isF, v, n, g, z) <-> z = nat_case(v, %m. F(g ` m), n)
lemma iterates_imp_wfrec_replacement:
[| PROP M_basic(M); relation1(M, isF, F); n ∈ nat; iterates_replacement(M, isF, v) |] ==> wfrec_replacement (M, %n f z. z = nat_case(v, %m. F(f ` m), n), Memrel(succ(n)))
theorem iterates_abs:
[| PROP M_trancl(M); iterates_replacement(M, isF, v); relation1(M, isF, F); n ∈ nat; M(v); M(z); ∀x[M]. M(F(x)) |] ==> is_iterates(M, isF, v, n, z) <-> z = F^n (v)
lemma iterates_closed:
[| PROP M_trancl(M); iterates_replacement(M, isF, v); relation1(M, isF, F); n ∈ nat; M(v); ∀x[M]. M(F(x)) |] ==> M(F^n (v))
lemmas datatype_univs:
a ∈ univ(A) ==> Inl(a) ∈ univ(A)
b ∈ univ(A) ==> Inr(b) ∈ univ(A)
[| a ∈ univ(A); b ∈ univ(A) |] ==> 〈a, b〉 ∈ univ(A)
c ∈ nat ==> c ∈ univ(A)
c ∈ A ==> c ∈ univ(A)
lemmas datatype_univs:
a ∈ univ(A) ==> Inl(a) ∈ univ(A)
b ∈ univ(A) ==> Inr(b) ∈ univ(A)
[| a ∈ univ(A); b ∈ univ(A) |] ==> 〈a, b〉 ∈ univ(A)
c ∈ nat ==> c ∈ univ(A)
c ∈ A ==> c ∈ univ(A)
lemma list_fun_bnd_mono:
bnd_mono(univ(A), %X. {0} + A × X)
lemma list_fun_contin:
contin(%X. {0} + A × X)
lemma list_eq_lfp2:
list(A) = lfp(univ(A), %X. {0} + A × X)
lemma list_eq_Union:
list(A) = (\<Union>n∈nat. (%X. {0} + A × X)^n (0))
lemma list_functor_abs:
[| PROP M_basic(M); M(A); M(X); M(Z) |] ==> is_list_functor(M, A, X, Z) <-> Z = {0} + A × X
lemma formula_fun_bnd_mono:
bnd_mono(univ(0), %X. (nat × nat + nat × nat) + X × X + X)
lemma formula_fun_contin:
contin(%X. (nat × nat + nat × nat) + X × X + X)
lemma formula_eq_lfp2:
formula = lfp(univ(0), %X. (nat × nat + nat × nat) + X × X + X)
lemma formula_eq_Union:
formula = (\<Union>n∈nat. (%X. (nat × nat + nat × nat) + X × X + X)^n (0))
lemma formula_functor_abs:
[| PROP M_basic(M); M(X); M(Z) |] ==> is_formula_functor(M, X, Z) <-> Z = (nat × nat + nat × nat) + X × X + X
lemma Nil_in_list_N:
[] ∈ list_N(A, succ(n))
lemma Cons_in_list_N:
Cons(a, l) ∈ list_N(A, succ(n)) <-> a ∈ A ∧ l ∈ list_N(A, n)
lemma list_N_0:
list_N(A, 0) = 0
lemma list_N_succ:
list_N(A, succ(n)) = {0} + A × list_N(A, n)
lemma list_N_imp_list:
[| l ∈ list_N(A, n); n ∈ nat |] ==> l ∈ list(A)
lemma list_N_imp_length_lt:
[| n ∈ nat; l ∈ list_N(A, n) |] ==> length(l) < n
lemma list_imp_list_N:
[| l ∈ list(A); n ∈ nat; length(l) < n |] ==> l ∈ list_N(A, n)
lemma list_N_imp_eq_length:
[| n ∈ nat; l ∉ list_N(A, n); l ∈ list_N(A, succ(n)) |] ==> n = length(l)
lemma list_rec_eq:
[| PROP M_trivial(M); l ∈ list(A) |] ==> list_rec(a, g, l) = transrec (succ(length(l)), %x h. Lambda (list(A), list_case'(a, %a l. g(a, l, h ` succ(length(l)) ` l)))) ` l
lemma depth_type:
p ∈ formula ==> depth(p) ∈ nat
lemma Member_in_formula_N:
Member(x, y) ∈ formula_N(succ(n)) <-> x ∈ nat ∧ y ∈ nat
lemma Equal_in_formula_N:
Equal(x, y) ∈ formula_N(succ(n)) <-> x ∈ nat ∧ y ∈ nat
lemma Nand_in_formula_N:
Nand(x, y) ∈ formula_N(succ(n)) <-> x ∈ formula_N(n) ∧ y ∈ formula_N(n)
lemma Forall_in_formula_N:
Forall(x) ∈ formula_N(succ(n)) <-> x ∈ formula_N(n)
lemma formula_N_0:
formula_N(0) = 0
lemma formula_N_succ:
formula_N(succ(n)) = (nat × nat + nat × nat) + formula_N(n) × formula_N(n) + formula_N(n)
lemma formula_N_imp_formula:
[| p ∈ formula_N(n); n ∈ nat |] ==> p ∈ formula
lemma formula_N_imp_depth_lt:
[| n ∈ nat; p ∈ formula_N(n) |] ==> depth(p) < n
lemma formula_imp_formula_N:
[| p ∈ formula; n ∈ nat; depth(p) < n |] ==> p ∈ formula_N(n)
lemma formula_N_imp_eq_depth:
[| n ∈ nat; p ∉ formula_N(n); p ∈ formula_N(succ(n)) |] ==> n = depth(p)
lemma formula_N_mono:
[| m ∈ nat; n ∈ nat; m ≤ n |] ==> formula_N(m) ⊆ formula_N(n)
lemma formula_N_distrib:
[| m ∈ nat; n ∈ nat |] ==> formula_N(m ∪ n) = formula_N(m) ∪ formula_N(n)
lemma list_replacement2':
[| PROP M_datatypes(M); M(A) |] ==> strong_replacement(M, %n y. n ∈ nat ∧ y = (%X. {0} + A × X)^n (0))
lemma list_closed:
[| PROP M_datatypes(M); M(A) |] ==> M(list(A))
lemmas list_into_M:
[| PROP M_datatypes(M); y ∈ list(A); M(A) |] ==> M(y)
lemma list_N_abs:
[| PROP M_datatypes(M); M(A); n ∈ nat; M(Z) |] ==> is_list_N(M, A, n, Z) <-> Z = list_N(A, n)
lemma list_N_closed:
[| PROP M_datatypes(M); M(A); n ∈ nat |] ==> M(list_N(A, n))
lemma mem_list_abs:
[| PROP M_datatypes(M); M(A) |] ==> mem_list(M, A, l) <-> l ∈ list(A)
lemma list_abs:
[| PROP M_datatypes(M); M(A); M(Z) |] ==> is_list(M, A, Z) <-> Z = list(A)
lemma formula_replacement2':
PROP M_datatypes(M) ==> strong_replacement (M, %n y. n ∈ nat ∧ y = (%X. (nat × nat + nat × nat) + X × X + X)^n (0))
lemma formula_closed:
PROP M_datatypes(M) ==> M(formula)
lemmas formula_into_M:
[| PROP M_datatypes(M); y ∈ formula |] ==> M(y)
lemma formula_N_abs:
[| PROP M_datatypes(M); n ∈ nat; M(Z) |] ==> is_formula_N(M, n, Z) <-> Z = formula_N(n)
lemma formula_N_closed:
[| PROP M_datatypes(M); n ∈ nat |] ==> M(formula_N(n))
lemma mem_formula_abs:
PROP M_datatypes(M) ==> mem_formula(M, l) <-> l ∈ formula
lemma formula_abs:
[| PROP M_datatypes(M); M(Z) |] ==> is_formula(M, Z) <-> Z = formula
lemma eclose_eq_Union:
eclose(A) = (\<Union>n∈nat. Union^n (A))
lemma eclose_replacement2':
[| PROP M_eclose(M); M(A) |] ==> strong_replacement(M, %n y. n ∈ nat ∧ y = Union^n (A))
lemma eclose_closed:
[| PROP M_eclose(M); M(A) |] ==> M(eclose(A))
lemma is_eclose_n_abs:
[| PROP M_eclose(M); M(A); n ∈ nat; M(Z) |] ==> is_eclose_n(M, A, n, Z) <-> Z = Union^n (A)
lemma mem_eclose_abs:
[| PROP M_eclose(M); M(A) |] ==> mem_eclose(M, A, l) <-> l ∈ eclose(A)
lemma eclose_abs:
[| PROP M_eclose(M); M(A); M(Z) |] ==> is_eclose(M, A, Z) <-> Z = eclose(A)
theorem transrec_abs:
[| PROP M_eclose(M); transrec_replacement(M, MH, i); relation2(M, MH, H); Ord(i); M(i); M(z); ∀x[M]. ∀g[M]. function(g) --> M(H(x, g)) |] ==> is_transrec(M, MH, i, z) <-> z = transrec(i, H)
theorem transrec_closed:
[| PROP M_eclose(M); transrec_replacement(M, MH, i); relation2(M, MH, H); Ord(i); M(i); ∀x[M]. ∀g[M]. function(g) --> M(H(x, g)) |] ==> M(transrec(i, H))
lemma transrec_replacementI:
[| PROP M_eclose(M); M(a); strong_replacement (M, %x z. ∃y[M]. pair(M, x, y, z) ∧ is_wfrec(M, MH, Memrel(eclose({a})), x, y)) |] ==> transrec_replacement(M, MH, a)
lemma length_abs:
[| PROP M_datatypes(M); M(A); l ∈ list(A); n ∈ nat |] ==> is_length(M, A, l, n) <-> n = length(l)
lemma length_closed:
[| PROP M_trivial(M); l ∈ list(A) |] ==> M(length(l))
lemma nth_eq_hd_iterates_tl:
[| xs ∈ list(A); n ∈ nat |] ==> nth(n, xs) = hd'(tl'^n (xs))
lemma iterates_tl'_closed:
[| PROP M_basic(M); n ∈ nat; M(x) |] ==> M(tl'^n (x))
lemma nth_closed:
[| PROP M_datatypes(M); xs ∈ list(A); n ∈ nat; M(A) |] ==> M(nth(n, xs))
lemma nth_abs:
[| PROP M_datatypes(M); M(A); n ∈ nat; l ∈ list(A); M(Z) |] ==> is_nth(M, n, l, Z) <-> Z = nth(n, l)
lemma Member_abs:
[| PROP M_trivial(M); M(x); M(y); M(Z) |] ==> is_Member(M, x, y, Z) <-> Z = Member(x, y)
lemma Member_in_M_iff:
PROP M_trivial(M) ==> M(Member(x, y)) <-> M(x) ∧ M(y)
lemma Equal_abs:
[| PROP M_trivial(M); M(x); M(y); M(Z) |] ==> is_Equal(M, x, y, Z) <-> Z = Equal(x, y)
lemma Equal_in_M_iff:
PROP M_trivial(M) ==> M(Equal(x, y)) <-> M(x) ∧ M(y)
lemma Nand_abs:
[| PROP M_trivial(M); M(x); M(y); M(Z) |] ==> is_Nand(M, x, y, Z) <-> Z = Nand(x, y)
lemma Nand_in_M_iff:
PROP M_trivial(M) ==> M(Nand(x, y)) <-> M(x) ∧ M(y)
lemma Forall_abs:
[| PROP M_trivial(M); M(x); M(Z) |] ==> is_Forall(M, x, Z) <-> Z = Forall(x)
lemma Forall_in_M_iff:
PROP M_trivial(M) ==> M(Forall(x)) <-> M(x)
lemma formula_rec_eq:
[| PROP M_trivial(M); p ∈ formula |] ==> formula_rec(a, b, c, d, p) = transrec (succ(depth(p)), %x h. Lambda(formula, formula_rec_case(a, b, c, d, h))) ` p
lemma depth_abs:
[| PROP M_datatypes(M); p ∈ formula; n ∈ nat |] ==> is_depth(M, p, n) <-> n = depth(p)
lemma depth_closed:
[| PROP M_trivial(M); p ∈ formula |] ==> M(depth(p))
lemma formula_case_abs:
[| PROP M_datatypes(M); Relation2(M, nat, nat, is_a, a); Relation2(M, nat, nat, is_b, b); Relation2(M, formula, formula, is_c, c); Relation1(M, formula, is_d, d); p ∈ 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)
lemma formula_case_closed:
[| PROP M_datatypes(M); p ∈ formula; ∀x[M]. ∀y[M]. x ∈ nat --> y ∈ nat --> M(a(x, y)); ∀x[M]. ∀y[M]. x ∈ nat --> y ∈ nat --> M(b(x, y)); ∀x[M]. ∀y[M]. x ∈ formula --> y ∈ formula --> M(c(x, y)); ∀x[M]. x ∈ formula --> M(d(x)) |] ==> M(formula_case(a, b, c, d, p))
lemma Relation1_formula_rec_case:
[| PROP M_datatypes(M); Relation2(M, nat, nat, is_a, a); Relation2(M, nat, nat, is_b, b); Relation2 (M, formula, formula, is_c, %u v. c(u, v, h ` succ(depth(u)) ` u, h ` succ(depth(v)) ` v)); Relation1(M, formula, is_d, %u. d(u, h ` succ(depth(u)) ` u)); M(h) |] ==> Relation1 (M, formula, is_formula_case(M, is_a, is_b, is_c, is_d), formula_rec_case(a, b, c, d, h))
lemma formula_rec_case_closed:
[| PROP Formula_Rec(M, a, is_a, b, is_b, c, is_c, d, is_d); M(g); p ∈ formula |] ==> M(formula_rec_case(a, b, c, d, g, p))
lemma formula_rec_lam_closed:
[| PROP Formula_Rec(M, a, is_a, b, is_b, c, is_c, d, is_d); M(g) |] ==> M(Lambda(formula, formula_rec_case(a, b, c, d, g)))
lemma MH_rel2:
PROP Formula_Rec(M, a, is_a, b, is_b, c, is_c, d, is_d) ==> relation2 (M, %u f z. ∀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), %x h. Lambda(formula, formula_rec_case(a, b, c, d, h)))
lemma fr_transrec_closed:
[| PROP Formula_Rec(M, a, is_a, b, is_b, c, is_c, d, is_d); n ∈ nat |] ==> M(transrec(n, %x h. Lambda(formula, formula_rec_case(a, b, c, d, h))))
theorem formula_rec_closed:
[| PROP Formula_Rec(M, a, is_a, b, is_b, c, is_c, d, is_d); p ∈ formula |] ==> M(formula_rec(a, b, c, d, p))
theorem formula_rec_abs:
[| PROP Formula_Rec(M, a, is_a, b, is_b, c, is_c, d, is_d); p ∈ formula; M(z) |] ==> is_formula_rec (M, %u f z. ∀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), p, z) <-> z = formula_rec(a, b, c, d, p)