# Theory Datatype_absolute

```(*  Title:      ZF/Constructible/Datatype_absolute.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section ‹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›

definition
directed :: "i=>o" where
"directed(A) == A≠0 & (∀x∈A. ∀y∈A. x ∪ y ∈ A)"

definition
contin :: "(i=>i) => o" where
"contin(h) == (∀A. directed(A) ⟶ h(⋃A) = (⋃X∈A. h(X)))"

lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n ∈ nat|] ==> h^n (0) ⊆ D"
apply (induct_tac n)
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 (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(⋃n∈nat. h^n (0)) = (⋃n∈nat. h^n (0))"
apply (rule trans)
apply (rule equalityI)
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) ⊆ (⋃n∈nat. h^n(0))"
apply (rule lfp_lowerbound)
apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff)
done

lemma Union_subset_lfp:
"bnd_mono(D,h) ==> (⋃n∈nat. h^n(0)) ⊆ lfp(D,h)"
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) = (⋃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 (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))"

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 (rule equalityI)
prefer 2 apply blast
apply clarify
apply (rename_tac B C)
apply (rule_tac a="B ∪ C" in UN_I)
done

lemma const_contin: "contin(λX. A)"

lemma id_contin: "contin(λX. X)"

subsection ‹Absoluteness for "Iterates"›

definition
iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o" where
"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)"

definition
is_iterates :: "[i=>o, [i,i]=>o, i, i, i] => o" where
"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)"

definition
iterates_replacement :: "[i=>o, [i,i]=>o, i] => o" where
"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_trancl) 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)))"

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
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
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 (rule equalityI)
apply (rule lfp_lowerbound)
prefer 2 apply (rule lfp_subset)
apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
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 (blast intro: datatype_univs
dest: lfp_subset [THEN subsetD])
done

text‹Re-expresses lists using "iterates", no univ.›
lemma list_eq_Union:
"list(A) = (⋃n∈nat. (λX. {0} + A*X) ^ n (0))"
by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)

definition
is_list_functor :: "[i=>o,i,i,i] => o" where
"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 (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 =
(⋃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)

definition
is_formula_functor :: "[i=>o,i,i] => o" where
"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_trancl) formula_functor_abs [simp]:
"[| M(X); M(Z) |]
==> is_formula_functor(M,X,Z) ⟷
Z = ((nat*nat) + (nat*nat)) + (X*X + X)"

subsection‹\<^term>‹M› Contains the List and Formula Datatypes›

definition
list_N :: "[i,i] => i" where
"list_N(A,n) == (λX. {0} + A * X)^n (0)"

lemma Nil_in_list_N [simp]: "[] ∈ list_N(A,succ(n))"

lemma Cons_in_list_N [simp]:
"Cons(a,l) ∈ list_N(A,succ(n)) ⟷ a∈A & l ∈ list_N(A,n)"

text‹These two aren't simprules because they reveal the underlying
list representation.›
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)"
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 (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)
done

definition
is_list_N :: "[i=>o,i,i,i] => o" where
"is_list_N(M,A,n,Z) ==
∃zero[M]. empty(M,zero) &
is_iterates(M, is_list_functor(M,A), zero, n, Z)"

definition
mem_list :: "[i=>o,i,i] => o" where
"mem_list(M,A,l) ==
∃n[M]. ∃listn[M].
finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l ∈ listn"

definition
is_list :: "[i=>o,i,i] => o" where
"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)

definition
formula_N :: "i => i" where
"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"

lemma Equal_in_formula_N [simp]:
"Equal(x,y) ∈ formula_N(succ(n)) ⟷ x ∈ nat & y ∈ nat"

lemma Nand_in_formula_N [simp]:
"Nand(x,y) ∈ formula_N(succ(n)) ⟷ x ∈ formula_N(n) & y ∈ formula_N(n)"

lemma Forall_in_formula_N [simp]:
"Forall(x) ∈ formula_N(succ(n)) ⟷ x ∈ formula_N(n)"

text‹These two aren't simprules because they reveal the underlying
formula representation.›
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"
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 (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 (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

definition
is_formula_N :: "[i=>o,i,i] => o" where
"is_formula_N(M,n,Z) ==
∃zero[M]. empty(M,zero) &
is_iterates(M, is_formula_functor(M), zero, n, Z)"

definition
mem_formula :: "[i=>o,i] => o" where
"mem_formula(M,p) ==
∃n[M]. ∃formn[M].
finite_ordinal(M,n) & is_formula_N(M,n,formn) & p ∈ formn"

definition
is_formula :: "[i=>o,i] => o" where
"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)"]])
done

lemma (in M_datatypes) list_closed [intro,simp]:
"M(A) ==> M(list(A))"
apply (insert list_replacement1)
list_replacement2' relation1_def
iterates_closed [of "is_list_functor(M,A)"])

text‹WARNING: use only with ‹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 (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)"]])
done

lemma (in M_datatypes) formula_closed [intro,simp]:
"M(formula)"
apply (insert formula_replacement1)
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 (rule M_equalityI, simp_all)
done

subsection‹Absoluteness for ‹ε›-Closure: the \<^term>‹eclose› Operator›

text‹Re-expresses eclose using "iterates"›
lemma eclose_eq_Union:
"eclose(A) = (⋃n∈nat. Union^n (A))"
apply (rule UN_cong)
apply (rule refl)
apply (induct_tac n)
done

definition
is_eclose_n :: "[i=>o,i,i,i] => o" where
"is_eclose_n(M,A,n,Z) == is_iterates(M, big_union(M), A, n, Z)"

definition
mem_eclose :: "[i=>o,i,i] => o" where
"mem_eclose(M,A,l) ==
∃n[M]. ∃eclosen[M].
finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l ∈ eclosen"

definition
is_eclose :: "[i=>o,i,i] => o" where
"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)"]])
done

lemma (in M_eclose) eclose_closed [intro,simp]:
"M(A) ==> M(eclose(A))"
apply (insert eclose_replacement1)
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 (rule M_equalityI, simp_all)
done

subsection ‹Absoluteness for \<^term>‹transrec››

text‹\<^prop>‹transrec(a,H) ≡ wfrec(Memrel(eclose({a})), a, H)››

definition
is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o" where
"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)"

definition
transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o" where
"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
‹trans_wfrec_abs› rather than ‹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)"

subsection‹Absoluteness for the List Operator \<^term>‹length››
text‹But it is never used.›

definition
is_length :: "[i=>o,i,i,i] => o" where
"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 (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))"

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)
done

lemma (in M_basic) iterates_tl'_closed:
"[|n ∈ nat; M(x)|] ==> M(tl'^n (x))"
apply (induct_tac n, simp)
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)
done

definition
is_nth :: "[i=>o,i,i,i] => o" where
"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›

definition
is_Member :: "[i=>o,i,i,i] => o" where
― ‹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))"

lemma (in M_trivial) Member_in_M_iff [iff]:
"M(Member(x,y)) ⟷ M(x) & M(y)"

definition
is_Equal :: "[i=>o,i,i,i] => o" where
― ‹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))"

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

definition
is_Nand :: "[i=>o,i,i,i] => o" where
― ‹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))"

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

definition
is_Forall :: "[i=>o,i,i] => o" where
― ‹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))"

lemma (in M_trivial) Forall_in_M_iff [iff]: "M(Forall(x)) ⟷ M(x)"

subsection ‹Absoluteness for \<^term>‹formula_rec››

definition
formula_rec_case :: "[[i,i]=>i, [i,i]=>i, [i,i,i,i]=>i, [i,i]=>i, i, i] => i" where
― ‹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 (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)
txt‹Inductive step for \<^term>‹Forall››
apply (subst transrec)
done

subsubsection‹Absoluteness for the Formula Operator \<^term>‹depth››

definition
is_depth :: "[i=>o,i,i] => o" where
"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 (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))"

subsubsection‹\<^term>‹is_formula_case›: relativization of \<^term>‹formula_case››

definition
is_formula_case ::
"[i=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i]=>o, i, i] => o" where
― ‹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 (erule formula.cases)
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›

definition
is_formula_rec :: "[i=>o, [i,i,i]=>o, i, i] => o" where
― ‹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)
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 development.›
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))"