# Theory Univ

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

Standard notation for Vset(i) is V(i), but users might want V for a
variable.

NOTE: univ(A) could be a translation; would simplify many proofs!
But Ind_Syntax.univ refers to the constant "Univ.univ"
*)

section‹The Cumulative Hierarchy and a Small Universe for Recursive Types›

theory Univ imports Epsilon Cardinal begin

definition
Vfrom       :: "[i,i]⇒i"  where
"Vfrom(A,i) ≡ transrec(i, λx f. A ∪ (⋃y∈x. Pow(f`y)))"

abbreviation
Vset :: "i⇒i" where
"Vset(x) ≡ Vfrom(0,x)"

definition
Vrec        :: "[i, [i,i]⇒i] ⇒i"  where
"Vrec(a,H) ≡ transrec(rank(a), λx g. λz∈Vset(succ(x)).
H(z, λw∈Vset(x). g`rank(w)`w)) ` a"

definition
Vrecursor   :: "[[i,i]⇒i, i] ⇒i"  where
"Vrecursor(H,a) ≡ transrec(rank(a), λx g. λz∈Vset(succ(x)).
H(λw∈Vset(x). g`rank(w)`w, z)) ` a"

definition
univ        :: "i⇒i"  where
"univ(A) ≡ Vfrom(A,nat)"

subsection‹Immediate Consequences of the Definition of \<^term>‹Vfrom(A,i)››

text‹NOT SUITABLE FOR REWRITING -- RECURSIVE!›
lemma Vfrom: "Vfrom(A,i) = A ∪ (⋃j∈i. Pow(Vfrom(A,j)))"
by (subst Vfrom_def [THEN def_transrec], simp)

subsubsection‹Monotonicity›

lemma Vfrom_mono [rule_format]:
"A<=B ⟹ ∀j. i<=j ⟶ Vfrom(A,i) ⊆ Vfrom(B,j)"
apply (rule_tac a=i in eps_induct)
apply (rule impI [THEN allI])
apply (subst Vfrom [of A])
apply (subst Vfrom [of B])
apply (erule Un_mono)
apply (erule UN_mono, blast)
done

lemma VfromI: "⟦a ∈ Vfrom(A,j);  j<i⟧ ⟹ a ∈ Vfrom(A,i)"
by (blast dest: Vfrom_mono [OF subset_refl le_imp_subset [OF leI]])

subsubsection‹A fundamental equality: Vfrom does not require ordinals!›

lemma Vfrom_rank_subset1: "Vfrom(A,x) ⊆ Vfrom(A,rank(x))"
proof (induct x rule: eps_induct)
fix x
assume "∀y∈x. Vfrom(A,y) ⊆ Vfrom(A,rank(y))"
thus "Vfrom(A, x) ⊆ Vfrom(A, rank(x))"
by (simp add: Vfrom [of _ x] Vfrom [of _ "rank(x)"],
blast intro!: rank_lt [THEN ltD])
qed

lemma Vfrom_rank_subset2: "Vfrom(A,rank(x)) ⊆ Vfrom(A,x)"
apply (rule_tac a=x in eps_induct)
apply (subst Vfrom)
apply (subst Vfrom, rule subset_refl [THEN Un_mono])
apply (rule UN_least)
txt‹expand ‹rank(x1) = (⋃y∈x1. succ(rank(y)))› in assumptions›
apply (erule rank [THEN equalityD1, THEN subsetD, THEN UN_E])
apply (rule subset_trans)
apply (erule_tac [2] UN_upper)
apply (rule subset_refl [THEN Vfrom_mono, THEN subset_trans, THEN Pow_mono])
apply (erule ltI [THEN le_imp_subset])
apply (rule Ord_rank [THEN Ord_succ])
apply (erule bspec, assumption)
done

lemma Vfrom_rank_eq: "Vfrom(A,rank(x)) = Vfrom(A,x)"
apply (rule equalityI)
apply (rule Vfrom_rank_subset2)
apply (rule Vfrom_rank_subset1)
done

subsection‹Basic Closure Properties›

lemma zero_in_Vfrom: "y:x ⟹ 0 ∈ Vfrom(A,x)"
by (subst Vfrom, blast)

lemma i_subset_Vfrom: "i ⊆ Vfrom(A,i)"
apply (rule_tac a=i in eps_induct)
apply (subst Vfrom, blast)
done

lemma A_subset_Vfrom: "A ⊆ Vfrom(A,i)"
apply (subst Vfrom)
apply (rule Un_upper1)
done

lemmas A_into_Vfrom = A_subset_Vfrom [THEN subsetD]

lemma subset_mem_Vfrom: "a ⊆ Vfrom(A,i) ⟹ a ∈ Vfrom(A,succ(i))"
by (subst Vfrom, blast)

subsubsection‹Finite sets and ordered pairs›

lemma singleton_in_Vfrom: "a ∈ Vfrom(A,i) ⟹ {a} ∈ Vfrom(A,succ(i))"
by (rule subset_mem_Vfrom, safe)

lemma doubleton_in_Vfrom:
"⟦a ∈ Vfrom(A,i);  b ∈ Vfrom(A,i)⟧ ⟹ {a,b} ∈ Vfrom(A,succ(i))"
by (rule subset_mem_Vfrom, safe)

lemma Pair_in_Vfrom:
"⟦a ∈ Vfrom(A,i);  b ∈ Vfrom(A,i)⟧ ⟹ ⟨a,b⟩ ∈ Vfrom(A,succ(succ(i)))"
unfolding Pair_def
apply (blast intro: doubleton_in_Vfrom)
done

lemma succ_in_Vfrom: "a ⊆ Vfrom(A,i) ⟹ succ(a) ∈ Vfrom(A,succ(succ(i)))"
apply (intro subset_mem_Vfrom succ_subsetI, assumption)
apply (erule subset_trans)
apply (rule Vfrom_mono [OF subset_refl subset_succI])
done

subsection‹0, Successor and Limit Equations for \<^term>‹Vfrom››

lemma Vfrom_0: "Vfrom(A,0) = A"
by (subst Vfrom, blast)

lemma Vfrom_succ_lemma: "Ord(i) ⟹ Vfrom(A,succ(i)) = A ∪ Pow(Vfrom(A,i))"
apply (rule Vfrom [THEN trans])
apply (rule equalityI [THEN subst_context,
OF _ succI1 [THEN RepFunI, THEN Union_upper]])
apply (rule UN_least)
apply (rule subset_refl [THEN Vfrom_mono, THEN Pow_mono])
apply (erule ltI [THEN le_imp_subset])
apply (erule Ord_succ)
done

lemma Vfrom_succ: "Vfrom(A,succ(i)) = A ∪ Pow(Vfrom(A,i))"
apply (rule_tac x1 = "succ (i)" in Vfrom_rank_eq [THEN subst])
apply (rule_tac x1 = i in Vfrom_rank_eq [THEN subst])
apply (subst rank_succ)
apply (rule Ord_rank [THEN Vfrom_succ_lemma])
done

(*The premise distinguishes this from Vfrom(A,0);  allowing X=0 forces
the conclusion to be Vfrom(A,⋃(X)) = A ∪ (⋃y∈X. Vfrom(A,y)) *)
lemma Vfrom_Union: "y:X ⟹ Vfrom(A,⋃(X)) = (⋃y∈X. Vfrom(A,y))"
apply (subst Vfrom)
apply (rule equalityI)
txt‹first inclusion›
apply (rule Un_least)
apply (rule A_subset_Vfrom [THEN subset_trans])
apply (rule UN_upper, assumption)
apply (rule UN_least)
apply (erule UnionE)
apply (rule subset_trans)
apply (erule_tac [2] UN_upper,
subst Vfrom, erule subset_trans [OF UN_upper Un_upper2])
txt‹opposite inclusion›
apply (rule UN_least)
apply (subst Vfrom, blast)
done

subsection‹\<^term>‹Vfrom› applied to Limit Ordinals›

(*NB. limit ordinals are non-empty:
Vfrom(A,0) = A = A ∪ (⋃y∈0. Vfrom(A,y)) *)
lemma Limit_Vfrom_eq:
"Limit(i) ⟹ Vfrom(A,i) = (⋃y∈i. Vfrom(A,y))"
apply (rule Limit_has_0 [THEN ltD, THEN Vfrom_Union, THEN subst], assumption)
done

lemma Limit_VfromE:
"⟦a ∈ Vfrom(A,i);  ¬R ⟹ Limit(i);
⋀x. ⟦x<i;  a ∈ Vfrom(A,x)⟧ ⟹ R
⟧ ⟹ R"
apply (rule classical)
apply (rule Limit_Vfrom_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
prefer 2 apply assumption
apply blast
apply (blast intro: ltI Limit_is_Ord)
done

lemma singleton_in_VLimit:
"⟦a ∈ Vfrom(A,i);  Limit(i)⟧ ⟹ {a} ∈ Vfrom(A,i)"
apply (erule Limit_VfromE, assumption)
apply (erule singleton_in_Vfrom [THEN VfromI])
apply (blast intro: Limit_has_succ)
done

lemmas Vfrom_UnI1 =
Un_upper1 [THEN subset_refl [THEN Vfrom_mono, THEN subsetD]]
lemmas Vfrom_UnI2 =
Un_upper2 [THEN subset_refl [THEN Vfrom_mono, THEN subsetD]]

text‹Hard work is finding a single j:i such that {a,b}<=Vfrom(A,j)›
lemma doubleton_in_VLimit:
"⟦a ∈ Vfrom(A,i);  b ∈ Vfrom(A,i);  Limit(i)⟧ ⟹ {a,b} ∈ Vfrom(A,i)"
apply (erule Limit_VfromE, assumption)
apply (erule Limit_VfromE, assumption)
apply (blast intro:  VfromI [OF doubleton_in_Vfrom]
Vfrom_UnI1 Vfrom_UnI2 Limit_has_succ Un_least_lt)
done

lemma Pair_in_VLimit:
"⟦a ∈ Vfrom(A,i);  b ∈ Vfrom(A,i);  Limit(i)⟧ ⟹ ⟨a,b⟩ ∈ Vfrom(A,i)"
txt‹Infer that a, b occur at ordinals x,xa < i.›
apply (erule Limit_VfromE, assumption)
apply (erule Limit_VfromE, assumption)
txt‹Infer that \<^term>‹succ(succ(x ∪ xa)) < i››
apply (blast intro: VfromI [OF Pair_in_Vfrom]
Vfrom_UnI1 Vfrom_UnI2 Limit_has_succ Un_least_lt)
done

lemma product_VLimit: "Limit(i) ⟹ Vfrom(A,i) * Vfrom(A,i) ⊆ Vfrom(A,i)"
by (blast intro: Pair_in_VLimit)

lemmas Sigma_subset_VLimit =
subset_trans [OF Sigma_mono product_VLimit]

lemmas nat_subset_VLimit =
subset_trans [OF nat_le_Limit [THEN le_imp_subset] i_subset_Vfrom]

lemma nat_into_VLimit: "⟦n: nat;  Limit(i)⟧ ⟹ n ∈ Vfrom(A,i)"
by (blast intro: nat_subset_VLimit [THEN subsetD])

subsubsection‹Closure under Disjoint Union›

lemmas zero_in_VLimit = Limit_has_0 [THEN ltD, THEN zero_in_Vfrom]

lemma one_in_VLimit: "Limit(i) ⟹ 1 ∈ Vfrom(A,i)"
by (blast intro: nat_into_VLimit)

lemma Inl_in_VLimit:
"⟦a ∈ Vfrom(A,i); Limit(i)⟧ ⟹ Inl(a) ∈ Vfrom(A,i)"
unfolding Inl_def
apply (blast intro: zero_in_VLimit Pair_in_VLimit)
done

lemma Inr_in_VLimit:
"⟦b ∈ Vfrom(A,i); Limit(i)⟧ ⟹ Inr(b) ∈ Vfrom(A,i)"
unfolding Inr_def
apply (blast intro: one_in_VLimit Pair_in_VLimit)
done

lemma sum_VLimit: "Limit(i) ⟹ Vfrom(C,i)+Vfrom(C,i) ⊆ Vfrom(C,i)"
by (blast intro!: Inl_in_VLimit Inr_in_VLimit)

lemmas sum_subset_VLimit = subset_trans [OF sum_mono sum_VLimit]

subsection‹Properties assuming \<^term>‹Transset(A)››

lemma Transset_Vfrom: "Transset(A) ⟹ Transset(Vfrom(A,i))"
apply (rule_tac a=i in eps_induct)
apply (subst Vfrom)
apply (blast intro!: Transset_Union_family Transset_Un Transset_Pow)
done

lemma Transset_Vfrom_succ:
"Transset(A) ⟹ Vfrom(A, succ(i)) = Pow(Vfrom(A,i))"
apply (rule Vfrom_succ [THEN trans])
apply (rule equalityI [OF _ Un_upper2])
apply (rule Un_least [OF _ subset_refl])
apply (rule A_subset_Vfrom [THEN subset_trans])
apply (erule Transset_Vfrom [THEN Transset_iff_Pow [THEN iffD1]])
done

lemma Transset_Pair_subset: "⟦⟨a,b⟩ ⊆ C; Transset(C)⟧ ⟹ a: C ∧ b: C"
by (unfold Pair_def Transset_def, blast)

lemma Transset_Pair_subset_VLimit:
"⟦⟨a,b⟩ ⊆ Vfrom(A,i);  Transset(A);  Limit(i)⟧
⟹ ⟨a,b⟩ ∈ Vfrom(A,i)"
apply (erule Transset_Pair_subset [THEN conjE])
apply (erule Transset_Vfrom)
apply (blast intro: Pair_in_VLimit)
done

lemma Union_in_Vfrom:
"⟦X ∈ Vfrom(A,j);  Transset(A)⟧ ⟹ ⋃(X) ∈ Vfrom(A, succ(j))"
apply (drule Transset_Vfrom)
apply (rule subset_mem_Vfrom)
apply (unfold Transset_def, blast)
done

lemma Union_in_VLimit:
"⟦X ∈ Vfrom(A,i);  Limit(i);  Transset(A)⟧ ⟹ ⋃(X) ∈ Vfrom(A,i)"
apply (rule Limit_VfromE, assumption+)
apply (blast intro: Limit_has_succ VfromI Union_in_Vfrom)
done

(*** Closure under product/sum applied to elements -- thus Vfrom(A,i)
is a model of simple type theory provided A is a transitive set
and i is a limit ordinal
***)

text‹General theorem for membership in Vfrom(A,i) when i is a limit ordinal›
lemma in_VLimit:
"⟦a ∈ Vfrom(A,i);  b ∈ Vfrom(A,i);  Limit(i);
⋀x y j. ⟦j<i; 1:j; x ∈ Vfrom(A,j); y ∈ Vfrom(A,j)⟧
⟹ ∃k. h(x,y) ∈ Vfrom(A,k) ∧ k<i⟧
⟹ h(a,b) ∈ Vfrom(A,i)"
txt‹Infer that a, b occur at ordinals x,xa < i.›
apply (erule Limit_VfromE, assumption)
apply (erule Limit_VfromE, assumption, atomize)
apply (drule_tac x=a in spec)
apply (drule_tac x=b in spec)
apply (drule_tac x="x ∪ xa ∪ 2" in spec)
apply (simp add: Un_least_lt_iff lt_Ord Vfrom_UnI1 Vfrom_UnI2)
apply (blast intro: Limit_has_0 Limit_has_succ VfromI)
done

subsubsection‹Products›

lemma prod_in_Vfrom:
"⟦a ∈ Vfrom(A,j);  b ∈ Vfrom(A,j);  Transset(A)⟧
⟹ a*b ∈ Vfrom(A, succ(succ(succ(j))))"
apply (drule Transset_Vfrom)
apply (rule subset_mem_Vfrom)
unfolding Transset_def
apply (blast intro: Pair_in_Vfrom)
done

lemma prod_in_VLimit:
"⟦a ∈ Vfrom(A,i);  b ∈ Vfrom(A,i);  Limit(i);  Transset(A)⟧
⟹ a*b ∈ Vfrom(A,i)"
apply (erule in_VLimit, assumption+)
apply (blast intro: prod_in_Vfrom Limit_has_succ)
done

subsubsection‹Disjoint Sums, or Quine Ordered Pairs›

lemma sum_in_Vfrom:
"⟦a ∈ Vfrom(A,j);  b ∈ Vfrom(A,j);  Transset(A);  1:j⟧
⟹ a+b ∈ Vfrom(A, succ(succ(succ(j))))"
unfolding sum_def
apply (drule Transset_Vfrom)
apply (rule subset_mem_Vfrom)
unfolding Transset_def
apply (blast intro: zero_in_Vfrom Pair_in_Vfrom i_subset_Vfrom [THEN subsetD])
done

lemma sum_in_VLimit:
"⟦a ∈ Vfrom(A,i);  b ∈ Vfrom(A,i);  Limit(i);  Transset(A)⟧
⟹ a+b ∈ Vfrom(A,i)"
apply (erule in_VLimit, assumption+)
apply (blast intro: sum_in_Vfrom Limit_has_succ)
done

subsubsection‹Function Space!›

lemma fun_in_Vfrom:
"⟦a ∈ Vfrom(A,j);  b ∈ Vfrom(A,j);  Transset(A)⟧ ⟹
a->b ∈ Vfrom(A, succ(succ(succ(succ(j)))))"
unfolding Pi_def
apply (drule Transset_Vfrom)
apply (rule subset_mem_Vfrom)
apply (rule Collect_subset [THEN subset_trans])
apply (subst Vfrom)
apply (rule subset_trans [THEN subset_trans])
apply (rule_tac [3] Un_upper2)
apply (rule_tac [2] succI1 [THEN UN_upper])
apply (rule Pow_mono)
unfolding Transset_def
apply (blast intro: Pair_in_Vfrom)
done

lemma fun_in_VLimit:
"⟦a ∈ Vfrom(A,i);  b ∈ Vfrom(A,i);  Limit(i);  Transset(A)⟧
⟹ a->b ∈ Vfrom(A,i)"
apply (erule in_VLimit, assumption+)
apply (blast intro: fun_in_Vfrom Limit_has_succ)
done

lemma Pow_in_Vfrom:
"⟦a ∈ Vfrom(A,j);  Transset(A)⟧ ⟹ Pow(a) ∈ Vfrom(A, succ(succ(j)))"
apply (drule Transset_Vfrom)
apply (rule subset_mem_Vfrom)
unfolding Transset_def
apply (subst Vfrom, blast)
done

lemma Pow_in_VLimit:
"⟦a ∈ Vfrom(A,i);  Limit(i);  Transset(A)⟧ ⟹ Pow(a) ∈ Vfrom(A,i)"
by (blast elim: Limit_VfromE intro: Limit_has_succ Pow_in_Vfrom VfromI)

subsection‹The Set \<^term>‹Vset(i)››

lemma Vset: "Vset(i) = (⋃j∈i. Pow(Vset(j)))"
by (subst Vfrom, blast)

lemmas Vset_succ = Transset_0 [THEN Transset_Vfrom_succ]
lemmas Transset_Vset = Transset_0 [THEN Transset_Vfrom]

subsubsection‹Characterisation of the elements of \<^term>‹Vset(i)››

lemma VsetD [rule_format]: "Ord(i) ⟹ ∀b. b ∈ Vset(i) ⟶ rank(b) < i"
apply (erule trans_induct)
apply (subst Vset, safe)
apply (subst rank)
apply (blast intro: ltI UN_succ_least_lt)
done

lemma VsetI_lemma [rule_format]:
"Ord(i) ⟹ ∀b. rank(b) ∈ i ⟶ b ∈ Vset(i)"
apply (erule trans_induct)
apply (rule allI)
apply (subst Vset)
apply (blast intro!: rank_lt [THEN ltD])
done

lemma VsetI: "rank(x)<i ⟹ x ∈ Vset(i)"
by (blast intro: VsetI_lemma elim: ltE)

text‹Merely a lemma for the next result›
lemma Vset_Ord_rank_iff: "Ord(i) ⟹ b ∈ Vset(i) ⟷ rank(b) < i"
by (blast intro: VsetD VsetI)

lemma Vset_rank_iff [simp]: "b ∈ Vset(a) ⟷ rank(b) < rank(a)"
apply (rule Vfrom_rank_eq [THEN subst])
apply (rule Ord_rank [THEN Vset_Ord_rank_iff])
done

text‹This is rank(rank(a)) = rank(a)›
declare Ord_rank [THEN rank_of_Ord, simp]

lemma rank_Vset: "Ord(i) ⟹ rank(Vset(i)) = i"
apply (subst rank)
apply (rule equalityI, safe)
apply (blast intro: VsetD [THEN ltD])
apply (blast intro: VsetD [THEN ltD] Ord_trans)
apply (blast intro: i_subset_Vfrom [THEN subsetD]
Ord_in_Ord [THEN rank_of_Ord, THEN ssubst])
done

lemma Finite_Vset: "i ∈ nat ⟹ Finite(Vset(i))"
apply (erule nat_induct)
done

subsubsection‹Reasoning about Sets in Terms of Their Elements' Ranks›

lemma arg_subset_Vset_rank: "a ⊆ Vset(rank(a))"
apply (rule subsetI)
apply (erule rank_lt [THEN VsetI])
done

lemma Int_Vset_subset:
"⟦⋀i. Ord(i) ⟹ a ∩ Vset(i) ⊆ b⟧ ⟹ a ⊆ b"
apply (rule subset_trans)
apply (rule Int_greatest [OF subset_refl arg_subset_Vset_rank])
apply (blast intro: Ord_rank)
done

subsubsection‹Set Up an Environment for Simplification›

lemma rank_Inl: "rank(a) < rank(Inl(a))"
unfolding Inl_def
apply (rule rank_pair2)
done

lemma rank_Inr: "rank(a) < rank(Inr(a))"
unfolding Inr_def
apply (rule rank_pair2)
done

lemmas rank_rls = rank_Inl rank_Inr rank_pair1 rank_pair2

subsubsection‹Recursion over Vset Levels!›

text‹NOT SUITABLE FOR REWRITING: recursive!›
lemma Vrec: "Vrec(a,H) = H(a, λx∈Vset(rank(a)). Vrec(x,H))"
unfolding Vrec_def
apply (subst transrec, simp)
apply (rule refl [THEN lam_cong, THEN subst_context], simp add: lt_def)
done

text‹This form avoids giant explosions in proofs. NOTE the form of the premise!›
lemma def_Vrec:
"⟦⋀x. h(x)≡Vrec(x,H)⟧ ⟹
h(a) = H(a, λx∈Vset(rank(a)). h(x))"
apply simp
apply (rule Vrec)
done

text‹NOT SUITABLE FOR REWRITING: recursive!›
lemma Vrecursor:
"Vrecursor(H,a) = H(λx∈Vset(rank(a)). Vrecursor(H,x),  a)"
unfolding Vrecursor_def
apply (subst transrec, simp)
apply (rule refl [THEN lam_cong, THEN subst_context], simp add: lt_def)
done

text‹This form avoids giant explosions in proofs. NOTE the form of the premise!›
lemma def_Vrecursor:
"h ≡ Vrecursor(H) ⟹ h(a) = H(λx∈Vset(rank(a)). h(x),  a)"
apply simp
apply (rule Vrecursor)
done

subsection‹The Datatype Universe: \<^term>‹univ(A)››

lemma univ_mono: "A<=B ⟹ univ(A) ⊆ univ(B)"
unfolding univ_def
apply (erule Vfrom_mono)
apply (rule subset_refl)
done

lemma Transset_univ: "Transset(A) ⟹ Transset(univ(A))"
unfolding univ_def
apply (erule Transset_Vfrom)
done

subsubsection‹The Set \<^term>‹univ(A)› as a Limit›

lemma univ_eq_UN: "univ(A) = (⋃i∈nat. Vfrom(A,i))"
unfolding univ_def
apply (rule Limit_nat [THEN Limit_Vfrom_eq])
done

lemma subset_univ_eq_Int: "c ⊆ univ(A) ⟹ c = (⋃i∈nat. c ∩ Vfrom(A,i))"
apply (rule subset_UN_iff_eq [THEN iffD1])
apply (erule univ_eq_UN [THEN subst])
done

lemma univ_Int_Vfrom_subset:
"⟦a ⊆ univ(X);
⋀i. i:nat ⟹ a ∩ Vfrom(X,i) ⊆ b⟧
⟹ a ⊆ b"
apply (subst subset_univ_eq_Int, assumption)
apply (rule UN_least, simp)
done

lemma univ_Int_Vfrom_eq:
"⟦a ⊆ univ(X);   b ⊆ univ(X);
⋀i. i:nat ⟹ a ∩ Vfrom(X,i) = b ∩ Vfrom(X,i)
⟧ ⟹ a = b"
apply (rule equalityI)
apply (rule univ_Int_Vfrom_subset, assumption)
apply (blast elim: equalityCE)
apply (rule univ_Int_Vfrom_subset, assumption)
apply (blast elim: equalityCE)
done

subsection‹Closure Properties for \<^term>‹univ(A)››

lemma zero_in_univ: "0 ∈ univ(A)"
unfolding univ_def
apply (rule nat_0I [THEN zero_in_Vfrom])
done

lemma zero_subset_univ: "{0} ⊆ univ(A)"
by (blast intro: zero_in_univ)

lemma A_subset_univ: "A ⊆ univ(A)"
unfolding univ_def
apply (rule A_subset_Vfrom)
done

lemmas A_into_univ = A_subset_univ [THEN subsetD]

subsubsection‹Closure under Unordered and Ordered Pairs›

lemma singleton_in_univ: "a: univ(A) ⟹ {a} ∈ univ(A)"
unfolding univ_def
apply (blast intro: singleton_in_VLimit Limit_nat)
done

lemma doubleton_in_univ:
"⟦a: univ(A);  b: univ(A)⟧ ⟹ {a,b} ∈ univ(A)"
unfolding univ_def
apply (blast intro: doubleton_in_VLimit Limit_nat)
done

lemma Pair_in_univ:
"⟦a: univ(A);  b: univ(A)⟧ ⟹ ⟨a,b⟩ ∈ univ(A)"
unfolding univ_def
apply (blast intro: Pair_in_VLimit Limit_nat)
done

lemma Union_in_univ:
"⟦X: univ(A);  Transset(A)⟧ ⟹ ⋃(X) ∈ univ(A)"
unfolding univ_def
apply (blast intro: Union_in_VLimit Limit_nat)
done

lemma product_univ: "univ(A)*univ(A) ⊆ univ(A)"
unfolding univ_def
apply (rule Limit_nat [THEN product_VLimit])
done

subsubsection‹The Natural Numbers›

lemma nat_subset_univ: "nat ⊆ univ(A)"
unfolding univ_def
apply (rule i_subset_Vfrom)
done

lemma nat_into_univ: "n ∈ nat ⟹ n ∈ univ(A)"
by (rule nat_subset_univ [THEN subsetD])

subsubsection‹Instances for 1 and 2›

lemma one_in_univ: "1 ∈ univ(A)"
unfolding univ_def
apply (rule Limit_nat [THEN one_in_VLimit])
done

text‹unused!›
lemma two_in_univ: "2 ∈ univ(A)"
by (blast intro: nat_into_univ)

lemma bool_subset_univ: "bool ⊆ univ(A)"
unfolding bool_def
apply (blast intro!: zero_in_univ one_in_univ)
done

lemmas bool_into_univ = bool_subset_univ [THEN subsetD]

subsubsection‹Closure under Disjoint Union›

lemma Inl_in_univ: "a: univ(A) ⟹ Inl(a) ∈ univ(A)"
unfolding univ_def
apply (erule Inl_in_VLimit [OF _ Limit_nat])
done

lemma Inr_in_univ: "b: univ(A) ⟹ Inr(b) ∈ univ(A)"
unfolding univ_def
apply (erule Inr_in_VLimit [OF _ Limit_nat])
done

lemma sum_univ: "univ(C)+univ(C) ⊆ univ(C)"
unfolding univ_def
apply (rule Limit_nat [THEN sum_VLimit])
done

lemmas sum_subset_univ = subset_trans [OF sum_mono sum_univ]

lemma Sigma_subset_univ:
"⟦A ⊆ univ(D); ⋀x. x ∈ A ⟹ B(x) ⊆ univ(D)⟧ ⟹ Sigma(A,B) ⊆ univ(D)"
apply (blast intro: Sigma_subset_VLimit del: subsetI)
done

(*Closure under binary union -- use Un_least
Closure under Collect -- use  Collect_subset [THEN subset_trans]
Closure under RepFun -- use   RepFun_subset *)

subsection‹Finite Branching Closure Properties›

subsubsection‹Closure under Finite Powerset›

lemma Fin_Vfrom_lemma:
"⟦b: Fin(Vfrom(A,i));  Limit(i)⟧ ⟹ ∃j. b ⊆ Vfrom(A,j) ∧ j<i"
apply (erule Fin_induct)
apply (blast dest!: Limit_has_0, safe)
apply (erule Limit_VfromE, assumption)
apply (blast intro!: Un_least_lt intro: Vfrom_UnI1 Vfrom_UnI2)
done

lemma Fin_VLimit: "Limit(i) ⟹ Fin(Vfrom(A,i)) ⊆ Vfrom(A,i)"
apply (rule subsetI)
apply (drule Fin_Vfrom_lemma, safe)
apply (rule Vfrom [THEN ssubst])
apply (blast dest!: ltD)
done

lemmas Fin_subset_VLimit = subset_trans [OF Fin_mono Fin_VLimit]

lemma Fin_univ: "Fin(univ(A)) ⊆ univ(A)"
unfolding univ_def
apply (rule Limit_nat [THEN Fin_VLimit])
done

subsubsection‹Closure under Finite Powers: Functions from a Natural Number›

lemma nat_fun_VLimit:
"⟦n: nat;  Limit(i)⟧ ⟹ n -> Vfrom(A,i) ⊆ Vfrom(A,i)"
apply (erule nat_fun_subset_Fin [THEN subset_trans])
apply (blast del: subsetI
intro: subset_refl Fin_subset_VLimit Sigma_subset_VLimit nat_subset_VLimit)
done

lemmas nat_fun_subset_VLimit = subset_trans [OF Pi_mono nat_fun_VLimit]

lemma nat_fun_univ: "n: nat ⟹ n -> univ(A) ⊆ univ(A)"
unfolding univ_def
apply (erule nat_fun_VLimit [OF _ Limit_nat])
done

subsubsection‹Closure under Finite Function Space›

text‹General but seldom-used version; normally the domain is fixed›
lemma FiniteFun_VLimit1:
"Limit(i) ⟹ Vfrom(A,i) -||> Vfrom(A,i) ⊆ Vfrom(A,i)"
apply (rule FiniteFun.dom_subset [THEN subset_trans])
apply (blast del: subsetI
intro: Fin_subset_VLimit Sigma_subset_VLimit subset_refl)
done

lemma FiniteFun_univ1: "univ(A) -||> univ(A) ⊆ univ(A)"
unfolding univ_def
apply (rule Limit_nat [THEN FiniteFun_VLimit1])
done

text‹Version for a fixed domain›
lemma FiniteFun_VLimit:
"⟦W ⊆ Vfrom(A,i); Limit(i)⟧ ⟹ W -||> Vfrom(A,i) ⊆ Vfrom(A,i)"
apply (rule subset_trans)
apply (erule FiniteFun_mono [OF _ subset_refl])
apply (erule FiniteFun_VLimit1)
done

lemma FiniteFun_univ:
"W ⊆ univ(A) ⟹ W -||> univ(A) ⊆ univ(A)"
unfolding univ_def
apply (erule FiniteFun_VLimit [OF _ Limit_nat])
done

lemma FiniteFun_in_univ:
"⟦f: W -||> univ(A);  W ⊆ univ(A)⟧ ⟹ f ∈ univ(A)"
by (erule FiniteFun_univ [THEN subsetD], assumption)

text‹Remove ‹⊆› from the rule above›
lemmas FiniteFun_in_univ' = FiniteFun_in_univ [OF _ subsetI]

subsection‹* For QUniv.  Properties of Vfrom analogous to the "take-lemma" *›

text‹Intersecting a*b with Vfrom...›

text‹This version says a, b exist one level down, in the smaller set Vfrom(X,i)›
lemma doubleton_in_Vfrom_D:
"⟦{a,b} ∈ Vfrom(X,succ(i));  Transset(X)⟧
⟹ a ∈ Vfrom(X,i)  ∧  b ∈ Vfrom(X,i)"
by (drule Transset_Vfrom_succ [THEN equalityD1, THEN subsetD, THEN PowD],
assumption, fast)

text‹This weaker version says a, b exist at the same level›
lemmas Vfrom_doubleton_D = Transset_Vfrom [THEN Transset_doubleton_D]

(** Using only the weaker theorem would prove ⟨a,b⟩ ∈ Vfrom(X,i)
implies a, b ∈ Vfrom(X,i), which is useless for induction.
Using only the stronger theorem would prove ⟨a,b⟩ ∈ Vfrom(X,succ(succ(i)))
implies a, b ∈ Vfrom(X,i), leaving the succ(i) case untreated.
The combination gives a reduction by precisely one level, which is
most convenient for proofs.
**)

lemma Pair_in_Vfrom_D:
"⟦⟨a,b⟩ ∈ Vfrom(X,succ(i));  Transset(X)⟧
⟹ a ∈ Vfrom(X,i)  ∧  b ∈ Vfrom(X,i)"
unfolding Pair_def
apply (blast dest!: doubleton_in_Vfrom_D Vfrom_doubleton_D)
done

lemma product_Int_Vfrom_subset:
"Transset(X) ⟹
(a*b) ∩ Vfrom(X, succ(i)) ⊆ (a ∩ Vfrom(X,i)) * (b ∩ Vfrom(X,i))"
by (blast dest!: Pair_in_Vfrom_D)

ML
‹
val rank_ss =