# Theory QUniv

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

section‹A Small Universe for Lazy Recursive Types›

theory QUniv imports Univ QPair begin

(*Disjoint sums as a datatype*)
rep_datatype
elimination   sumE
induction     TrueI
case_eqns     case_Inl case_Inr

(*Variant disjoint sums as a datatype*)
rep_datatype
elimination   qsumE
induction     TrueI
case_eqns     qcase_QInl qcase_QInr

definition
quniv :: "i ⇒ i"  where
"quniv(A) ≡ Pow(univ(eclose(A)))"

subsection‹Properties involving Transset and Sum›

lemma Transset_includes_summands:
"⟦Transset(C); A+B ⊆ C⟧ ⟹ A ⊆ C ∧ B ⊆ C"
apply (blast dest: Transset_includes_range)
done

lemma Transset_sum_Int_subset:
"Transset(C) ⟹ (A+B) ∩ C ⊆ (A ∩ C) + (B ∩ C)"
apply (blast dest: Transset_Pair_D)
done

subsection‹Introduction and Elimination Rules›

lemma qunivI: "X ⊆ univ(eclose(A)) ⟹ X ∈ quniv(A)"

lemma qunivD: "X ∈ quniv(A) ⟹ X ⊆ univ(eclose(A))"

lemma quniv_mono: "A<=B ⟹ quniv(A) ⊆ quniv(B)"
unfolding quniv_def
apply (erule eclose_mono [THEN univ_mono, THEN Pow_mono])
done

subsection‹Closure Properties›

lemma univ_eclose_subset_quniv: "univ(eclose(A)) ⊆ quniv(A)"
apply (simp add: quniv_def Transset_iff_Pow [symmetric])
apply (rule Transset_eclose [THEN Transset_univ])
done

(*Key property for proving A_subset_quniv; requires eclose in definition of quniv*)
lemma univ_subset_quniv: "univ(A) ⊆ quniv(A)"
apply (rule arg_subset_eclose [THEN univ_mono, THEN subset_trans])
apply (rule univ_eclose_subset_quniv)
done

lemmas univ_into_quniv = univ_subset_quniv [THEN subsetD]

lemma Pow_univ_subset_quniv: "Pow(univ(A)) ⊆ quniv(A)"
unfolding quniv_def
apply (rule arg_subset_eclose [THEN univ_mono, THEN Pow_mono])
done

lemmas univ_subset_into_quniv =
PowI [THEN Pow_univ_subset_quniv [THEN subsetD]]

lemmas zero_in_quniv = zero_in_univ [THEN univ_into_quniv]
lemmas one_in_quniv = one_in_univ [THEN univ_into_quniv]
lemmas two_in_quniv = two_in_univ [THEN univ_into_quniv]

lemmas A_subset_quniv =  subset_trans [OF A_subset_univ univ_subset_quniv]

lemmas A_into_quniv = A_subset_quniv [THEN subsetD]

(*** univ(A) closure for Quine-inspired pairs and injections ***)

(*Quine ordered pairs*)
lemma QPair_subset_univ:
"⟦a ⊆ univ(A);  b ⊆ univ(A)⟧ ⟹ <a;b> ⊆ univ(A)"

subsection‹Quine Disjoint Sum›

lemma QInl_subset_univ: "a ⊆ univ(A) ⟹ QInl(a) ⊆ univ(A)"
unfolding QInl_def
apply (erule empty_subsetI [THEN QPair_subset_univ])
done

lemmas naturals_subset_nat =
Ord_nat [THEN Ord_is_Transset, unfolded Transset_def, THEN bspec]

lemmas naturals_subset_univ =
subset_trans [OF naturals_subset_nat nat_subset_univ]

lemma QInr_subset_univ: "a ⊆ univ(A) ⟹ QInr(a) ⊆ univ(A)"
unfolding QInr_def
apply (erule nat_1I [THEN naturals_subset_univ, THEN QPair_subset_univ])
done

subsection‹Closure for Quine-Inspired Products and Sums›

(*Quine ordered pairs*)
lemma QPair_in_quniv:
"⟦a: quniv(A);  b: quniv(A)⟧ ⟹ <a;b> ∈ quniv(A)"
by (simp add: quniv_def QPair_def sum_subset_univ)

lemma QSigma_quniv: "quniv(A) <*> quniv(A) ⊆ quniv(A)"
by (blast intro: QPair_in_quniv)

lemmas QSigma_subset_quniv =  subset_trans [OF QSigma_mono QSigma_quniv]

(*The opposite inclusion*)
lemma quniv_QPair_D:
"<a;b> ∈ quniv(A) ⟹ a: quniv(A) ∧ b: quniv(A)"
unfolding quniv_def QPair_def
apply (rule Transset_includes_summands [THEN conjE])
apply (rule Transset_eclose [THEN Transset_univ])
apply (erule PowD, blast)
done

lemmas quniv_QPair_E = quniv_QPair_D [THEN conjE]

lemma quniv_QPair_iff: "<a;b> ∈ quniv(A) ⟷ a: quniv(A) ∧ b: quniv(A)"
by (blast intro: QPair_in_quniv dest: quniv_QPair_D)

subsection‹Quine Disjoint Sum›

lemma QInl_in_quniv: "a: quniv(A) ⟹ QInl(a) ∈ quniv(A)"
by (simp add: QInl_def zero_in_quniv QPair_in_quniv)

lemma QInr_in_quniv: "b: quniv(A) ⟹ QInr(b) ∈ quniv(A)"
by (simp add: QInr_def one_in_quniv QPair_in_quniv)

lemma qsum_quniv: "quniv(C) <+> quniv(C) ⊆ quniv(C)"
by (blast intro: QInl_in_quniv QInr_in_quniv)

lemmas qsum_subset_quniv = subset_trans [OF qsum_mono qsum_quniv]

subsection‹The Natural Numbers›

lemmas nat_subset_quniv =  subset_trans [OF nat_subset_univ univ_subset_quniv]

(* n:nat ⟹ n:quniv(A) *)
lemmas nat_into_quniv = nat_subset_quniv [THEN subsetD]

lemmas bool_subset_quniv = subset_trans [OF bool_subset_univ univ_subset_quniv]

lemmas bool_into_quniv = bool_subset_quniv [THEN subsetD]

(*Intersecting <a;b> with Vfrom...*)

lemma QPair_Int_Vfrom_succ_subset:
"Transset(X) ⟹
<a;b> ∩ Vfrom(X, succ(i))  ⊆  <a ∩ Vfrom(X,i);  b ∩ Vfrom(X,i)>"
by (simp add: QPair_def sum_def Int_Un_distrib2 Un_mono
product_Int_Vfrom_subset [THEN subset_trans]
Sigma_mono [OF Int_lower1 subset_refl])

subsection‹"Take-Lemma" Rules›

(*for proving a=b by coinduction and c: quniv(A)*)

(*Rule for level i -- preserving the level, not decreasing it*)

lemma QPair_Int_Vfrom_subset:
"Transset(X) ⟹
<a;b> ∩ Vfrom(X,i)  ⊆  <a ∩ Vfrom(X,i);  b ∩ Vfrom(X,i)>"
unfolding QPair_def
apply (erule Transset_Vfrom [THEN Transset_sum_Int_subset])
done

(*@{term"⟦a ∩ Vset(i) ⊆ c; b ∩ Vset(i) ⊆ d⟧ ⟹ <a;b> ∩ Vset(i) ⊆ <c;d>"}*)
lemmas QPair_Int_Vset_subset_trans =
subset_trans [OF Transset_0 [THEN QPair_Int_Vfrom_subset] QPair_mono]

lemma QPair_Int_Vset_subset_UN:
"Ord(i) ⟹ <a;b> ∩ Vset(i) ⊆ (⋃j∈i. <a ∩ Vset(j); b ∩ Vset(j)>)"
apply (erule Ord_cases)
(*0 case*)