(* Title: ZF/QUniv.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section\<open>A Small Universe for Lazy Recursive Types\<close>
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\<open>Properties involving Transset and Sum\<close>
lemma Transset_includes_summands:
"[| Transset(C); A+B \<subseteq> C |] ==> A \<subseteq> C & B \<subseteq> C"
apply (simp add: sum_def Un_subset_iff)
apply (blast dest: Transset_includes_range)
done
lemma Transset_sum_Int_subset:
"Transset(C) ==> (A+B) \<inter> C \<subseteq> (A \<inter> C) + (B \<inter> C)"
apply (simp add: sum_def Int_Un_distrib2)
apply (blast dest: Transset_Pair_D)
done
subsection\<open>Introduction and Elimination Rules\<close>
lemma qunivI: "X \<subseteq> univ(eclose(A)) ==> X \<in> quniv(A)"
by (simp add: quniv_def)
lemma qunivD: "X \<in> quniv(A) ==> X \<subseteq> univ(eclose(A))"
by (simp add: quniv_def)
lemma quniv_mono: "A<=B ==> quniv(A) \<subseteq> quniv(B)"
apply (unfold quniv_def)
apply (erule eclose_mono [THEN univ_mono, THEN Pow_mono])
done
subsection\<open>Closure Properties\<close>
lemma univ_eclose_subset_quniv: "univ(eclose(A)) \<subseteq> 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) \<subseteq> 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)) \<subseteq> quniv(A)"
apply (unfold 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 \<subseteq> univ(A); b \<subseteq> univ(A) |] ==> <a;b> \<subseteq> univ(A)"
by (simp add: QPair_def sum_subset_univ)
subsection\<open>Quine Disjoint Sum\<close>
lemma QInl_subset_univ: "a \<subseteq> univ(A) ==> QInl(a) \<subseteq> univ(A)"
apply (unfold 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 \<subseteq> univ(A) ==> QInr(a) \<subseteq> univ(A)"
apply (unfold QInr_def)
apply (erule nat_1I [THEN naturals_subset_univ, THEN QPair_subset_univ])
done
subsection\<open>Closure for Quine-Inspired Products and Sums\<close>
(*Quine ordered pairs*)
lemma QPair_in_quniv:
"[| a: quniv(A); b: quniv(A) |] ==> <a;b> \<in> quniv(A)"
by (simp add: quniv_def QPair_def sum_subset_univ)
lemma QSigma_quniv: "quniv(A) <*> quniv(A) \<subseteq> 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> \<in> quniv(A) ==> a: quniv(A) & b: quniv(A)"
apply (unfold 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> \<in> quniv(A) \<longleftrightarrow> a: quniv(A) & b: quniv(A)"
by (blast intro: QPair_in_quniv dest: quniv_QPair_D)
subsection\<open>Quine Disjoint Sum\<close>
lemma QInl_in_quniv: "a: quniv(A) ==> QInl(a) \<in> quniv(A)"
by (simp add: QInl_def zero_in_quniv QPair_in_quniv)
lemma QInr_in_quniv: "b: quniv(A) ==> QInr(b) \<in> quniv(A)"
by (simp add: QInr_def one_in_quniv QPair_in_quniv)
lemma qsum_quniv: "quniv(C) <+> quniv(C) \<subseteq> quniv(C)"
by (blast intro: QInl_in_quniv QInr_in_quniv)
lemmas qsum_subset_quniv = subset_trans [OF qsum_mono qsum_quniv]
subsection\<open>The Natural Numbers\<close>
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> \<inter> Vfrom(X, succ(i)) \<subseteq> <a \<inter> Vfrom(X,i); b \<inter> 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\<open>"Take-Lemma" Rules\<close>
(*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> \<inter> Vfrom(X,i) \<subseteq> <a \<inter> Vfrom(X,i); b \<inter> Vfrom(X,i)>"
apply (unfold QPair_def)
apply (erule Transset_Vfrom [THEN Transset_sum_Int_subset])
done
(*@{term"[| a \<inter> Vset(i) \<subseteq> c; b \<inter> Vset(i) \<subseteq> d |] ==> <a;b> \<inter> Vset(i) \<subseteq> <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> \<inter> Vset(i) \<subseteq> (\<Union>j\<in>i. <a \<inter> Vset(j); b \<inter> Vset(j)>)"
apply (erule Ord_cases)
(*0 case*)
apply (simp add: Vfrom_0)
(*succ(j) case*)
apply (erule ssubst)
apply (rule Transset_0 [THEN QPair_Int_Vfrom_succ_subset, THEN subset_trans])
apply (rule succI1 [THEN UN_upper])
(*Limit(i) case*)
apply (simp del: UN_simps
add: Limit_Vfrom_eq Int_UN_distrib UN_mono QPair_Int_Vset_subset_trans)
done
end