(* Title: ZF/QUniv.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
header{*A Small Universe for Lazy Recursive Types*}
theory QUniv = Univ + QPair:
(*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
constdefs
quniv :: "i => i"
"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 (simp add: sum_def Un_subset_iff)
apply (blast dest: Transset_includes_range)
done
lemma Transset_sum_Int_subset:
"Transset(C) ==> (A+B) Int C <= (A Int C) + (B Int C)"
apply (simp add: sum_def Int_Un_distrib2)
apply (blast dest: Transset_Pair_D)
done
subsection{*Introduction and Elimination Rules*}
lemma qunivI: "X <= univ(eclose(A)) ==> X : quniv(A)"
by (simp add: quniv_def)
lemma qunivD: "X : quniv(A) ==> X <= univ(eclose(A))"
by (simp add: quniv_def)
lemma quniv_mono: "A<=B ==> quniv(A) <= quniv(B)"
apply (unfold 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 def 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, standard]
lemma Pow_univ_subset_quniv: "Pow(univ(A)) <= 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], standard]
lemmas zero_in_quniv = zero_in_univ [THEN univ_into_quniv, standard]
lemmas one_in_quniv = one_in_univ [THEN univ_into_quniv, standard]
lemmas two_in_quniv = two_in_univ [THEN univ_into_quniv, standard]
lemmas A_subset_quniv = subset_trans [OF A_subset_univ univ_subset_quniv]
lemmas A_into_quniv = A_subset_quniv [THEN subsetD, standard]
(*** 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)"
by (simp add: QPair_def sum_subset_univ)
subsection{*Quine Disjoint Sum*}
lemma QInl_subset_univ: "a <= univ(A) ==> QInl(a) <= 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, standard]
lemmas naturals_subset_univ =
subset_trans [OF naturals_subset_nat nat_subset_univ]
lemma QInr_subset_univ: "a <= univ(A) ==> QInr(a) <= univ(A)"
apply (unfold 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)"
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, standard]
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, standard]
lemmas bool_subset_quniv = subset_trans [OF bool_subset_univ univ_subset_quniv]
lemmas bool_into_quniv = bool_subset_quniv [THEN subsetD, standard]
(*Intersecting <a;b> with Vfrom...*)
lemma QPair_Int_Vfrom_succ_subset:
"Transset(X) ==>
<a;b> Int Vfrom(X, succ(i)) <= <a Int Vfrom(X,i); b Int 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> Int Vfrom(X,i) <= <a Int Vfrom(X,i); b Int Vfrom(X,i)>"
apply (unfold QPair_def)
apply (erule Transset_Vfrom [THEN Transset_sum_Int_subset])
done
(*[| a Int Vset(i) <= c; b Int Vset(i) <= d |] ==> <a;b> Int 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> Int Vset(i) <= (\<Union>j\<in>i. <a Int Vset(j); b Int 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
ML
{*
val Transset_includes_summands = thm "Transset_includes_summands";
val Transset_sum_Int_subset = thm "Transset_sum_Int_subset";
val qunivI = thm "qunivI";
val qunivD = thm "qunivD";
val quniv_mono = thm "quniv_mono";
val univ_eclose_subset_quniv = thm "univ_eclose_subset_quniv";
val univ_subset_quniv = thm "univ_subset_quniv";
val univ_into_quniv = thm "univ_into_quniv";
val Pow_univ_subset_quniv = thm "Pow_univ_subset_quniv";
val univ_subset_into_quniv = thm "univ_subset_into_quniv";
val zero_in_quniv = thm "zero_in_quniv";
val one_in_quniv = thm "one_in_quniv";
val two_in_quniv = thm "two_in_quniv";
val A_subset_quniv = thm "A_subset_quniv";
val A_into_quniv = thm "A_into_quniv";
val QPair_subset_univ = thm "QPair_subset_univ";
val QInl_subset_univ = thm "QInl_subset_univ";
val naturals_subset_nat = thm "naturals_subset_nat";
val naturals_subset_univ = thm "naturals_subset_univ";
val QInr_subset_univ = thm "QInr_subset_univ";
val QPair_in_quniv = thm "QPair_in_quniv";
val QSigma_quniv = thm "QSigma_quniv";
val QSigma_subset_quniv = thm "QSigma_subset_quniv";
val quniv_QPair_D = thm "quniv_QPair_D";
val quniv_QPair_E = thm "quniv_QPair_E";
val quniv_QPair_iff = thm "quniv_QPair_iff";
val QInl_in_quniv = thm "QInl_in_quniv";
val QInr_in_quniv = thm "QInr_in_quniv";
val qsum_quniv = thm "qsum_quniv";
val qsum_subset_quniv = thm "qsum_subset_quniv";
val nat_subset_quniv = thm "nat_subset_quniv";
val nat_into_quniv = thm "nat_into_quniv";
val bool_subset_quniv = thm "bool_subset_quniv";
val bool_into_quniv = thm "bool_into_quniv";
val QPair_Int_Vfrom_succ_subset = thm "QPair_Int_Vfrom_succ_subset";
val QPair_Int_Vfrom_subset = thm "QPair_Int_Vfrom_subset";
val QPair_Int_Vset_subset_trans = thm "QPair_Int_Vset_subset_trans";
val QPair_Int_Vset_subset_UN = thm "QPair_Int_Vset_subset_UN";
*}
end