--- a/src/ZF/QUniv.ML Tue Jul 02 22:46:23 2002 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,196 +0,0 @@
-(* Title: ZF/QUniv
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1993 University of Cambridge
-
-A small universe for lazy recursive types
-*)
-
-(** Properties involving Transset and Sum **)
-
-Goalw [sum_def] "[| Transset(C); A+B <= C |] ==> A <= C & B <= C";
-by (dtac (Un_subset_iff RS iffD1) 1);
-by (blast_tac (claset() addDs [Transset_includes_range]) 1);
-qed "Transset_includes_summands";
-
-Goalw [sum_def] "Transset(C) ==> (A+B) Int C <= (A Int C) + (B Int C)";
-by (stac Int_Un_distrib2 1);
-by (blast_tac (claset() addDs [Transset_Pair_D]) 1);
-qed "Transset_sum_Int_subset";
-
-(** Introduction and elimination rules avoid tiresome folding/unfolding **)
-
-Goalw [quniv_def]
- "X <= univ(eclose(A)) ==> X : quniv(A)";
-by (etac PowI 1);
-qed "qunivI";
-
-Goalw [quniv_def]
- "X : quniv(A) ==> X <= univ(eclose(A))";
-by (etac PowD 1);
-qed "qunivD";
-
-Goalw [quniv_def] "A<=B ==> quniv(A) <= quniv(B)";
-by (etac (eclose_mono RS univ_mono RS Pow_mono) 1);
-qed "quniv_mono";
-
-(*** Closure properties ***)
-
-Goalw [quniv_def] "univ(eclose(A)) <= quniv(A)";
-by (rtac (Transset_iff_Pow RS iffD1) 1);
-by (rtac (Transset_eclose RS Transset_univ) 1);
-qed "univ_eclose_subset_quniv";
-
-(*Key property for proving A_subset_quniv; requires eclose in def of quniv*)
-Goal "univ(A) <= quniv(A)";
-by (rtac (arg_subset_eclose RS univ_mono RS subset_trans) 1);
-by (rtac univ_eclose_subset_quniv 1);
-qed "univ_subset_quniv";
-
-bind_thm ("univ_into_quniv", univ_subset_quniv RS subsetD);
-
-Goalw [quniv_def] "Pow(univ(A)) <= quniv(A)";
-by (rtac (arg_subset_eclose RS univ_mono RS Pow_mono) 1);
-qed "Pow_univ_subset_quniv";
-
-bind_thm ("univ_subset_into_quniv",
- PowI RS (Pow_univ_subset_quniv RS subsetD));
-
-bind_thm ("zero_in_quniv", zero_in_univ RS univ_into_quniv);
-bind_thm ("one_in_quniv", one_in_univ RS univ_into_quniv);
-bind_thm ("two_in_quniv", two_in_univ RS univ_into_quniv);
-
-bind_thm ("A_subset_quniv",
- [A_subset_univ, univ_subset_quniv] MRS subset_trans);
-
-bind_thm ("A_into_quniv", A_subset_quniv RS subsetD);
-
-(*** univ(A) closure for Quine-inspired pairs and injections ***)
-
-(*Quine ordered pairs*)
-Goalw [QPair_def]
- "[| a <= univ(A); b <= univ(A) |] ==> <a;b> <= univ(A)";
-by (REPEAT (ares_tac [sum_subset_univ] 1));
-qed "QPair_subset_univ";
-
-(** Quine disjoint sum **)
-
-Goalw [QInl_def] "a <= univ(A) ==> QInl(a) <= univ(A)";
-by (etac (empty_subsetI RS QPair_subset_univ) 1);
-qed "QInl_subset_univ";
-
-bind_thm ("naturals_subset_nat",
- rewrite_rule [Transset_def] (Ord_nat RS Ord_is_Transset)
- RS bspec);
-
-bind_thm ("naturals_subset_univ",
- [naturals_subset_nat, nat_subset_univ] MRS subset_trans);
-
-Goalw [QInr_def] "a <= univ(A) ==> QInr(a) <= univ(A)";
-by (etac (nat_1I RS naturals_subset_univ RS QPair_subset_univ) 1);
-qed "QInr_subset_univ";
-
-(*** Closure for Quine-inspired products and sums ***)
-
-(*Quine ordered pairs*)
-Goalw [quniv_def,QPair_def]
- "[| a: quniv(A); b: quniv(A) |] ==> <a;b> : quniv(A)";
-by (REPEAT (dtac PowD 1));
-by (REPEAT (ares_tac [PowI, sum_subset_univ] 1));
-qed "QPair_in_quniv";
-
-Goal "quniv(A) <*> quniv(A) <= quniv(A)";
-by (REPEAT (ares_tac [subsetI, QPair_in_quniv] 1
- ORELSE eresolve_tac [QSigmaE, ssubst] 1));
-qed "QSigma_quniv";
-
-bind_thm ("QSigma_subset_quniv",
- [QSigma_mono, QSigma_quniv] MRS subset_trans);
-
-(*The opposite inclusion*)
-Goalw [quniv_def,QPair_def]
- "<a;b> : quniv(A) ==> a: quniv(A) & b: quniv(A)";
-by (etac ([Transset_eclose RS Transset_univ, PowD] MRS
- Transset_includes_summands RS conjE) 1);
-by (REPEAT (ares_tac [conjI,PowI] 1));
-qed "quniv_QPair_D";
-
-bind_thm ("quniv_QPair_E", quniv_QPair_D RS conjE);
-
-Goal "<a;b> : quniv(A) <-> a: quniv(A) & b: quniv(A)";
-by (REPEAT (ares_tac [iffI, QPair_in_quniv, quniv_QPair_D] 1
- ORELSE etac conjE 1));
-qed "quniv_QPair_iff";
-
-
-(** Quine disjoint sum **)
-
-Goalw [QInl_def] "a: quniv(A) ==> QInl(a) : quniv(A)";
-by (REPEAT (ares_tac [zero_in_quniv,QPair_in_quniv] 1));
-qed "QInl_in_quniv";
-
-Goalw [QInr_def] "b: quniv(A) ==> QInr(b) : quniv(A)";
-by (REPEAT (ares_tac [one_in_quniv, QPair_in_quniv] 1));
-qed "QInr_in_quniv";
-
-Goal "quniv(C) <+> quniv(C) <= quniv(C)";
-by (REPEAT (ares_tac [subsetI, QInl_in_quniv, QInr_in_quniv] 1
- ORELSE eresolve_tac [qsumE, ssubst] 1));
-qed "qsum_quniv";
-
-bind_thm ("qsum_subset_quniv", [qsum_mono, qsum_quniv] MRS subset_trans);
-
-
-(*** The natural numbers ***)
-
-bind_thm ("nat_subset_quniv",
- [nat_subset_univ, univ_subset_quniv] MRS subset_trans);
-
-(* n:nat ==> n:quniv(A) *)
-bind_thm ("nat_into_quniv", (nat_subset_quniv RS subsetD));
-
-bind_thm ("bool_subset_quniv",
- [bool_subset_univ, univ_subset_quniv] MRS subset_trans);
-
-bind_thm ("bool_into_quniv", bool_subset_quniv RS subsetD);
-
-
-(*** Intersecting <a;b> with Vfrom... ***)
-
-Goalw [QPair_def,sum_def]
- "Transset(X) ==> \
-\ <a;b> Int Vfrom(X, succ(i)) <= <a Int Vfrom(X,i); b Int Vfrom(X,i)>";
-by (stac Int_Un_distrib2 1);
-by (rtac Un_mono 1);
-by (REPEAT (ares_tac [product_Int_Vfrom_subset RS subset_trans,
- [Int_lower1, subset_refl] MRS Sigma_mono] 1));
-qed "QPair_Int_Vfrom_succ_subset";
-
-(**** "Take-lemma" rules for proving a=b by coinduction and c: quniv(A) ****)
-
-(*Rule for level i -- preserving the level, not decreasing it*)
-
-Goalw [QPair_def]
- "Transset(X) ==> \
-\ <a;b> Int Vfrom(X,i) <= <a Int Vfrom(X,i); b Int Vfrom(X,i)>";
-by (etac (Transset_Vfrom RS Transset_sum_Int_subset) 1);
-qed "QPair_Int_Vfrom_subset";
-
-(*[| a Int Vset(i) <= c; b Int Vset(i) <= d |] ==> <a;b> Int Vset(i) <= <c;d>*)
-bind_thm ("QPair_Int_Vset_subset_trans",
- [Transset_0 RS QPair_Int_Vfrom_subset, QPair_mono] MRS subset_trans);
-
-Goal "Ord(i) ==> <a;b> Int Vset(i) <= (UN j:i. <a Int Vset(j); b Int Vset(j)>)";
-by (etac Ord_cases 1 THEN REPEAT_FIRST hyp_subst_tac);
-(*0 case*)
-by (stac Vfrom_0 1);
-by (Fast_tac 1);
-(*succ(j) case*)
-by (rtac (Transset_0 RS QPair_Int_Vfrom_succ_subset RS subset_trans) 1);
-by (rtac (succI1 RS UN_upper) 1);
-(*Limit(i) case*)
-by (asm_simp_tac
- (simpset() delsimps UN_simps
- addsimps [Limit_Vfrom_eq, Int_UN_distrib,
- UN_mono, QPair_Int_Vset_subset_trans]) 1);
-qed "QPair_Int_Vset_subset_UN";