diff -r e0b172d01c37 -r f0200e91b272 src/ZF/QUniv.ML --- a/src/ZF/QUniv.ML Wed Dec 07 12:34:47 1994 +0100 +++ b/src/ZF/QUniv.ML Wed Dec 07 13:12:04 1994 +0100 @@ -13,35 +13,35 @@ goalw QUniv.thy [quniv_def] "!!X A. X <= univ(eclose(A)) ==> X : quniv(A)"; by (etac PowI 1); -val qunivI = result(); +qed "qunivI"; goalw QUniv.thy [quniv_def] "!!X A. X : quniv(A) ==> X <= univ(eclose(A))"; by (etac PowD 1); -val qunivD = result(); +qed "qunivD"; goalw QUniv.thy [quniv_def] "!!A B. A<=B ==> quniv(A) <= quniv(B)"; by (etac (eclose_mono RS univ_mono RS Pow_mono) 1); -val quniv_mono = result(); +qed "quniv_mono"; (*** Closure properties ***) goalw QUniv.thy [quniv_def] "univ(eclose(A)) <= quniv(A)"; by (rtac (Transset_iff_Pow RS iffD1) 1); by (rtac (Transset_eclose RS Transset_univ) 1); -val univ_eclose_subset_quniv = result(); +qed "univ_eclose_subset_quniv"; (*Key property for proving A_subset_quniv; requires eclose in def of quniv*) goal QUniv.thy "univ(A) <= quniv(A)"; by (rtac (arg_subset_eclose RS univ_mono RS subset_trans) 1); by (rtac univ_eclose_subset_quniv 1); -val univ_subset_quniv = result(); +qed "univ_subset_quniv"; val univ_into_quniv = standard (univ_subset_quniv RS subsetD); goalw QUniv.thy [quniv_def] "Pow(univ(A)) <= quniv(A)"; by (rtac (arg_subset_eclose RS univ_mono RS Pow_mono) 1); -val Pow_univ_subset_quniv = result(); +qed "Pow_univ_subset_quniv"; val univ_subset_into_quniv = standard (PowI RS (Pow_univ_subset_quniv RS subsetD)); @@ -61,13 +61,13 @@ goalw QUniv.thy [QPair_def] "!!A a. [| a <= univ(A); b <= univ(A) |] ==> <= univ(A)"; by (REPEAT (ares_tac [sum_subset_univ] 1)); -val QPair_subset_univ = result(); +qed "QPair_subset_univ"; (** Quine disjoint sum **) goalw QUniv.thy [QInl_def] "!!A a. a <= univ(A) ==> QInl(a) <= univ(A)"; by (etac (empty_subsetI RS QPair_subset_univ) 1); -val QInl_subset_univ = result(); +qed "QInl_subset_univ"; val naturals_subset_nat = rewrite_rule [Transset_def] (Ord_nat RS Ord_is_Transset) @@ -78,7 +78,7 @@ goalw QUniv.thy [QInr_def] "!!A a. a <= univ(A) ==> QInr(a) <= univ(A)"; by (etac (nat_1I RS naturals_subset_univ RS QPair_subset_univ) 1); -val QInr_subset_univ = result(); +qed "QInr_subset_univ"; (*** Closure for Quine-inspired products and sums ***) @@ -87,12 +87,12 @@ "!!A a. [| a: quniv(A); b: quniv(A) |] ==> : quniv(A)"; by (REPEAT (dtac PowD 1)); by (REPEAT (ares_tac [PowI, sum_subset_univ] 1)); -val QPair_in_quniv = result(); +qed "QPair_in_quniv"; goal QUniv.thy "quniv(A) <*> quniv(A) <= quniv(A)"; by (REPEAT (ares_tac [subsetI, QPair_in_quniv] 1 ORELSE eresolve_tac [QSigmaE, ssubst] 1)); -val QSigma_quniv = result(); +qed "QSigma_quniv"; val QSigma_subset_quniv = standard (QSigma_mono RS (QSigma_quniv RSN (2,subset_trans))); @@ -103,30 +103,30 @@ by (etac ([Transset_eclose RS Transset_univ, PowD] MRS Transset_includes_summands RS conjE) 1); by (REPEAT (ares_tac [conjI,PowI] 1)); -val quniv_QPair_D = result(); +qed "quniv_QPair_D"; val quniv_QPair_E = standard (quniv_QPair_D RS conjE); goal QUniv.thy " : quniv(A) <-> a: quniv(A) & b: quniv(A)"; by (REPEAT (ares_tac [iffI, QPair_in_quniv, quniv_QPair_D] 1 ORELSE etac conjE 1)); -val quniv_QPair_iff = result(); +qed "quniv_QPair_iff"; (** Quine disjoint sum **) goalw QUniv.thy [QInl_def] "!!A a. a: quniv(A) ==> QInl(a) : quniv(A)"; by (REPEAT (ares_tac [zero_in_quniv,QPair_in_quniv] 1)); -val QInl_in_quniv = result(); +qed "QInl_in_quniv"; goalw QUniv.thy [QInr_def] "!!A b. b: quniv(A) ==> QInr(b) : quniv(A)"; by (REPEAT (ares_tac [one_in_quniv, QPair_in_quniv] 1)); -val QInr_in_quniv = result(); +qed "QInr_in_quniv"; goal QUniv.thy "quniv(C) <+> quniv(C) <= quniv(C)"; by (REPEAT (ares_tac [subsetI, QInl_in_quniv, QInr_in_quniv] 1 ORELSE eresolve_tac [qsumE, ssubst] 1)); -val qsum_quniv = result(); +qed "qsum_quniv"; val qsum_subset_quniv = standard (qsum_mono RS (qsum_quniv RSN (2,subset_trans))); @@ -156,7 +156,7 @@ by (dtac (Transset_Vfrom_succ RS equalityD1 RS subsetD RS PowD) 1); by (assume_tac 1); by (fast_tac ZF_cs 1); -val doubleton_in_Vfrom_D = result(); +qed "doubleton_in_Vfrom_D"; (*This weaker version says a, b exist at the same level*) val Vfrom_doubleton_D = standard (Transset_Vfrom RS Transset_doubleton_D); @@ -173,13 +173,13 @@ "!!X. [| : Vfrom(X,succ(i)); Transset(X) |] ==> \ \ a: Vfrom(X,i) & b: Vfrom(X,i)"; by (fast_tac (ZF_cs addSDs [doubleton_in_Vfrom_D, Vfrom_doubleton_D]) 1); -val Pair_in_Vfrom_D = result(); +qed "Pair_in_Vfrom_D"; goal Univ.thy "!!X. Transset(X) ==> \ \ (a*b) Int Vfrom(X, succ(i)) <= (a Int Vfrom(X,i)) * (b Int Vfrom(X,i))"; by (fast_tac (ZF_cs addSDs [Pair_in_Vfrom_D]) 1); -val product_Int_Vfrom_subset = result(); +qed "product_Int_Vfrom_subset"; (*** Intersecting with Vfrom... ***) @@ -190,7 +190,7 @@ by (rtac Un_mono 1); by (REPEAT (ares_tac [product_Int_Vfrom_subset RS subset_trans, [Int_lower1, subset_refl] MRS Sigma_mono] 1)); -val QPair_Int_Vfrom_succ_subset = result(); +qed "QPair_Int_Vfrom_succ_subset"; (**** "Take-lemma" rules for proving a=b by coinduction and c: quniv(A) ****) @@ -200,7 +200,7 @@ "!!X. Transset(X) ==> \ \ Int Vfrom(X,i) <= "; by (etac (Transset_Vfrom RS Transset_sum_Int_subset) 1); -val QPair_Int_Vfrom_subset = result(); +qed "QPair_Int_Vfrom_subset"; (*[| a Int Vset(i) <= c; b Int Vset(i) <= d |] ==> Int Vset(i) <= *) val QPair_Int_Vset_subset_trans = standard @@ -219,4 +219,4 @@ (*Limit(i) case*) by (asm_simp_tac (ZF_ss addsimps [Limit_Vfrom_eq, Int_UN_distrib, subset_refl, UN_mono, QPair_Int_Vset_subset_trans]) 1); -val QPair_Int_Vset_subset_UN = result(); +qed "QPair_Int_Vset_subset_UN";