diff -r e0e5c1581e4c -r 2ca12511676d src/CCL/Gfp.ML --- a/src/CCL/Gfp.ML Wed Nov 30 13:18:42 1994 +0100 +++ b/src/CCL/Gfp.ML Wed Nov 30 13:53:46 1994 +0100 @@ -18,27 +18,27 @@ val prems = goalw Gfp.thy [gfp_def] "[| A <= f(A) |] ==> A <= gfp(f)"; by (rtac (CollectI RS Union_upper) 1); by (resolve_tac prems 1); -val gfp_upperbound = result(); +qed "gfp_upperbound"; val prems = goalw Gfp.thy [gfp_def] "[| !!u. u <= f(u) ==> u<=A |] ==> gfp(f) <= A"; by (REPEAT (ares_tac ([Union_least]@prems) 1)); by (etac CollectD 1); -val gfp_least = result(); +qed "gfp_least"; val [mono] = goal Gfp.thy "mono(f) ==> gfp(f) <= f(gfp(f))"; by (EVERY1 [rtac gfp_least, rtac subset_trans, atac, rtac (mono RS monoD), rtac gfp_upperbound, atac]); -val gfp_lemma2 = result(); +qed "gfp_lemma2"; val [mono] = goal Gfp.thy "mono(f) ==> f(gfp(f)) <= gfp(f)"; by (EVERY1 [rtac gfp_upperbound, rtac (mono RS monoD), rtac gfp_lemma2, rtac mono]); -val gfp_lemma3 = result(); +qed "gfp_lemma3"; val [mono] = goal Gfp.thy "mono(f) ==> gfp(f) = f(gfp(f))"; by (REPEAT (resolve_tac [equalityI,gfp_lemma2,gfp_lemma3,mono] 1)); -val gfp_Tarski = result(); +qed "gfp_Tarski"; (*** Coinduction rules for greatest fixed points ***) @@ -47,7 +47,7 @@ "[| a: A; A <= f(A) |] ==> a : gfp(f)"; by (rtac (gfp_upperbound RS subsetD) 1); by (REPEAT (ares_tac prems 1)); -val coinduct = result(); +qed "coinduct"; val [prem,mono] = goal Gfp.thy "[| A <= f(A) Un gfp(f); mono(f) |] ==> \ @@ -57,7 +57,7 @@ by (rtac (mono RS gfp_Tarski RS subst) 1); by (rtac (prem RS Un_least) 1); by (rtac Un_upper2 1); -val coinduct2_lemma = result(); +qed "coinduct2_lemma"; (*strong version, thanks to Martin Coen*) val ainA::prems = goal Gfp.thy @@ -65,7 +65,7 @@ by (rtac coinduct 1); by (rtac (prems MRS coinduct2_lemma) 2); by (resolve_tac [ainA RS UnI1] 1); -val coinduct2 = result(); +qed "coinduct2"; (*** Even Stronger version of coinduct [by Martin Coen] - instead of the condition A <= f(A) @@ -73,7 +73,7 @@ val [prem] = goal Gfp.thy "mono(f) ==> mono(%x.f(x) Un A Un B)"; by (REPEAT (ares_tac [subset_refl, monoI, Un_mono, prem RS monoD] 1)); -val coinduct3_mono_lemma= result(); +qed "coinduct3_mono_lemma"; val [prem,mono] = goal Gfp.thy "[| A <= f(lfp(%x.f(x) Un A Un gfp(f))); mono(f) |] ==> \ @@ -87,7 +87,7 @@ by (rtac (mono RS monoD) 1); by (rtac (mono RS coinduct3_mono_lemma RS lfp_Tarski RS ssubst) 1); by (rtac Un_upper2 1); -val coinduct3_lemma = result(); +qed "coinduct3_lemma"; val ainA::prems = goal Gfp.thy "[| a:A; A <= f(lfp(%x.f(x) Un A Un gfp(f))); mono(f) |] ==> a : gfp(f)"; @@ -95,7 +95,7 @@ by (rtac (prems MRS coinduct3_lemma) 2); by (resolve_tac (prems RL [coinduct3_mono_lemma RS lfp_Tarski RS ssubst]) 1); by (rtac (ainA RS UnI2 RS UnI1) 1); -val coinduct3 = result(); +qed "coinduct3"; (** Definition forms of gfp_Tarski, to control unfolding **) @@ -103,25 +103,25 @@ val [rew,mono] = goal Gfp.thy "[| h==gfp(f); mono(f) |] ==> h = f(h)"; by (rewtac rew); by (rtac (mono RS gfp_Tarski) 1); -val def_gfp_Tarski = result(); +qed "def_gfp_Tarski"; val rew::prems = goal Gfp.thy "[| h==gfp(f); a:A; A <= f(A) |] ==> a: h"; by (rewtac rew); by (REPEAT (ares_tac (prems @ [coinduct]) 1)); -val def_coinduct = result(); +qed "def_coinduct"; val rew::prems = goal Gfp.thy "[| h==gfp(f); a:A; A <= f(A) Un h; mono(f) |] ==> a: h"; by (rewtac rew); by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct2]) 1)); -val def_coinduct2 = result(); +qed "def_coinduct2"; val rew::prems = goal Gfp.thy "[| h==gfp(f); a:A; A <= f(lfp(%x.f(x) Un A Un h)); mono(f) |] ==> a: h"; by (rewtac rew); by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct3]) 1)); -val def_coinduct3 = result(); +qed "def_coinduct3"; (*Monotonicity of gfp!*) val prems = goal Gfp.thy @@ -131,4 +131,4 @@ by (rtac gfp_lemma2 1); by (resolve_tac prems 1); by (resolve_tac prems 1); -val gfp_mono = result(); +qed "gfp_mono";