--- a/src/CCL/Gfp.ML Mon Jul 17 18:42:38 2006 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,125 +0,0 @@
-(* Title: CCL/Gfp.ML
- ID: $Id$
-*)
-
-(*** Proof of Knaster-Tarski Theorem using gfp ***)
-
-(* gfp(f) is the least upper bound of {u. u <= f(u)} *)
-
-val prems = goalw (the_context ()) [gfp_def] "[| A <= f(A) |] ==> A <= gfp(f)";
-by (rtac (CollectI RS Union_upper) 1);
-by (resolve_tac prems 1);
-qed "gfp_upperbound";
-
-val prems = goalw (the_context ()) [gfp_def]
- "[| !!u. u <= f(u) ==> u<=A |] ==> gfp(f) <= A";
-by (REPEAT (ares_tac ([Union_least]@prems) 1));
-by (etac CollectD 1);
-qed "gfp_least";
-
-val [mono] = goal (the_context ()) "mono(f) ==> gfp(f) <= f(gfp(f))";
-by (EVERY1 [rtac gfp_least, rtac subset_trans, atac,
- rtac (mono RS monoD), rtac gfp_upperbound, atac]);
-qed "gfp_lemma2";
-
-val [mono] = goal (the_context ()) "mono(f) ==> f(gfp(f)) <= gfp(f)";
-by (EVERY1 [rtac gfp_upperbound, rtac (mono RS monoD),
- rtac gfp_lemma2, rtac mono]);
-qed "gfp_lemma3";
-
-val [mono] = goal (the_context ()) "mono(f) ==> gfp(f) = f(gfp(f))";
-by (REPEAT (resolve_tac [equalityI,gfp_lemma2,gfp_lemma3,mono] 1));
-qed "gfp_Tarski";
-
-(*** Coinduction rules for greatest fixed points ***)
-
-(*weak version*)
-val prems = goal (the_context ())
- "[| a: A; A <= f(A) |] ==> a : gfp(f)";
-by (rtac (gfp_upperbound RS subsetD) 1);
-by (REPEAT (ares_tac prems 1));
-qed "coinduct";
-
-val [prem,mono] = goal (the_context ())
- "[| A <= f(A) Un gfp(f); mono(f) |] ==> \
-\ A Un gfp(f) <= f(A Un gfp(f))";
-by (rtac subset_trans 1);
-by (rtac (mono RS mono_Un) 2);
-by (rtac (mono RS gfp_Tarski RS subst) 1);
-by (rtac (prem RS Un_least) 1);
-by (rtac Un_upper2 1);
-qed "coinduct2_lemma";
-
-(*strong version, thanks to Martin Coen*)
-val ainA::prems = goal (the_context ())
- "[| a: A; A <= f(A) Un gfp(f); mono(f) |] ==> a : gfp(f)";
-by (rtac coinduct 1);
-by (rtac (prems MRS coinduct2_lemma) 2);
-by (resolve_tac [ainA RS UnI1] 1);
-qed "coinduct2";
-
-(*** Even Stronger version of coinduct [by Martin Coen]
- - instead of the condition A <= f(A)
- consider A <= (f(A) Un f(f(A)) ...) Un gfp(A) ***)
-
-val [prem] = goal (the_context ()) "mono(f) ==> mono(%x. f(x) Un A Un B)";
-by (REPEAT (ares_tac [subset_refl, monoI, Un_mono, prem RS monoD] 1));
-qed "coinduct3_mono_lemma";
-
-val [prem,mono] = goal (the_context ())
- "[| A <= f(lfp(%x. f(x) Un A Un gfp(f))); mono(f) |] ==> \
-\ lfp(%x. f(x) Un A Un gfp(f)) <= f(lfp(%x. f(x) Un A Un gfp(f)))";
-by (rtac subset_trans 1);
-by (rtac (mono RS coinduct3_mono_lemma RS lfp_lemma3) 1);
-by (rtac (Un_least RS Un_least) 1);
-by (rtac subset_refl 1);
-by (rtac prem 1);
-by (rtac (mono RS gfp_Tarski RS equalityD1 RS subset_trans) 1);
-by (rtac (mono RS monoD) 1);
-by (stac (mono RS coinduct3_mono_lemma RS lfp_Tarski) 1);
-by (rtac Un_upper2 1);
-qed "coinduct3_lemma";
-
-val ainA::prems = goal (the_context ())
- "[| a:A; A <= f(lfp(%x. f(x) Un A Un gfp(f))); mono(f) |] ==> a : gfp(f)";
-by (rtac coinduct 1);
-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);
-qed "coinduct3";
-
-
-(** Definition forms of gfp_Tarski, to control unfolding **)
-
-val [rew,mono] = goal (the_context ()) "[| h==gfp(f); mono(f) |] ==> h = f(h)";
-by (rewtac rew);
-by (rtac (mono RS gfp_Tarski) 1);
-qed "def_gfp_Tarski";
-
-val rew::prems = goal (the_context ())
- "[| h==gfp(f); a:A; A <= f(A) |] ==> a: h";
-by (rewtac rew);
-by (REPEAT (ares_tac (prems @ [coinduct]) 1));
-qed "def_coinduct";
-
-val rew::prems = goal (the_context ())
- "[| 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));
-qed "def_coinduct2";
-
-val rew::prems = goal (the_context ())
- "[| 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));
-qed "def_coinduct3";
-
-(*Monotonicity of gfp!*)
-val prems = goal (the_context ())
- "[| mono(f); !!Z. f(Z)<=g(Z) |] ==> gfp(f) <= gfp(g)";
-by (rtac gfp_upperbound 1);
-by (rtac subset_trans 1);
-by (rtac gfp_lemma2 1);
-by (resolve_tac prems 1);
-by (resolve_tac prems 1);
-qed "gfp_mono";