--- a/src/CCL/Lfp.ML Mon Jul 17 18:42:38 2006 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,72 +0,0 @@
-(* Title: CCL/Lfp.ML
- ID: $Id$
-*)
-
-(*** Proof of Knaster-Tarski Theorem ***)
-
-(* lfp(f) is the greatest lower bound of {u. f(u) <= u} *)
-
-val prems = goalw (the_context ()) [lfp_def] "[| f(A) <= A |] ==> lfp(f) <= A";
-by (rtac (CollectI RS Inter_lower) 1);
-by (resolve_tac prems 1);
-qed "lfp_lowerbound";
-
-val prems = goalw (the_context ()) [lfp_def]
- "[| !!u. f(u) <= u ==> A<=u |] ==> A <= lfp(f)";
-by (REPEAT (ares_tac ([Inter_greatest]@prems) 1));
-by (etac CollectD 1);
-qed "lfp_greatest";
-
-val [mono] = goal (the_context ()) "mono(f) ==> f(lfp(f)) <= lfp(f)";
-by (EVERY1 [rtac lfp_greatest, rtac subset_trans,
- rtac (mono RS monoD), rtac lfp_lowerbound, atac, atac]);
-qed "lfp_lemma2";
-
-val [mono] = goal (the_context ()) "mono(f) ==> lfp(f) <= f(lfp(f))";
-by (EVERY1 [rtac lfp_lowerbound, rtac (mono RS monoD),
- rtac lfp_lemma2, rtac mono]);
-qed "lfp_lemma3";
-
-val [mono] = goal (the_context ()) "mono(f) ==> lfp(f) = f(lfp(f))";
-by (REPEAT (resolve_tac [equalityI,lfp_lemma2,lfp_lemma3,mono] 1));
-qed "lfp_Tarski";
-
-
-(*** General induction rule for least fixed points ***)
-
-val [lfp,mono,indhyp] = goal (the_context ())
- "[| a: lfp(f); mono(f); \
-\ !!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x) \
-\ |] ==> P(a)";
-by (res_inst_tac [("a","a")] (Int_lower2 RS subsetD RS CollectD) 1);
-by (rtac (lfp RSN (2, lfp_lowerbound RS subsetD)) 1);
-by (EVERY1 [rtac Int_greatest, rtac subset_trans,
- rtac (Int_lower1 RS (mono RS monoD)),
- rtac (mono RS lfp_lemma2),
- rtac (CollectI RS subsetI), rtac indhyp, atac]);
-qed "induct";
-
-(** Definition forms of lfp_Tarski and induct, to control unfolding **)
-
-val [rew,mono] = goal (the_context ()) "[| h==lfp(f); mono(f) |] ==> h = f(h)";
-by (rewtac rew);
-by (rtac (mono RS lfp_Tarski) 1);
-qed "def_lfp_Tarski";
-
-val rew::prems = goal (the_context ())
- "[| A == lfp(f); a:A; mono(f); \
-\ !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x) \
-\ |] ==> P(a)";
-by (EVERY1 [rtac induct, (*backtracking to force correct induction*)
- REPEAT1 o (ares_tac (map (rewrite_rule [rew]) prems))]);
-qed "def_induct";
-
-(*Monotonicity of lfp!*)
-val prems = goal (the_context ())
- "[| mono(g); !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)";
-by (rtac lfp_lowerbound 1);
-by (rtac subset_trans 1);
-by (resolve_tac prems 1);
-by (rtac lfp_lemma2 1);
-by (resolve_tac prems 1);
-qed "lfp_mono";