src/HOL/Lfp.ML

 (*  Title:      HOL/lfp.ML
     ID:         $Id$
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1992  University of Cambridge
 
-For lfp.thy.  The Knaster-Tarski Theorem
+The Knaster-Tarski Theorem
 *)
 
 open Lfp;
 
 (*** Proof of Knaster-Tarski Theorem ***)
 
 (* lfp(f) is the greatest lower bound of {u. f(u) <= u} *)
 
-val prems = goalw Lfp.thy [lfp_def] "[| f(A) <= A |] ==> lfp(f) <= A";
+Goalw [lfp_def] "f(A) <= A ==> lfp(f) <= A";
 by (rtac (CollectI RS Inter_lower) 1);
-by (resolve_tac prems 1);
+by (assume_tac 1);
 qed "lfp_lowerbound";
 
-val prems = goalw Lfp.thy [lfp_def]
+val prems = Goalw [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 Lfp.thy "mono(f) ==> f(lfp(f)) <= lfp(f)";
+Goal "mono(f) ==> f(lfp(f)) <= lfp(f)";
 by (EVERY1 [rtac lfp_greatest, rtac subset_trans,
-            rtac (mono RS monoD), rtac lfp_lowerbound, atac, atac]);
+            etac monoD, rtac lfp_lowerbound, atac, atac]);
 qed "lfp_lemma2";
 
-val [mono] = goal Lfp.thy "mono(f) ==> lfp(f) <= f(lfp(f))";
-by (EVERY1 [rtac lfp_lowerbound, rtac (mono RS monoD), 
-            rtac lfp_lemma2, rtac mono]);
+Goal "mono(f) ==> lfp(f) <= f(lfp(f))";
+by (EVERY1 [rtac lfp_lowerbound, rtac monoD, assume_tac,
+            etac lfp_lemma2]);
 qed "lfp_lemma3";
 
-val [mono] = goal Lfp.thy "mono(f) ==> lfp(f) = f(lfp(f))";
-by (REPEAT (resolve_tac [equalityI,lfp_lemma2,lfp_lemma3,mono] 1));
+Goal "mono(f) ==> lfp(f) = f(lfp(f))";
+by (REPEAT (ares_tac [equalityI,lfp_lemma2,lfp_lemma3] 1));
 qed "lfp_Tarski";
 
 (*** General induction rule for least fixed points ***)
 
-val [lfp,mono,indhyp] = goal Lfp.thy
+val [lfp,mono,indhyp] = Goal
     "[| 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);

 val [rew,mono] = goal Lfp.thy "[| 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 Lfp.thy
+val rew::prems = Goal
     "[| A == lfp(f);  mono(f);   a:A;                   \
 \       !!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 [prem] = goal Lfp.thy "[| !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)";
+val [prem] = Goal "[| !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)";
 by (rtac (lfp_lowerbound RS lfp_greatest) 1);
 by (etac (prem RS subset_trans) 1);
 qed "lfp_mono";