--- a/src/HOL/Lfp.ML Tue Jan 30 15:19:20 1996 +0100
+++ b/src/HOL/Lfp.ML Tue Jan 30 15:24:36 1996 +0100
@@ -1,6 +1,6 @@
-(* Title: HOL/lfp.ML
+(* Title: HOL/lfp.ML
ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
For lfp.thy. The Knaster-Tarski Theorem
@@ -25,12 +25,12 @@
val [mono] = goal Lfp.thy "mono(f) ==> f(lfp(f)) <= lfp(f)";
by (EVERY1 [rtac lfp_greatest, rtac subset_trans,
- rtac (mono RS monoD), rtac lfp_lowerbound, atac, atac]);
+ rtac (mono RS 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]);
+ rtac lfp_lemma2, rtac mono]);
qed "lfp_lemma3";
val [mono] = goal Lfp.thy "mono(f) ==> lfp(f) = f(lfp(f))";
@@ -40,23 +40,23 @@
(*** General induction rule for least fixed points ***)
val [lfp,mono,indhyp] = goal Lfp.thy
- "[| a: lfp(f); mono(f); \
-\ !!x. [| x: f(lfp(f) Int {x.P(x)}) |] ==> P(x) \
+ "[| 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]);
+ rtac (Int_lower1 RS (mono RS monoD)),
+ rtac (mono RS lfp_lemma2),
+ rtac (CollectI RS subsetI), rtac indhyp, atac]);
qed "induct";
val major::prems = goal Lfp.thy
"[| (a,b) : lfp f; mono f; \
\ !!a b. (a,b) : f(lfp f Int Collect(split P)) ==> P a b |] ==> P a b";
by(res_inst_tac [("c1","P")] (split RS subst) 1);
-br (major RS induct) 1;
-brs prems 1;
+by (rtac (major RS induct) 1);
+by (resolve_tac prems 1);
by(res_inst_tac[("p","x")]PairE 1);
by(hyp_subst_tac 1);
by(asm_simp_tac (!simpset addsimps prems) 1);
@@ -64,7 +64,7 @@
(*** Fixpoint induction a la David Park ***)
goal Lfp.thy "!!f. [| mono f; f A <= A |] ==> lfp(f) <= A";
-br subsetI 1;
+by (rtac subsetI 1);
by(EVERY[etac induct 1, atac 1, etac subsetD 1, rtac subsetD 1,
atac 2, fast_tac (set_cs addSEs [monoD]) 1]);
qed "Park_induct";
@@ -77,15 +77,15 @@
qed "def_lfp_Tarski";
val rew::prems = goal Lfp.thy
- "[| A == lfp(f); mono(f); a:A; \
-\ !!x. [| x: f(A Int {x.P(x)}) |] ==> P(x) \
+ "[| 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))]);
+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)";
-br (lfp_lowerbound RS lfp_greatest) 1;
-be (prem RS subset_trans) 1;
+by (rtac (lfp_lowerbound RS lfp_greatest) 1);
+by (etac (prem RS subset_trans) 1);
qed "lfp_mono";