src/HOL/Lfp.ML

-(*  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
 *)
 

 by (etac CollectD 1);
 qed "lfp_greatest";
 
 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))";
 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 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);
 qed"induct2";
 
 (*** 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";
 
 (** Definition forms of lfp_Tarski and induct, to control unfolding **)

 by (rewtac rew);
 by (rtac (mono RS lfp_Tarski) 1);
 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";