src/ZF/Trancl.ML

 (*  Title:      ZF/trancl.ML
     ID:         $Id$
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1992  University of Cambridge
 
-For trancl.thy.  Transitive closure of a relation
+Transitive closure of a relation
 *)
 
 open Trancl;
 
 goal Trancl.thy "bnd_mono(field(r)*field(r), %s. id(field(r)) Un (r O s))";
 by (rtac bnd_monoI 1);
 by (REPEAT (ares_tac [subset_refl, Un_mono, comp_mono] 2));
-by (Fast_tac 1);
+by (Blast_tac 1);
 qed "rtrancl_bnd_mono";
 
 val [prem] = goalw Trancl.thy [rtrancl_def] "r<=s ==> r^* <= s^*";
 by (rtac lfp_mono 1);
 by (REPEAT (resolve_tac [rtrancl_bnd_mono, prem, subset_refl, id_mono,

 qed "r_into_rtrancl";
 
 (*The premise ensures that r consists entirely of pairs*)
 val prems = goal Trancl.thy "r <= Sigma(A,B) ==> r <= r^*";
 by (cut_facts_tac prems 1);
-by (fast_tac (!claset addIs [r_into_rtrancl]) 1);
+by (blast_tac (!claset addIs [r_into_rtrancl]) 1);
 qed "r_subset_rtrancl";
 
 goal Trancl.thy "field(r^*) = field(r)";
-by (fast_tac (!claset addIs [r_into_rtrancl] 
+by (blast_tac (!claset addIs [r_into_rtrancl] 
                     addSDs [rtrancl_type RS subsetD]) 1);
 qed "rtrancl_field";
 
 
 (** standard induction rule **)

 qed "r_into_trancl";
 
 (*The premise ensures that r consists entirely of pairs*)
 val prems = goal Trancl.thy "r <= Sigma(A,B) ==> r <= r^+";
 by (cut_facts_tac prems 1);
-by (fast_tac (!claset addIs [r_into_trancl]) 1);
+by (blast_tac (!claset addIs [r_into_trancl]) 1);
 qed "r_subset_trancl";
 
 (*intro rule by definition: from r^* and r  *)
 val prems = goalw Trancl.thy [trancl_def]
     "[| <a,b> : r^*;  <b,c> : r |]   ==>  <a,c> : r^+";

 \  |] ==> P(b)";
 by (rtac (rewrite_rule [trancl_def] major  RS  compEpair) 1);
 (*by induction on this formula*)
 by (subgoal_tac "ALL z. <y,z> : r --> P(z)" 1);
 (*now solve first subgoal: this formula is sufficient*)
-by (Fast_tac 1);
+by (Blast_tac 1);
 by (etac rtrancl_induct 1);
 by (ALLGOALS (fast_tac (!claset addIs (rtrancl_into_trancl1::prems))));
 qed "trancl_induct";
 
 (*elimination of r^+ -- NOT an induction rule*)

 \    |] ==> P";
 by (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+  &  <y,b> : r)" 1);
 by (fast_tac (!claset addIs prems) 1);
 by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1);
 by (etac rtranclE 1);
-by (ALLGOALS (fast_tac (!claset addIs [rtrancl_into_trancl1])));
+by (ALLGOALS (blast_tac (!claset addIs [rtrancl_into_trancl1])));
 qed "tranclE";
 
 goalw Trancl.thy [trancl_def] "r^+ <= field(r)*field(r)";
-by (fast_tac (!claset addEs [rtrancl_type RS subsetD RS SigmaE2]) 1);
+by (blast_tac (!claset addEs [rtrancl_type RS subsetD RS SigmaE2]) 1);
 qed "trancl_type";
 
 val [prem] = goalw Trancl.thy [trancl_def] "r<=s ==> r^+ <= s^+";
 by (REPEAT (resolve_tac [prem, comp_mono, rtrancl_mono] 1));
 qed "trancl_mono";