Gfp.ML

--- a/Gfp.ML	Fri Nov 11 10:35:03 1994 +0100
+++ b/Gfp.ML	Mon Nov 21 17:50:34 1994 +0100
@@ -15,27 +15,27 @@
 val prems = goalw Gfp.thy [gfp_def] "[| X <= f(X) |] ==> X <= gfp(f)";
 by (rtac (CollectI RS Union_upper) 1);
 by (resolve_tac prems 1);
-val gfp_upperbound = result();
+qed "gfp_upperbound";
 
 val prems = goalw Gfp.thy [gfp_def]
     "[| !!u. u <= f(u) ==> u<=X |] ==> gfp(f) <= X";
 by (REPEAT (ares_tac ([Union_least]@prems) 1));
 by (etac CollectD 1);
-val gfp_least = result();
+qed "gfp_least";
 
 val [mono] = goal Gfp.thy "mono(f) ==> gfp(f) <= f(gfp(f))";
 by (EVERY1 [rtac gfp_least, rtac subset_trans, atac,
 	    rtac (mono RS monoD), rtac gfp_upperbound, atac]);
-val gfp_lemma2 = result();
+qed "gfp_lemma2";
 
 val [mono] = goal Gfp.thy "mono(f) ==> f(gfp(f)) <= gfp(f)";
 by (EVERY1 [rtac gfp_upperbound, rtac (mono RS monoD), 
 	    rtac gfp_lemma2, rtac mono]);
-val gfp_lemma3 = result();
+qed "gfp_lemma3";
 
 val [mono] = goal Gfp.thy "mono(f) ==> gfp(f) = f(gfp(f))";
 by (REPEAT (resolve_tac [equalityI,gfp_lemma2,gfp_lemma3,mono] 1));
-val gfp_Tarski = result();
+qed "gfp_Tarski";
 
 (*** Coinduction rules for greatest fixed points ***)
 
@@ -44,7 +44,7 @@
     "[| a: X;  X <= f(X) |] ==> a : gfp(f)";
 by (rtac (gfp_upperbound RS subsetD) 1);
 by (REPEAT (ares_tac prems 1));
-val weak_coinduct = result();
+qed "weak_coinduct";
 
 val [prem,mono] = goal Gfp.thy
     "[| X <= f(X Un gfp(f));  mono(f) |] ==>  \
@@ -53,21 +53,21 @@
 by (rtac (mono RS gfp_lemma2 RS subset_trans) 1);
 by (rtac (Un_upper2 RS subset_trans) 1);
 by (rtac (mono RS mono_Un) 1);
-val coinduct_lemma = result();
+qed "coinduct_lemma";
 
 (*strong version, thanks to Coen & Frost*)
 goal Gfp.thy
     "!!X. [| mono(f);  a: X;  X <= f(X Un gfp(f)) |] ==> a : gfp(f)";
 by (rtac (coinduct_lemma RSN (2, weak_coinduct)) 1);
 by (REPEAT (ares_tac [UnI1, Un_least] 1));
-val coinduct = result();
+qed "coinduct";
 
 val [mono,prem] = goal Gfp.thy
     "[| mono(f);  a: gfp(f) |] ==> a: f(X Un gfp(f))";
 br (mono RS mono_Un RS subsetD) 1;
 br (mono RS gfp_lemma2 RS subsetD RS UnI2) 1;
 by (rtac prem 1);
-val gfp_fun_UnI2 = result();
+qed "gfp_fun_UnI2";
 
 (***  Even Stronger version of coinduct  [by Martin Coen]
          - instead of the condition  X <= f(X)
@@ -75,7 +75,7 @@
 
 val [prem] = goal Gfp.thy "mono(f) ==> mono(%x.f(x) Un X Un B)";
 by (REPEAT (ares_tac [subset_refl, monoI, Un_mono, prem RS monoD] 1));
-val coinduct3_mono_lemma= result();
+qed "coinduct3_mono_lemma";
 
 val [prem,mono] = goal Gfp.thy
     "[| X <= f(lfp(%x.f(x) Un X Un gfp(f)));  mono(f) |] ==> \
@@ -89,7 +89,7 @@
 by (rtac (mono RS monoD) 1);
 by (rtac (mono RS coinduct3_mono_lemma RS lfp_Tarski RS ssubst) 1);
 by (rtac Un_upper2 1);
-val coinduct3_lemma = result();
+qed "coinduct3_lemma";
 
 val prems = goal Gfp.thy
     "[| mono(f);  a:X;  X <= f(lfp(%x.f(x) Un X Un gfp(f))) |] ==> a : gfp(f)";
@@ -97,7 +97,7 @@
 by (resolve_tac (prems RL [coinduct3_mono_lemma RS lfp_Tarski RS ssubst]) 1);
 by (rtac (UnI2 RS UnI1) 1);
 by (REPEAT (resolve_tac prems 1));
-val coinduct3 = result();
+qed "coinduct3";
 
 
 (** Definition forms of gfp_Tarski and coinduct, to control unfolding **)
@@ -105,13 +105,13 @@
 val [rew,mono] = goal Gfp.thy "[| A==gfp(f);  mono(f) |] ==> A = f(A)";
 by (rewtac rew);
 by (rtac (mono RS gfp_Tarski) 1);
-val def_gfp_Tarski = result();
+qed "def_gfp_Tarski";
 
 val rew::prems = goal Gfp.thy
     "[| A==gfp(f);  mono(f);  a:X;  X <= f(X Un A) |] ==> a: A";
 by (rewtac rew);
 by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct]) 1));
-val def_coinduct = result();
+qed "def_coinduct";
 
 (*The version used in the induction/coinduction package*)
 val prems = goal Gfp.thy
@@ -120,13 +120,13 @@
 \    a : A";
 by (rtac def_coinduct 1);
 by (REPEAT (ares_tac (prems @ [subsetI,CollectI]) 1));
-val def_Collect_coinduct = result();
+qed "def_Collect_coinduct";
 
 val rew::prems = goal Gfp.thy
     "[| A==gfp(f); mono(f);  a:X;  X <= f(lfp(%x.f(x) Un X Un A)) |] ==> a: A";
 by (rewtac rew);
 by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct3]) 1));
-val def_coinduct3 = result();
+qed "def_coinduct3";
 
 (*Monotonicity of gfp!*)
 val prems = goal Gfp.thy
@@ -142,4 +142,4 @@
 val [prem] = goal Gfp.thy "[| !!Z. f(Z)<=g(Z) |] ==> gfp(f) <= gfp(g)";
 br (gfp_upperbound RS gfp_least) 1;
 be (prem RSN (2,subset_trans)) 1;
-val gfp_mono = result();
+qed "gfp_mono";