diff -r 142f1eb707b4 -r befa4e9f7c90 gfp.ML
--- a/gfp.ML	Thu Sep 16 14:29:14 1993 +0200
+++ b/gfp.ML	Wed Sep 22 15:43:05 1993 +0200
@@ -12,13 +12,13 @@
 
 (* gfp(f) is the least upper bound of {u. u <= f(u)} *)
 
-val prems = goalw Gfp.thy [gfp_def] "[| A <= f(A) |] ==> A <= gfp(f)";
+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();
 
 val prems = goalw Gfp.thy [gfp_def]
-    "[| !!u. u <= f(u) ==> u<=A |] ==> gfp(f) <= A";
+    "[| !!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();
@@ -41,39 +41,45 @@
 
 (*weak version*)
 val prems = goal Gfp.thy
-    "[| a: A;  A <= f(A) |] ==> a : gfp(f)";
+    "[| a: X;  X <= f(X) |] ==> a : gfp(f)";
 by (rtac (gfp_upperbound RS subsetD) 1);
 by (REPEAT (ares_tac prems 1));
-val coinduct = result();
+val weak_coinduct = result();
 
 val [prem,mono] = goal Gfp.thy
-    "[| A <= f(A) Un gfp(f);  mono(f) |] ==>  \
-\    A Un gfp(f) <= f(A Un gfp(f))";
-by (rtac subset_trans 1);
-by (rtac (mono RS mono_Un) 2);
-by (rtac (mono RS gfp_Tarski RS subst) 1);
+    "[| X <= f(X Un gfp(f));  mono(f) |] ==>  \
+\    X Un gfp(f) <= f(X Un gfp(f))";
 by (rtac (prem RS Un_least) 1);
-by (rtac Un_upper2 1);
-val coinduct2_lemma = result();
+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();
 
-(*strong version, thanks to Martin Coen*)
-val prems = goal Gfp.thy
-    "[| a: A;  A <= f(A) Un gfp(f);  mono(f) |] ==> a : gfp(f)";
-by (rtac (coinduct2_lemma RSN (2,coinduct)) 1);
-by (REPEAT (resolve_tac (prems@[UnI1]) 1));
-val coinduct2 = result();
+(*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();
+
+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();
 
 (***  Even Stronger version of coinduct  [by Martin Coen]
-         - instead of the condition  A <= f(A)
-                           consider  A <= (f(A) Un f(f(A)) ...) Un gfp(A) ***)
+         - instead of the condition  X <= f(X)
+                           consider  X <= (f(X) Un f(f(X)) ...) Un gfp(X) ***)
 
-val [prem] = goal Gfp.thy "mono(f) ==> mono(%x.f(x) Un A Un B)";
+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();
 
 val [prem,mono] = goal Gfp.thy
-    "[| A <= f(lfp(%x.f(x) Un A Un gfp(f)));  mono(f) |] ==> \
-\    lfp(%x.f(x) Un A Un gfp(f)) <= f(lfp(%x.f(x) Un A Un gfp(f)))";
+    "[| X <= f(lfp(%x.f(x) Un X Un gfp(f)));  mono(f) |] ==> \
+\    lfp(%x.f(x) Un X Un gfp(f)) <= f(lfp(%x.f(x) Un X Un gfp(f)))";
 by (rtac subset_trans 1);
 by (rtac (mono RS coinduct3_mono_lemma RS lfp_lemma3) 1);
 by (rtac (Un_least RS Un_least) 1);
@@ -86,8 +92,8 @@
 val coinduct3_lemma = result();
 
 val prems = goal Gfp.thy
-    "[| a:A;  A <= f(lfp(%x.f(x) Un A Un gfp(f))); mono(f) |] ==> a : gfp(f)";
-by (rtac (coinduct3_lemma RSN (2,coinduct)) 1);
+    "[| mono(f);  a:X;  X <= f(lfp(%x.f(x) Un X Un gfp(f))) |] ==> a : gfp(f)";
+by (rtac (coinduct3_lemma RSN (2,weak_coinduct)) 1);
 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));
@@ -96,25 +102,19 @@
 
 (** Definition forms of gfp_Tarski and coinduct, to control unfolding **)
 
-val [rew,mono] = goal Gfp.thy "[| h==gfp(f);  mono(f) |] ==> h = f(h)";
+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();
 
 val rew::prems = goal Gfp.thy
-    "[| h==gfp(f);  a:A;  A <= f(A) |] ==> a: h";
+    "[| A==gfp(f);  mono(f);  a:X;  X <= f(X Un A) |] ==> a: A";
 by (rewtac rew);
-by (REPEAT (ares_tac (prems @ [coinduct]) 1));
+by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct]) 1));
 val def_coinduct = result();
 
 val rew::prems = goal Gfp.thy
-    "[| h==gfp(f);  a:A;  A <= f(A) Un h; mono(f) |] ==> a: h";
-by (rewtac rew);
-by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct2]) 1));
-val def_coinduct2 = result();
-
-val rew::prems = goal Gfp.thy
-    "[| h==gfp(f);  a:A;  A <= f(lfp(%x.f(x) Un A Un h)); mono(f) |] ==> a: h";
+    "[| 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();