src/HOL/Real/RealDef.thy
 changeset 15169 2b5da07a0b89 parent 15140 322485b816ac child 15229 1eb23f805c06
```--- a/src/HOL/Real/RealDef.thy	Wed Sep 01 15:03:41 2004 +0200
+++ b/src/HOL/Real/RealDef.thy	Wed Sep 01 15:04:01 2004 +0200
@@ -154,8 +154,8 @@
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
Abs_Real (realrel``{(x+u, y+v)})"
proof -
-  have "congruent2 realrel realrel
-        (\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)"
+  have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
+        respects2 realrel"
thus ?thesis
@@ -181,7 +181,7 @@

lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
proof -
-  have "congruent realrel (\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})})"
+  have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
thus ?thesis
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
@@ -203,9 +203,10 @@
done

lemma real_mult_congruent2:
-    "congruent2 realrel realrel (%p1 p2.
+    "(%p1 p2.
(%(x1,y1). (%(x2,y2).
-          { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)"
+          { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
+     respects2 realrel"
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)