--- 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"
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma)
thus ?thesis
by (simp add: real_add_def UN_UN_split_split_eq
@@ -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"
by (simp add: congruent_def preal_add_commute)
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)
apply (simp add: preal_mult_commute preal_add_commute)
apply (auto simp add: real_mult_congruent2_lemma)