--- a/src/HOL/ex/Lagrange.thy Tue Jul 20 14:24:23 2004 +0200
+++ b/src/HOL/ex/Lagrange.thy Tue Jul 20 16:07:45 2004 +0200
@@ -23,7 +23,7 @@
(*once a slow step, but now (2001) just three seconds!*)
lemma Lagrange_lemma:
- "!!x1::'a::comm_ring_1.
+ "!!x1::'a::comm_ring.
(sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) =
sq(x1*y1 - x2*y2 - x3*y3 - x4*y4) +
sq(x1*y2 + x2*y1 + x3*y4 - x4*y3) +
@@ -34,7 +34,7 @@
(* A challenge by John Harrison. Takes about 4 mins on a 3GHz machine.
-lemma "!!p1::'a::comm_ring_1.
+lemma "!!p1::'a::comm_ring.
(sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) *
(sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2)
= sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) +