1.1 --- a/src/HOL/ex/Lagrange.thy	Tue May 11 14:00:02 2004 +0200
     1.2 +++ b/src/HOL/ex/Lagrange.thy	Tue May 11 20:11:08 2004 +0200
     1.3 @@ -23,7 +23,7 @@
     1.4  
     1.5  (*once a slow step, but now (2001) just three seconds!*)
     1.6  lemma Lagrange_lemma:
     1.7 - "!!x1::'a::ring.
     1.8 + "!!x1::'a::comm_ring_1.
     1.9    (sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) =
    1.10    sq(x1*y1 - x2*y2 - x3*y3 - x4*y4)  +
    1.11    sq(x1*y2 + x2*y1 + x3*y4 - x4*y3)  +
    1.12 @@ -34,7 +34,7 @@
    1.13  
    1.14  (* A challenge by John Harrison. Takes about 4 mins on a 3GHz machine.
    1.15  
    1.16 -lemma "!!p1::'a::ring.
    1.17 +lemma "!!p1::'a::comm_ring_1.
    1.18   (sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) * 
    1.19   (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2) 
    1.20    = sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) +