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src/HOL/ex/Lagrange.ML

changeset 11375 | a6730c90e753 |

parent 5078 | 7b5ea59c0275 |

--- a/src/HOL/ex/Lagrange.ML Wed Jun 13 16:29:51 2001 +0200 +++ b/src/HOL/ex/Lagrange.ML Wed Jun 13 16:30:12 2001 +0200 @@ -4,21 +4,21 @@ Copyright 1996 TU Muenchen -The following lemma essentially shows that all composite natural numbers are -sums of fours squares, provided all prime numbers are. However, this is an -abstract thm about commutative rings and has a priori nothing to do with nat. -*) +The following lemma essentially shows that every natural number is the sum of +four squares, provided all prime numbers are. However, this is an abstract +theorem about commutative rings. It has, a priori, nothing to do with nat.*) -Goalw [Lagrange.sq_def] "!!x1::'a::cring. \ +Goalw [Lagrange.sq_def] + "!!x1::'a::cring. \ \ (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) + \ \ sq(x1*y3 - x2*y4 + x3*y1 + x4*y2) + \ \ sq(x1*y4 + x2*y3 - x3*y2 + x4*y1)"; -(*Takes up to three minutes...*) -by (cring_tac 1); +by (cring_tac 1); (*once a slow step, but now (2001) just three seconds!*) qed "Lagrange_lemma"; + (* A challenge by John Harrison. Takes forever because of the naive bottom-up strategy of the rewriter. @@ -33,5 +33,5 @@ \ sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +\ \ sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +\ \ sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)"; - +by (cring_tac 1); *)