src/HOL/ex/Lagrange.thy
 author bulwahn Fri Oct 21 11:17:14 2011 +0200 (2011-10-21) changeset 45231 d85a2fdc586c parent 37885 c43805c80eb6 child 58776 95e58e04e534 permissions -rw-r--r--
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
```     1 (*  Title:      HOL/ex/Lagrange.thy
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```     2     Author:     Tobias Nipkow
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```     3     Copyright   1996 TU Muenchen
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```     4 *)
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```     5
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```     6 header {* A lemma for Lagrange's theorem *}
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```     7
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```     8 theory Lagrange imports Main begin
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```     9
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```    10 text {* This theory only contains a single theorem, which is a lemma
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```    11 in Lagrange's proof that every natural number is the sum of 4 squares.
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```    12 Its sole purpose is to demonstrate ordered rewriting for commutative
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```    13 rings.
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```    14
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```    15 The enterprising reader might consider proving all of Lagrange's
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```    16 theorem.  *}
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```    17
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```    18 definition sq :: "'a::times => 'a" where "sq x == x*x"
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```    19
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```    20 text {* The following lemma essentially shows that every natural
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```    21 number is the sum of four squares, provided all prime numbers are.
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```    22 However, this is an abstract theorem about commutative rings.  It has,
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```    23 a priori, nothing to do with nat. *}
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```    24
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```    25 lemma Lagrange_lemma: fixes x1 :: "'a::comm_ring" shows
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```    26   "(sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) =
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```    27    sq (x1*y1 - x2*y2 - x3*y3 - x4*y4)  +
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```    28    sq (x1*y2 + x2*y1 + x3*y4 - x4*y3)  +
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```    29    sq (x1*y3 - x2*y4 + x3*y1 + x4*y2)  +
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```    30    sq (x1*y4 + x2*y3 - x3*y2 + x4*y1)"
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```    31 by (simp only: sq_def field_simps)
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```    32
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```    33
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```    34 text {* A challenge by John Harrison. Takes about 12s on a 1.6GHz machine. *}
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```    35
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```    36 lemma fixes p1 :: "'a::comm_ring" shows
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```    37   "(sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) *
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```    38    (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2)
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```    39     = sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) +
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```    40       sq (p1*q2 + q1*p2 + r1*s2 - s1*r2 + t1*u2 - u1*t2 - v1*w2 + w1*v2) +
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```    41       sq (p1*r2 - q1*s2 + r1*p2 + s1*q2 + t1*v2 + u1*w2 - v1*t2 - w1*u2) +
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```    42       sq (p1*s2 + q1*r2 - r1*q2 + s1*p2 + t1*w2 - u1*v2 + v1*u2 - w1*t2) +
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```    43       sq (p1*t2 - q1*u2 - r1*v2 - s1*w2 + t1*p2 + u1*q2 + v1*r2 + w1*s2) +
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```    44       sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +
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```    45       sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +
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```    46       sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)"
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```    47 by (simp only: sq_def field_simps)
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```    48
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```    49 end
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