src/HOL/ex/Groebner_Examples.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 26317 01a98fd72eae child 31021 53642251a04f permissions -rw-r--r--
```     1 (*  Title:      HOL/ex/Groebner_Examples.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     Amine Chaieb, TU Muenchen
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```     4 *)
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```     5
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```     6 header {* Groebner Basis Examples *}
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```     7
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```     8 theory Groebner_Examples
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```     9 imports Groebner_Basis
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```    10 begin
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```    11
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```    12 subsection {* Basic examples *}
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```    13
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```    14 lemma "3 ^ 3 == (?X::'a::{number_ring,recpower})"
```
```    15   by sring_norm
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```    16
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```    17 lemma "(x - (-2))^5 == ?X::int"
```
```    18   by sring_norm
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```    19
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```    20 lemma "(x - (-2))^5  * (y - 78) ^ 8 == ?X::int"
```
```    21   by sring_norm
```
```    22
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```    23 lemma "((-3) ^ (Suc (Suc (Suc 0)))) == (X::'a::{number_ring,recpower})"
```
```    24   apply (simp only: power_Suc power_0)
```
```    25   apply (simp only: comp_arith)
```
```    26   oops
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```    27
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```    28 lemma "((x::int) + y)^3 - 1 = (x - z)^2 - 10 \<Longrightarrow> x = z + 3 \<Longrightarrow> x = - y"
```
```    29   by algebra
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```    30
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```    31 lemma "(4::nat) + 4 = 3 + 5"
```
```    32   by algebra
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```    33
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```    34 lemma "(4::int) + 0 = 4"
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```    35   apply algebra?
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```    36   by simp
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```    37
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```    38 lemma
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```    39   assumes "a * x^2 + b * x + c = (0::int)" and "d * x^2 + e * x + f = 0"
```
```    40   shows "d^2*c^2 - 2*d*c*a*f + a^2*f^2 - e*d*b*c - e*b*a*f + a*e^2*c + f*d*b^2 = 0"
```
```    41   using assms by algebra
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```    42
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```    43 lemma "(x::int)^3  - x^2  - 5*x - 3 = 0 \<longleftrightarrow> (x = 3 \<or> x = -1)"
```
```    44   by algebra
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```    45
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```    46 theorem "x* (x\<twosuperior> - x  - 5) - 3 = (0::int) \<longleftrightarrow> (x = 3 \<or> x = -1)"
```
```    47   by algebra
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```    48
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```    49 lemma
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```    50   fixes x::"'a::{idom,recpower,number_ring}"
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```    51   shows "x^2*y = x^2 & x*y^2 = y^2 \<longleftrightarrow>  x=1 & y=1 | x=0 & y=0"
```
```    52   by algebra
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```    53
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```    54 subsection {* Lemmas for Lagrange's theorem *}
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```    55
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```    56 definition
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```    57   sq :: "'a::times => 'a" where
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```    58   "sq x == x*x"
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```    59
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```    60 lemma
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```    61   fixes x1 :: "'a::{idom,recpower,number_ring}"
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```    62   shows
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```    63   "(sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) =
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```    64     sq (x1*y1 - x2*y2 - x3*y3 - x4*y4)  +
```
```    65     sq (x1*y2 + x2*y1 + x3*y4 - x4*y3)  +
```
```    66     sq (x1*y3 - x2*y4 + x3*y1 + x4*y2)  +
```
```    67     sq (x1*y4 + x2*y3 - x3*y2 + x4*y1)"
```
```    68   by (algebra add: sq_def)
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```    69
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```    70 lemma
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```    71   fixes p1 :: "'a::{idom,recpower,number_ring}"
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```    72   shows
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```    73   "(sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) *
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```    74    (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2)
```
```    75     = sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) +
```
```    76       sq (p1*q2 + q1*p2 + r1*s2 - s1*r2 + t1*u2 - u1*t2 - v1*w2 + w1*v2) +
```
```    77       sq (p1*r2 - q1*s2 + r1*p2 + s1*q2 + t1*v2 + u1*w2 - v1*t2 - w1*u2) +
```
```    78       sq (p1*s2 + q1*r2 - r1*q2 + s1*p2 + t1*w2 - u1*v2 + v1*u2 - w1*t2) +
```
```    79       sq (p1*t2 - q1*u2 - r1*v2 - s1*w2 + t1*p2 + u1*q2 + v1*r2 + w1*s2) +
```
```    80       sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +
```
```    81       sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +
```
```    82       sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)"
```
```    83   by (algebra add: sq_def)
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```    84
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```    85
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```    86 subsection {* Colinearity is invariant by rotation *}
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```    87
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```    88 types point = "int \<times> int"
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```    89
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```    90 definition collinear ::"point \<Rightarrow> point \<Rightarrow> point \<Rightarrow> bool" where
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```    91   "collinear \<equiv> \<lambda>(Ax,Ay) (Bx,By) (Cx,Cy).
```
```    92     ((Ax - Bx) * (By - Cy) = (Ay - By) * (Bx - Cx))"
```
```    93
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```    94 lemma collinear_inv_rotation:
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```    95   assumes "collinear (Ax, Ay) (Bx, By) (Cx, Cy)" and "c\<twosuperior> + s\<twosuperior> = 1"
```
```    96   shows "collinear (Ax * c - Ay * s, Ay * c + Ax * s)
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```    97     (Bx * c - By * s, By * c + Bx * s) (Cx * c - Cy * s, Cy * c + Cx * s)"
```
```    98   using assms
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```    99   by (algebra add: collinear_def split_def fst_conv snd_conv)
```
```   100
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```   101 lemma "EX (d::int). a*y - a*x = n*d \<Longrightarrow> EX u v. a*u + n*v = 1 \<Longrightarrow> EX e. y - x = n*e"
```
```   102   by algebra
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```   103
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```   104 end
```