src/HOL/Decision_Procs/ex/Commutative_Ring_Ex.thy
 author wenzelm Sun Nov 02 18:21:45 2014 +0100 (2014-11-02) changeset 58889 5b7a9633cfa8 parent 49070 f00fee6d21d4 child 60533 1e7ccd864b62 permissions -rw-r--r--
```     1 (*  Author:     Bernhard Haeupler *)
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```     2
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```     3 section {* Some examples demonstrating the comm-ring method *}
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```     4
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```     5 theory Commutative_Ring_Ex
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```     6 imports "../Commutative_Ring"
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```     7 begin
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```     8
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```     9 lemma "4*(x::int)^5*y^3*x^2*3 + x*z + 3^5 = 12*x^7*y^3 + z*x + 243"
```
```    10   by comm_ring
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```    11
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```    12 lemma "((x::int) + y)^2  = x^2 + y^2 + 2*x*y"
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```    13   by comm_ring
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```    14
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```    15 lemma "((x::int) + y)^3  = x^3 + y^3 + 3*x^2*y + 3*y^2*x"
```
```    16   by comm_ring
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```    17
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```    18 lemma "((x::int) - y)^3  = x^3 + 3*x*y^2 + (-3)*y*x^2 - y^3"
```
```    19   by comm_ring
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```    20
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```    21 lemma "((x::int) - y)^2  = x^2 + y^2 - 2*x*y"
```
```    22   by comm_ring
```
```    23
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```    24 lemma " ((a::int) + b + c)^2 = a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*a*c"
```
```    25   by comm_ring
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```    26
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```    27 lemma "((a::int) - b - c)^2 = a^2 + b^2 + c^2 - 2*a*b + 2*b*c - 2*a*c"
```
```    28   by comm_ring
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```    29
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```    30 lemma "(a::int)*b + a*c = a*(b+c)"
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```    31   by comm_ring
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```    32
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```    33 lemma "(a::int)^2 - b^2 = (a - b) * (a + b)"
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```    34   by comm_ring
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```    35
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```    36 lemma "(a::int)^3 - b^3 = (a - b) * (a^2 + a*b + b^2)"
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```    37   by comm_ring
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```    38
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```    39 lemma "(a::int)^3 + b^3 = (a + b) * (a^2 - a*b + b^2)"
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```    40   by comm_ring
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```    41
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```    42 lemma "(a::int)^4 - b^4 = (a - b) * (a + b)*(a^2 + b^2)"
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```    43   by comm_ring
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```    44
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```    45 lemma "(a::int)^10 - b^10 =
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```    46   (a - b) * (a^9 + a^8*b + a^7*b^2 + a^6*b^3 + a^5*b^4 + a^4*b^5 + a^3*b^6 + a^2*b^7 + a*b^8 + b^9)"
```
```    47   by comm_ring
```
```    48
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```    49 end
```