src/HOL/Library/Sum_of_Squares_Remote.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 48934 f9a800f21434 child 58881 b9556a055632 permissions -rw-r--r--
simpler proof
```     1 (*  Title:      HOL/Library/Sum_of_Squares_Remote.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
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```     3     Author:     Philipp Meyer, TU Muenchen
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```     4 *)
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```     5
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```     6 header {* Examples with remote CSDP *}
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```     7
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```     8 theory Sum_of_Squares_Remote
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```     9 imports Sum_of_Squares
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```    10 begin
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```    11
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```    12 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0"
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```    13   by (sos remote_csdp)
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```    14
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```    15 lemma "a1 >= 0 & a2 >= 0 \<and> (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) \<and> (a1 * b1 + a2 * b2 = 0) --> a1 * a2 - b1 * b2 >= (0::real)"
```
```    16   by (sos remote_csdp)
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```    17
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```    18 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0"
```
```    19   by (sos remote_csdp)
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```    20
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```    21 lemma "(0::real) <= x & x <= 1 & 0 <= y & y <= 1  --> x^2 + y^2 < 1 |(x - 1)^2 + y^2 < 1 | x^2 + (y - 1)^2 < 1 | (x - 1)^2 + (y - 1)^2 < 1"
```
```    22   by (sos remote_csdp)
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```    23
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```    24 lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z"
```
```    25   by (sos remote_csdp)
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```    26
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```    27 lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3"
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```    28   by (sos remote_csdp)
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```    29
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```    30 lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)"
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```    31   by (sos remote_csdp)
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```    32
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```    33 lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1"
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```    34   by (sos remote_csdp)
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```    35
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```    36 end
```