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1 (* Title: HOL/ex/SOS_Remote.thy |
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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 Examples for Sum_of_Squares: remote CSDP server. |
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6 *) |
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7 |
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8 theory SOS_Remote |
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9 imports "~~/src/HOL/Library/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)" |
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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" |
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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" |
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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" |
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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 |
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37 |
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