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1 (* Title: HOL/Library/Sum_of_Squares_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 |
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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)" |
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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 lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1" |
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37 by (sos remote_csdp) |
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38 |
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39 lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)" |
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40 by (sos remote_csdp) |
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41 |
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42 (* ------------------------------------------------------------------------- *) |
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43 (* One component of denominator in dodecahedral example. *) |
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44 (* ------------------------------------------------------------------------- *) |
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45 |
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46 lemma "2 <= x & x <= 125841 / 50000 & 2 <= y & y <= 125841 / 50000 & 2 <= z & z <= 125841 / 50000 --> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= (0::real)" |
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47 by (sos remote_csdp) |
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48 |
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49 (* ------------------------------------------------------------------------- *) |
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50 (* Over a larger but simpler interval. *) |
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51 (* ------------------------------------------------------------------------- *) |
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52 |
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53 lemma "(2::real) <= x & x <= 4 & 2 <= y & y <= 4 & 2 <= z & z <= 4 --> 0 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)" |
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54 by (sos remote_csdp) |
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55 |
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56 (* ------------------------------------------------------------------------- *) |
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57 (* We can do 12. I think 12 is a sharp bound; see PP's certificate. *) |
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58 (* ------------------------------------------------------------------------- *) |
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59 |
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60 lemma "2 <= (x::real) & x <= 4 & 2 <= y & y <= 4 & 2 <= z & z <= 4 --> 12 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)" |
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61 by (sos remote_csdp) |
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62 |
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63 (* ------------------------------------------------------------------------- *) |
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64 (* Inequality from sci.math (see "Leon-Sotelo, por favor"). *) |
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65 (* ------------------------------------------------------------------------- *) |
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66 |
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67 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2" |
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68 by (sos remote_csdp) |
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69 |
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70 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2" |
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71 by (sos remote_csdp) |
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72 |
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73 lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2" |
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74 by (sos remote_csdp) |
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75 |
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76 lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x" |
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77 by (sos remote_csdp) |
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78 |
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79 lemma "(0::real) < x --> 0 < 1 + x + x^2" |
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80 by (sos remote_csdp) |
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81 |
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82 lemma "(0::real) <= x --> 0 < 1 + x + x^2" |
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83 by (sos remote_csdp) |
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84 |
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85 lemma "(0::real) < 1 + x^2" |
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86 by (sos remote_csdp) |
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87 |
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88 lemma "(0::real) <= 1 + 2 * x + x^2" |
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89 by (sos remote_csdp) |
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90 |
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91 lemma "(0::real) < 1 + abs x" |
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92 by (sos remote_csdp) |
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93 |
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94 lemma "(0::real) < 1 + (1 + x)^2 * (abs x)" |
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95 by (sos remote_csdp) |
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96 |
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97 |
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98 |
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99 lemma "abs ((1::real) + x^2) = (1::real) + x^2" |
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100 by (sos remote_csdp) |
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101 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0" |
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102 by (sos remote_csdp) |
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103 |
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104 lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z" |
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105 by (sos remote_csdp) |
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106 lemma "(1::real) < x --> x^2 < y --> 1 < y" |
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107 by (sos remote_csdp) |
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108 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" |
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109 by (sos remote_csdp) |
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110 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" |
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111 by (sos remote_csdp) |
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112 lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c" |
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113 by (sos remote_csdp) |
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114 lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x" |
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115 by (sos remote_csdp) |
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116 lemma "abs(x - z) <= e & abs(y - z) <= e & 0 <= u & 0 <= v & (u + v = 1) --> abs((u * x + v * y) - z) <= (e::real)" |
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117 by (sos remote_csdp) |
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118 |
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119 |
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120 (* lemma "((x::real) - y - 2 * x^4 = 0) & 0 <= x & x <= 2 & 0 <= y & y <= 3 --> y^2 - 7 * y - 12 * x + 17 >= 0" by sos *) (* Too hard?*) |
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121 |
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122 lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x" |
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123 by (sos remote_csdp) |
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124 |
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125 lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)" |
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126 by (sos remote_csdp) |
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127 |
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128 lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)" |
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129 by (sos remote_csdp) |
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130 |
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131 lemma "4*r^2 = p^2 - 4*q & r >= (0::real) & x^2 + p*x + q = 0 --> 2*(x::real) = - p + 2*r | 2*x = -p - 2*r" |
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132 by (sos remote_csdp) |
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133 |
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134 end |