1 (* Title: HOL/Library/Sum_Of_Squares.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 {* A decision method for universal multivariate real arithmetic with addition, |
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7 multiplication and ordering using semidefinite programming *} |
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8 |
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9 theory Sum_Of_Squares |
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10 imports Complex_Main |
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11 uses |
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12 "positivstellensatz.ML" |
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13 "Sum_Of_Squares/sum_of_squares.ML" |
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14 "Sum_Of_Squares/positivstellensatz_tools.ML" |
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15 "Sum_Of_Squares/sos_wrapper.ML" |
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16 begin |
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17 |
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18 text {* |
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19 In order to use the method sos, call it with @{text "(sos |
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20 remote_csdp)"} to use the remote solver. Or install CSDP |
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21 (https://projects.coin-or.org/Csdp), configure the Isabelle setting |
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22 @{text CSDP_EXE}, and call it with @{text "(sos csdp)"}. By |
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23 default, sos calls @{text remote_csdp}. This can take of the order |
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24 of a minute for one sos call, because sos calls CSDP repeatedly. If |
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25 you install CSDP locally, sos calls typically takes only a few |
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26 seconds. |
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27 sos generates a certificate which can be used to repeat the proof |
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28 without calling an external prover. |
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29 *} |
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30 |
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31 setup Sum_Of_Squares.setup |
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32 setup SOS_Wrapper.setup |
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33 |
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34 text {* Tests *} |
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35 |
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36 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0" |
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37 by (sos_cert "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))") |
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38 |
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39 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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40 by (sos_cert "(((A<0 * R<1) + (([~1/2*a1*b2 + ~1/2*a2*b1] * A=0) + (([~1/2*a1*a2 + 1/2*b1*b2] * A=1) + (((A<0 * R<1) * ((R<1/2 * [b2]^2) + (R<1/2 * [b1]^2))) + ((A<=0 * (A<=1 * R<1)) * ((R<1/2 * [b2]^2) + ((R<1/2 * [b1]^2) + ((R<1/2 * [a2]^2) + (R<1/2 * [a1]^2))))))))))") |
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41 |
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42 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0" |
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43 by (sos_cert "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))") |
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44 |
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45 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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46 by (sos_cert "((R<1 + (((A<=3 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=2 * (A<=7 * R<1)) * (R<1 * [1]^2)) + (((A<=1 * (A<=6 * R<1)) * (R<1 * [1]^2)) + ((A<=0 * (A<=5 * R<1)) * (R<1 * [1]^2)))))))") |
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47 |
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48 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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49 by (sos_cert "(((A<0 * R<1) + (((A<0 * R<1) * (R<1/2 * [1]^2)) + (((A<=2 * R<1) * (R<1/2 * [~1*x + y]^2)) + (((A<=1 * R<1) * (R<1/2 * [~1*x + z]^2)) + (((A<=1 * (A<=2 * (A<=3 * R<1))) * (R<1/2 * [1]^2)) + (((A<=0 * R<1) * (R<1/2 * [~1*y + z]^2)) + (((A<=0 * (A<=2 * (A<=3 * R<1))) * (R<1/2 * [1]^2)) + ((A<=0 * (A<=1 * (A<=3 * R<1))) * (R<1/2 * [1]^2))))))))))") |
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50 |
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51 lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3" |
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52 by (sos_cert "(((A<0 * R<1) + (([~3] * A=0) + (R<1 * ((R<2 * [~1/2*x + ~1/2*y + z]^2) + (R<3/2 * [~1*x + y]^2))))))") |
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53 |
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54 lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)" |
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55 by (sos_cert "(((A<0 * R<1) + (([~4] * A=0) + (R<1 * ((R<3 * [~1/3*w + ~1/3*x + ~1/3*y + z]^2) + ((R<8/3 * [~1/2*w + ~1/2*x + y]^2) + (R<2 * [~1*w + x]^2)))))))") |
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56 |
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57 lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1" |
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58 by (sos_cert "(((A<0 * R<1) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))") |
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59 |
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60 lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1" |
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61 by (sos_cert "((((A<0 * A<1) * R<1) + ((A<=0 * R<1) * (R<1 * [1]^2))))") |
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62 |
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63 lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)" |
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64 by (sos_cert "((((A<0 * R<1) + ((A<=1 * R<1) * (R<1 * [~8*x^3 + ~4*x^2 + 4*x + 1]^2)))) & ((((A<0 * A<1) * R<1) + ((A<=1 * (A<0 * R<1)) * (R<1 * [8*x^3 + ~4*x^2 + ~4*x + 1]^2)))))") |
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65 |
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66 (* ------------------------------------------------------------------------- *) |
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67 (* One component of denominator in dodecahedral example. *) |
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68 (* ------------------------------------------------------------------------- *) |
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69 |
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70 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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71 by (sos_cert "(((A<0 * R<1) + ((R<1 * ((R<5749028157/5000000000 * [~25000/222477*x + ~25000/222477*y + ~25000/222477*z + 1]^2) + ((R<864067/1779816 * [419113/864067*x + 419113/864067*y + z]^2) + ((R<320795/864067 * [419113/1283180*x + y]^2) + (R<1702293/5132720 * [x]^2))))) + (((A<=4 * (A<=5 * R<1)) * (R<3/2 * [1]^2)) + (((A<=3 * (A<=5 * R<1)) * (R<1/2 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<3/2 * [1]^2)) + (((A<=1 * (A<=5 * R<1)) * (R<1/2 * [1]^2)) + (((A<=1 * (A<=3 * R<1)) * (R<1/2 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<1 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<3/2 * [1]^2)))))))))))))") |
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72 |
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73 (* ------------------------------------------------------------------------- *) |
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74 (* Over a larger but simpler interval. *) |
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75 (* ------------------------------------------------------------------------- *) |
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76 |
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77 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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78 by (sos_cert "((R<1 + ((R<1 * ((R<1 * [~1/6*x + ~1/6*y + ~1/6*z + 1]^2) + ((R<1/18 * [~1/2*x + ~1/2*y + z]^2) + (R<1/24 * [~1*x + y]^2)))) + (((A<0 * R<1) * (R<1/12 * [1]^2)) + (((A<=4 * (A<=5 * R<1)) * (R<1/6 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<1/6 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<1/6 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<1/6 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<1/6 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1/6 * [1]^2)))))))))))") |
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79 |
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80 (* ------------------------------------------------------------------------- *) |
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81 (* We can do 12. I think 12 is a sharp bound; see PP's certificate. *) |
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82 (* ------------------------------------------------------------------------- *) |
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83 |
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84 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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85 by (sos_cert "(((A<0 * R<1) + (((A<=4 * R<1) * (R<2/3 * [1]^2)) + (((A<=4 * (A<=5 * R<1)) * (R<1 * [1]^2)) + (((A<=3 * (A<=4 * R<1)) * (R<1/3 * [1]^2)) + (((A<=2 * R<1) * (R<2/3 * [1]^2)) + (((A<=2 * (A<=5 * R<1)) * (R<1/3 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<8/3 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<1 * [1]^2)) + (((A<=1 * (A<=4 * R<1)) * (R<1/3 * [1]^2)) + (((A<=1 * (A<=2 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * R<1) * (R<2/3 * [1]^2)) + (((A<=0 * (A<=5 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<8/3 * [1]^2)) + (((A<=0 * (A<=3 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<8/3 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))))))))))))))))") |
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86 |
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87 (* ------------------------------------------------------------------------- *) |
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88 (* Inequality from sci.math (see "Leon-Sotelo, por favor"). *) |
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89 (* ------------------------------------------------------------------------- *) |
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90 |
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91 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2" |
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92 by (sos_cert "(((A<0 * R<1) + (([1] * A=0) + (R<1 * ((R<1 * [~1/2*x + ~1/2*y + 1]^2) + (R<3/4 * [~1*x + y]^2))))))") |
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93 |
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94 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2" |
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95 by (sos_cert "(((A<0 * R<1) + (([~1*x + ~1*y + 1] * A=0) + (R<1 * ((R<1 * [~1/2*x + ~1/2*y + 1]^2) + (R<3/4 * [~1*x + y]^2))))))") |
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96 |
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97 lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2" |
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98 by (sos_cert "(((A<0 * R<1) + (R<1 * ((R<1 * [~1/2*x^2 + y^2 + ~1/2*x*y]^2) + (R<3/4 * [~1*x^2 + x*y]^2)))))") |
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99 |
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100 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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101 by (sos_cert "(((A<0 * R<1) + (((A<=3 * R<1) * (R<1 * [1]^2)) + (((A<=1 * (A<=2 * R<1)) * (R<1/27 * [~1*a + b]^2)) + ((A<=0 * (A<=2 * R<1)) * (R<8/27 * [~1*a + b]^2))))))") |
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102 |
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103 lemma "(0::real) < x --> 0 < 1 + x + x^2" |
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104 by (sos_cert "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))") |
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105 |
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106 lemma "(0::real) <= x --> 0 < 1 + x + x^2" |
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107 by (sos_cert "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))") |
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108 |
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109 lemma "(0::real) < 1 + x^2" |
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110 by (sos_cert "((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))") |
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111 |
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112 lemma "(0::real) <= 1 + 2 * x + x^2" |
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113 by (sos_cert "(((A<0 * R<1) + (R<1 * (R<1 * [x + 1]^2))))") |
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114 |
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115 lemma "(0::real) < 1 + abs x" |
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116 by (sos_cert "((R<1 + (((A<=1 * R<1) * (R<1/2 * [1]^2)) + ((A<=0 * R<1) * (R<1/2 * [1]^2)))))") |
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117 |
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118 lemma "(0::real) < 1 + (1 + x)^2 * (abs x)" |
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119 by (sos_cert "(((R<1 + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [x + 1]^2))))) & ((R<1 + (((A<0 * R<1) * (R<1 * [x + 1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))") |
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120 |
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121 |
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122 |
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123 lemma "abs ((1::real) + x^2) = (1::real) + x^2" |
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124 by (sos_cert "(() & (((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<1 * R<1) * (R<1/2 * [1]^2))))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<0 * R<1) * (R<1 * [1]^2)))))))") |
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125 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0" |
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126 by (sos_cert "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))") |
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127 |
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128 lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z" |
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129 by (sos_cert "((((A<0 * A<1) * R<1) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))") |
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130 lemma "(1::real) < x --> x^2 < y --> 1 < y" |
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131 by (sos_cert "((((A<0 * A<1) * R<1) + ((R<1 * ((R<1/10 * [~2*x + y + 1]^2) + (R<1/10 * [~1*x + y]^2))) + (((A<1 * R<1) * (R<1/2 * [1]^2)) + (((A<0 * R<1) * (R<1 * [x]^2)) + (((A<=0 * R<1) * ((R<1/10 * [x + 1]^2) + (R<1/10 * [x]^2))) + (((A<=0 * (A<1 * R<1)) * (R<1/5 * [1]^2)) + ((A<=0 * (A<0 * R<1)) * (R<1/5 * [1]^2)))))))))") |
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132 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" |
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133 by (sos_cert "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))") |
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134 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" |
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135 by (sos_cert "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))") |
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136 lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c" |
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137 by (sos_cert "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))") |
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138 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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139 by (sos_cert "(((A<0 * (A<0 * R<1)) + (((A<=2 * (A<=3 * (A<0 * R<1))) * (R<2 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2)))))") |
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140 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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141 by (sos_cert "((((A<0 * R<1) + (((A<=3 * (A<=6 * R<1)) * (R<1 * [1]^2)) + ((A<=1 * (A<=5 * R<1)) * (R<1 * [1]^2))))) & ((((A<0 * A<1) * R<1) + (((A<=3 * (A<=5 * (A<0 * R<1))) * (R<1 * [1]^2)) + ((A<=1 * (A<=4 * (A<0 * R<1))) * (R<1 * [1]^2))))))") |
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142 |
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143 |
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144 (* 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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145 |
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146 lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x" |
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147 by (sos_cert "(((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2)))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<0 * R<1) * (R<1 * [1]^2))))))") |
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148 |
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149 lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)" |
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150 by (sos_cert "(((R<1 + (([~4/3] * A=0) + ((R<1 * ((R<1/3 * [3/2*x + 1]^2) + (R<7/12 * [x]^2))) + ((A<=0 * R<1) * (R<1/3 * [1]^2)))))) & (((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2)))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))))") |
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151 |
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152 lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)" |
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153 by (sos_cert "((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2))))") |
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154 |
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155 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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156 by (sos_cert "((((((A<0 * A<1) * R<1) + ([~4] * A=0))) & ((((A<0 * A<1) * R<1) + ([4] * A=0)))) & (((((A<0 * A<1) * R<1) + ([4] * A=0))) & ((((A<0 * A<1) * R<1) + ([~4] * A=0)))))") |
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157 |
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158 end |
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159 |
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