src/HOL/ex/SOS.thy
 author wenzelm Mon Aug 31 21:28:08 2015 +0200 (2015-08-31) changeset 61070 b72a990adfe2 parent 58630 71cdb885b3bb child 61156 931b732617a2 permissions -rw-r--r--
prefer symbols;
```     1 (*  Title:      HOL/ex/SOS.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 Examples for Sum_of_Squares.
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```     6 *)
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```     7
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```     8 theory SOS
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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"
```
```    13   by sos
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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
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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
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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
```
```    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
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```    26
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```    27 lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3"
```
```    28   by sos
```
```    29
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```    30 lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)"
```
```    31   by sos
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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
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```    35
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```    36 lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1"
```
```    37   by sos
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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
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```    41
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```    42
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```    43 text \<open>One component of denominator in dodecahedral example.\<close>
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```    44
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```    45 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)"
```
```    46   by sos
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```    47
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```    48
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```    49 text \<open>Over a larger but simpler interval.\<close>
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```    50
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```    51 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)"
```
```    52   by sos
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```    53
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```    54
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```    55 text \<open>We can do 12. I think 12 is a sharp bound; see PP's certificate.\<close>
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```    56
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```    57 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)"
```
```    58   by sos
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```    59
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```    60
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```    61 text \<open>Inequality from sci.math (see "Leon-Sotelo, por favor").\<close>
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```    62
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```    63 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2"
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```    64   by sos
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```    65
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```    66 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2"
```
```    67   by sos
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```    68
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```    69 lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2"
```
```    70   by sos
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```    71
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```    72 lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x"
```
```    73   by sos
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```    74
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```    75 lemma "(0::real) < x --> 0 < 1 + x + x^2"
```
```    76   by sos
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```    77
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```    78 lemma "(0::real) <= x --> 0 < 1 + x + x^2"
```
```    79   by sos
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```    80
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```    81 lemma "(0::real) < 1 + x^2"
```
```    82   by sos
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```    83
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```    84 lemma "(0::real) <= 1 + 2 * x + x^2"
```
```    85   by sos
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```    86
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```    87 lemma "(0::real) < 1 + abs x"
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```    88   by sos
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```    89
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```    90 lemma "(0::real) < 1 + (1 + x)^2 * (abs x)"
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```    91   by sos
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```    92
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```    93
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```    94 lemma "abs ((1::real) + x^2) = (1::real) + x^2"
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```    95   by sos
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```    96 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0"
```
```    97   by sos
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```    98
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```    99 lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z"
```
```   100   by sos
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```   101 lemma "(1::real) < x --> x^2 < y --> 1 < y"
```
```   102   by sos
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```   103 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
```
```   104   by sos
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```   105 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
```
```   106   by sos
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```   107 lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c"
```
```   108   by sos
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```   109 lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x"
```
```   110   by sos
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```   111 lemma "abs(x - z) <= e & abs(y - z) <= e & 0 <= u & 0 <= v & (u + v = 1) --> abs((u * x + v * y) - z) <= (e::real)"
```
```   112   by sos
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```   113
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```   114
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```   115 (* 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?*)
```
```   116
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```   117 lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
```
```   118   by sos
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```   119
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```   120 lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
```
```   121   by sos
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```   122
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```   123 lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
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```   124   by sos
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```   125
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```   126 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"
```
```   127   by sos
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```   128
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```   129 end
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```   130
```