src/HOL/ex/SOS.thy
 author wenzelm Wed Jun 22 10:09:20 2016 +0200 (2016-06-22) changeset 63343 fb5d8a50c641 parent 61156 931b732617a2 child 66453 cc19f7ca2ed6 permissions -rw-r--r--
bundle lifting_syntax;
```     1 (*  Title:      HOL/ex/SOS.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3     Author:     Philipp Meyer, TU Muenchen
```
```     4
```
```     5 Examples for Sum_of_Squares.
```
```     6 *)
```
```     7
```
```     8 theory SOS
```
```     9 imports "~~/src/HOL/Library/Sum_of_Squares"
```
```    10 begin
```
```    11
```
```    12 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0"
```
```    13   by sos
```
```    14
```
```    15 lemma "a1 \<ge> 0 \<and> a2 \<ge> 0 \<and> (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) \<and> (a1 * b1 + a2 * b2 = 0) \<longrightarrow>
```
```    16     a1 * a2 - b1 * b2 \<ge> (0::real)"
```
```    17   by sos
```
```    18
```
```    19 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<longrightarrow> a < 0"
```
```    20   by sos
```
```    21
```
```    22 lemma "(0::real) \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow>
```
```    23     x\<^sup>2 + y\<^sup>2 < 1 \<or> (x - 1)\<^sup>2 + y\<^sup>2 < 1 \<or> x\<^sup>2 + (y - 1)\<^sup>2 < 1 \<or> (x - 1)\<^sup>2 + (y - 1)\<^sup>2 < 1"
```
```    24   by sos
```
```    25
```
```    26 lemma "(0::real) \<le> x \<and> 0 \<le> y \<and> 0 \<le> z \<and> x + y + z \<le> 3 \<longrightarrow> x * y + x * z + y * z \<ge> 3 * x * y * z"
```
```    27   by sos
```
```    28
```
```    29 lemma "(x::real)\<^sup>2 + y\<^sup>2 + z\<^sup>2 = 1 \<longrightarrow> (x + y + z)\<^sup>2 \<le> 3"
```
```    30   by sos
```
```    31
```
```    32 lemma "w\<^sup>2 + x\<^sup>2 + y\<^sup>2 + z\<^sup>2 = 1 \<longrightarrow> (w + x + y + z)\<^sup>2 \<le> (4::real)"
```
```    33   by sos
```
```    34
```
```    35 lemma "(x::real) \<ge> 1 \<and> y \<ge> 1 \<longrightarrow> x * y \<ge> x + y - 1"
```
```    36   by sos
```
```    37
```
```    38 lemma "(x::real) > 1 \<and> y > 1 \<longrightarrow> x * y > x + y - 1"
```
```    39   by sos
```
```    40
```
```    41 lemma "\<bar>x\<bar> \<le> 1 \<longrightarrow> \<bar>64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x\<bar> \<le> (1::real)"
```
```    42   by sos
```
```    43
```
```    44
```
```    45 text \<open>One component of denominator in dodecahedral example.\<close>
```
```    46
```
```    47 lemma "2 \<le> x \<and> x \<le> 125841 / 50000 \<and> 2 \<le> y \<and> y \<le> 125841 / 50000 \<and> 2 \<le> z \<and> z \<le> 125841 / 50000 \<longrightarrow>
```
```    48     2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) \<ge> (0::real)"
```
```    49   by sos
```
```    50
```
```    51
```
```    52 text \<open>Over a larger but simpler interval.\<close>
```
```    53
```
```    54 lemma "(2::real) \<le> x \<and> x \<le> 4 \<and> 2 \<le> y \<and> y \<le> 4 \<and> 2 \<le> z \<and> z \<le> 4 \<longrightarrow>
```
```    55     0 \<le> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)"
```
```    56   by sos
```
```    57
```
```    58
```
```    59 text \<open>We can do 12. I think 12 is a sharp bound; see PP's certificate.\<close>
```
```    60
```
```    61 lemma "2 \<le> (x::real) \<and> x \<le> 4 \<and> 2 \<le> y \<and> y \<le> 4 \<and> 2 \<le> z \<and> z \<le> 4 \<longrightarrow>
```
```    62     12 \<le> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)"
```
```    63   by sos
```
```    64
```
```    65
```
```    66 text \<open>Inequality from sci.math (see "Leon-Sotelo, por favor").\<close>
```
```    67
```
```    68 lemma "0 \<le> (x::real) \<and> 0 \<le> y \<and> x * y = 1 \<longrightarrow> x + y \<le> x\<^sup>2 + y\<^sup>2"
```
```    69   by sos
```
```    70
```
```    71 lemma "0 \<le> (x::real) \<and> 0 \<le> y \<and> x * y = 1 \<longrightarrow> x * y * (x + y) \<le> x\<^sup>2 + y\<^sup>2"
```
```    72   by sos
```
```    73
```
```    74 lemma "0 \<le> (x::real) \<and> 0 \<le> y \<longrightarrow> x * y * (x + y)\<^sup>2 \<le> (x\<^sup>2 + y\<^sup>2)\<^sup>2"
```
```    75   by sos
```
```    76
```
```    77 lemma "(0::real) \<le> a \<and> 0 \<le> b \<and> 0 \<le> c \<and> c * (2 * a + b)^3 / 27 \<le> x \<longrightarrow> c * a\<^sup>2 * b \<le> x"
```
```    78   by sos
```
```    79
```
```    80 lemma "(0::real) < x \<longrightarrow> 0 < 1 + x + x\<^sup>2"
```
```    81   by sos
```
```    82
```
```    83 lemma "(0::real) \<le> x \<longrightarrow> 0 < 1 + x + x\<^sup>2"
```
```    84   by sos
```
```    85
```
```    86 lemma "(0::real) < 1 + x\<^sup>2"
```
```    87   by sos
```
```    88
```
```    89 lemma "(0::real) \<le> 1 + 2 * x + x\<^sup>2"
```
```    90   by sos
```
```    91
```
```    92 lemma "(0::real) < 1 + \<bar>x\<bar>"
```
```    93   by sos
```
```    94
```
```    95 lemma "(0::real) < 1 + (1 + x)\<^sup>2 * \<bar>x\<bar>"
```
```    96   by sos
```
```    97
```
```    98
```
```    99 lemma "\<bar>(1::real) + x\<^sup>2\<bar> = (1::real) + x\<^sup>2"
```
```   100   by sos
```
```   101
```
```   102 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0"
```
```   103   by sos
```
```   104
```
```   105 lemma "(0::real) < x \<longrightarrow> 1 < y \<longrightarrow> y * x \<le> z \<longrightarrow> x < z"
```
```   106   by sos
```
```   107
```
```   108 lemma "(1::real) < x \<longrightarrow> x\<^sup>2 < y \<longrightarrow> 1 < y"
```
```   109   by sos
```
```   110
```
```   111 lemma "(b::real)\<^sup>2 < 4 * a * c \<longrightarrow> a * x\<^sup>2 + b * x + c \<noteq> 0"
```
```   112   by sos
```
```   113
```
```   114 lemma "(b::real)\<^sup>2 < 4 * a * c \<longrightarrow> a * x\<^sup>2 + b * x + c \<noteq> 0"
```
```   115   by sos
```
```   116
```
```   117 lemma "(a::real) * x\<^sup>2 + b * x + c = 0 \<longrightarrow> b\<^sup>2 \<ge> 4 * a * c"
```
```   118   by sos
```
```   119
```
```   120 lemma "(0::real) \<le> b \<and> 0 \<le> c \<and> 0 \<le> x \<and> 0 \<le> y \<and> x\<^sup>2 = c \<and> y\<^sup>2 = a\<^sup>2 * c + b \<longrightarrow> a * c \<le> y * x"
```
```   121   by sos
```
```   122
```
```   123 lemma "\<bar>x - z\<bar> \<le> e \<and> \<bar>y - z\<bar> \<le> e \<and> 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1 --> \<bar>(u * x + v * y) - z\<bar> \<le> (e::real)"
```
```   124   by sos
```
```   125
```
```   126 lemma "(x::real) - y - 2 * x^4 = 0 \<and> 0 \<le> x \<and> x \<le> 2 \<and> 0 \<le> y \<and> y \<le> 3 \<longrightarrow> y\<^sup>2 - 7 * y - 12 * x + 17 \<ge> 0"
```
```   127   oops (*Too hard?*)
```
```   128
```
```   129 lemma "(0::real) \<le> x \<longrightarrow> (1 + x + x\<^sup>2) / (1 + x\<^sup>2) \<le> 1 + x"
```
```   130   by sos
```
```   131
```
```   132 lemma "(0::real) \<le> x \<longrightarrow> 1 - x \<le> 1 / (1 + x + x\<^sup>2)"
```
```   133   by sos
```
```   134
```
```   135 lemma "(x::real) \<le> 1 / 2 \<longrightarrow> - x - 2 * x\<^sup>2 \<le> - x / (1 - x)"
```
```   136   by sos
```
```   137
```
```   138 lemma "4 * r\<^sup>2 = p\<^sup>2 - 4 * q \<and> r \<ge> (0::real) \<and> x\<^sup>2 + p * x + q = 0 \<longrightarrow>
```
```   139     2 * (x::real) = - p + 2 * r \<or> 2 * x = - p - 2 * r"
```
```   140   by sos
```
```   141
```
```   142 end
```