src/HOL/ex/SOS_Cert.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_Cert.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3     Author:     Philipp Meyer, TU Muenchen
```
```     4
```
```     5 Examples for Sum_of_Squares: replay of certificates.
```
```     6 *)
```
```     7
```
```     8 theory SOS_Cert
```
```     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 "((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))))))")
```
```    14
```
```    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 "(((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))))))))))")
```
```    17
```
```    18 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0"
```
```    19   by (sos "((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))))))")
```
```    20
```
```    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 "((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)))))))")
```
```    23
```
```    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 "(((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))))))))))")
```
```    26
```
```    27 lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3"
```
```    28   by (sos "(((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))))))")
```
```    29
```
```    30 lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)"
```
```    31   by (sos "(((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)))))))")
```
```    32
```
```    33 lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1"
```
```    34   by (sos "(((A<0 * R<1) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))")
```
```    35
```
```    36 lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1"
```
```    37   by (sos "((((A<0 * A<1) * R<1) + ((A<=0 * R<1) * (R<1 * [1]^2))))")
```
```    38
```
```    39 lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)"
```
```    40   by (sos "((((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)))))")
```
```    41
```
```    42
```
```    43 text \<open>One component of denominator in dodecahedral example.\<close>
```
```    44
```
```    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 "(((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)))))))))))))")
```
```    47
```
```    48
```
```    49 text \<open>Over a larger but simpler interval.\<close>
```
```    50
```
```    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 "((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)))))))))))")
```
```    53
```
```    54
```
```    55 text \<open>We can do 12. I think 12 is a sharp bound; see PP's certificate.\<close>
```
```    56
```
```    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 "(((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))))))))))))))))))")
```
```    59
```
```    60
```
```    61 text \<open>Inequality from sci.math (see "Leon-Sotelo, por favor").\<close>
```
```    62
```
```    63 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2"
```
```    64   by (sos "(((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))))))")
```
```    65
```
```    66 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2"
```
```    67   by (sos "(((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))))))")
```
```    68
```
```    69 lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2"
```
```    70   by (sos "(((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)))))")
```
```    71
```
```    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 "(((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))))))")
```
```    74
```
```    75 lemma "(0::real) < x --> 0 < 1 + x + x^2"
```
```    76   by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")
```
```    77
```
```    78 lemma "(0::real) <= x --> 0 < 1 + x + x^2"
```
```    79   by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")
```
```    80
```
```    81 lemma "(0::real) < 1 + x^2"
```
```    82   by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))")
```
```    83
```
```    84 lemma "(0::real) <= 1 + 2 * x + x^2"
```
```    85   by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [x + 1]^2))))")
```
```    86
```
```    87 lemma "(0::real) < 1 + abs x"
```
```    88   by (sos "((R<1 + (((A<=1 * R<1) * (R<1/2 * [1]^2)) + ((A<=0 * R<1) * (R<1/2 * [1]^2)))))")
```
```    89
```
```    90 lemma "(0::real) < 1 + (1 + x)^2 * (abs x)"
```
```    91   by (sos "(((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))))))")
```
```    92
```
```    93
```
```    94 lemma "abs ((1::real) + x^2) = (1::real) + x^2"
```
```    95   by (sos "(() & (((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)))))))")
```
```    96 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0"
```
```    97   by (sos "((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))))))")
```
```    98
```
```    99 lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z"
```
```   100   by (sos "((((A<0 * A<1) * R<1) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))")
```
```   101 lemma "(1::real) < x --> x^2 < y --> 1 < y"
```
```   102   by (sos "((((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)))))))))")
```
```   103 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
```
```   104   by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
```
```   105 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
```
```   106   by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
```
```   107 lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c"
```
```   108   by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
```
```   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 "(((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)))))")
```
```   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 "((((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))))))")
```
```   113
```
```   114
```
```   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
```
```   117 lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
```
```   118   by (sos "(((((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))))))")
```
```   119
```
```   120 lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
```
```   121   by (sos "(((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))))))))")
```
```   122
```
```   123 lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
```
```   124   by (sos "((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2))))")
```
```   125
```
```   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 "((((((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)))))")
```
```   128
```
```   129 end
```
```   130
```