src/HOL/Library/Sum_Of_Squares.thy
 author huffman Thu Jun 11 09:03:24 2009 -0700 (2009-06-11) changeset 31563 ded2364d14d4 parent 31512 27118561c2e0 child 32268 d50f0cb67578 permissions -rw-r--r--
cleaned up some proofs
```     1 (* Title:      Library/Sum_Of_Squares
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```     2    Author:     Amine Chaieb, University of Cambridge
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```     3 *)
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```     4
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```     5 header {* A decision method for universal multivariate real arithmetic with addition,
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```     6           multiplication and ordering using semidefinite programming*}
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```     7 theory Sum_Of_Squares
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```     8   imports Complex_Main (* "~~/src/HOL/Decision_Procs/Dense_Linear_Order" *)
```
```     9   uses "positivstellensatz.ML" "sum_of_squares.ML"
```
```    10   begin
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```    11
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```    12 (* Note:
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```    13
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```    14 In order to use the method sos, install CSDP (https://projects.coin-or.org/Csdp/) and put the executable csdp on your path.
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```    15
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```    16 *)
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```    17
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```    18 method_setup sos = {* Scan.succeed (SIMPLE_METHOD' o Sos.sos_tac) *}
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```    19   "Prove universal problems over the reals using sums of squares"
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```    20
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```    21 text{* Tests -- commented since they work only when csdp is installed -- see above *}
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```    22 (*
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```    23 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0" by sos
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```    24
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```    25 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)" by sos
```
```    26
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```    27 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0" by sos
```
```    28
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```    29 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" by sos
```
```    30
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```    31 lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z" by sos
```
```    32
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```    33 lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3" by sos
```
```    34
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```    35 lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)" by sos
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```    36
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```    37 lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1" by sos
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```    38
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```    39 lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1" by sos;
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```    40
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```    41 lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)" by sos
```
```    42 *)
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```    43 (* ------------------------------------------------------------------------- *)
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```    44 (* One component of denominator in dodecahedral example.                     *)
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```    45 (* ------------------------------------------------------------------------- *)
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```    46 (*
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```    47 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)" by sos;
```
```    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)" by sos
```
```    54 *)
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```    55 (* ------------------------------------------------------------------------- *)
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```    56 (* We can do 12. I think 12 is a sharp bound; see PP's certificate.          *)
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```    57 (* ------------------------------------------------------------------------- *)
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```    58 (*
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```    59 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)" by sos
```
```    60 *)
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```    61
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```    62 (* ------------------------------------------------------------------------- *)
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```    63 (* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
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```    64 (* ------------------------------------------------------------------------- *)
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```    65 (*
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```    66 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2" by sos
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```    67
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```    68 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2" by sos
```
```    69
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```    70 lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2" by sos
```
```    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" by sos
```
```    73
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```    74 lemma "(0::real) < x --> 0 < 1 + x + x^2" by sos
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```    75
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```    76 lemma "(0::real) <= x --> 0 < 1 + x + x^2" by sos
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```    77
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```    78 lemma "(0::real) < 1 + x^2" by sos
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```    79
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```    80 lemma "(0::real) <= 1 + 2 * x + x^2" by sos
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```    81
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```    82 lemma "(0::real) < 1 + abs x" by sos
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```    83
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```    84 lemma "(0::real) < 1 + (1 + x)^2 * (abs x)" by sos
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```    85
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```    86
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```    87
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```    88 lemma "abs ((1::real) + x^2) = (1::real) + x^2" by sos
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```    89 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0" by sos
```
```    90
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```    91 lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z" by sos
```
```    92 lemma "(1::real) < x --> x^2 < y --> 1 < y" by sos
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```    93 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" by sos
```
```    94 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" by sos
```
```    95 lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c" by sos
```
```    96 lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x" by sos
```
```    97 lemma "abs(x - z) <= e & abs(y - z) <= e & 0 <= u & 0 <= v & (u + v = 1) --> abs((u * x + v * y) - z) <= (e::real)" by sos
```
```    98 *)
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```    99 (*
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```   100 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?*)
```
```   101 (*
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```   102 lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
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```   103 apply sos
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```   104 done
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```   105
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```   106 lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
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```   107 apply sos
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```   108 done
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```   109
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```   110 lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
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```   111 apply sos
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```   112 done
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```   113
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```   114 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" by sos
```
```   115 *)
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```   116
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```   117 end
```