8 theory SOS |
8 theory SOS |
9 imports "~~/src/HOL/Library/Sum_of_Squares" |
9 imports "~~/src/HOL/Library/Sum_of_Squares" |
10 begin |
10 begin |
11 |
11 |
12 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0" |
12 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0" |
13 by (sos csdp) |
13 by sos |
14 |
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)" |
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 csdp) |
16 by sos |
17 |
17 |
18 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0" |
18 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0" |
19 by (sos csdp) |
19 by sos |
20 |
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" |
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 csdp) |
22 by sos |
23 |
23 |
24 lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z" |
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 csdp) |
25 by sos |
26 |
26 |
27 lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3" |
27 lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3" |
28 by (sos csdp) |
28 by sos |
29 |
29 |
30 lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)" |
30 lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)" |
31 by (sos csdp) |
31 by sos |
32 |
32 |
33 lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1" |
33 lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1" |
34 by (sos csdp) |
34 by sos |
35 |
35 |
36 lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1" |
36 lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1" |
37 by (sos csdp) |
37 by sos |
38 |
38 |
39 lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)" |
39 lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)" |
40 by (sos csdp) |
40 by sos |
41 |
41 |
42 |
42 |
43 text \<open>One component of denominator in dodecahedral example.\<close> |
43 text \<open>One component of denominator in dodecahedral example.\<close> |
44 |
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)" |
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 csdp) |
46 by sos |
47 |
47 |
48 |
48 |
49 text \<open>Over a larger but simpler interval.\<close> |
49 text \<open>Over a larger but simpler interval.\<close> |
50 |
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)" |
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 csdp) |
52 by sos |
53 |
53 |
54 |
54 |
55 text \<open>We can do 12. I think 12 is a sharp bound; see PP's certificate.\<close> |
55 text \<open>We can do 12. I think 12 is a sharp bound; see PP's certificate.\<close> |
56 |
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)" |
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 csdp) |
58 by sos |
59 |
59 |
60 |
60 |
61 text \<open>Inequality from sci.math (see "Leon-Sotelo, por favor").\<close> |
61 text \<open>Inequality from sci.math (see "Leon-Sotelo, por favor").\<close> |
62 |
62 |
63 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2" |
63 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2" |
64 by (sos csdp) |
64 by sos |
65 |
65 |
66 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2" |
66 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2" |
67 by (sos csdp) |
67 by sos |
68 |
68 |
69 lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2" |
69 lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2" |
70 by (sos csdp) |
70 by sos |
71 |
71 |
72 lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x" |
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 csdp) |
73 by sos |
74 |
74 |
75 lemma "(0::real) < x --> 0 < 1 + x + x^2" |
75 lemma "(0::real) < x --> 0 < 1 + x + x^2" |
76 by (sos csdp) |
76 by sos |
77 |
77 |
78 lemma "(0::real) <= x --> 0 < 1 + x + x^2" |
78 lemma "(0::real) <= x --> 0 < 1 + x + x^2" |
79 by (sos csdp) |
79 by sos |
80 |
80 |
81 lemma "(0::real) < 1 + x^2" |
81 lemma "(0::real) < 1 + x^2" |
82 by (sos csdp) |
82 by sos |
83 |
83 |
84 lemma "(0::real) <= 1 + 2 * x + x^2" |
84 lemma "(0::real) <= 1 + 2 * x + x^2" |
85 by (sos csdp) |
85 by sos |
86 |
86 |
87 lemma "(0::real) < 1 + abs x" |
87 lemma "(0::real) < 1 + abs x" |
88 by (sos csdp) |
88 by sos |
89 |
89 |
90 lemma "(0::real) < 1 + (1 + x)^2 * (abs x)" |
90 lemma "(0::real) < 1 + (1 + x)^2 * (abs x)" |
91 by (sos csdp) |
91 by sos |
92 |
92 |
93 |
93 |
94 lemma "abs ((1::real) + x^2) = (1::real) + x^2" |
94 lemma "abs ((1::real) + x^2) = (1::real) + x^2" |
95 by (sos csdp) |
95 by sos |
96 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0" |
96 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0" |
97 by (sos csdp) |
97 by sos |
98 |
98 |
99 lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z" |
99 lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z" |
100 by (sos csdp) |
100 by sos |
101 lemma "(1::real) < x --> x^2 < y --> 1 < y" |
101 lemma "(1::real) < x --> x^2 < y --> 1 < y" |
102 by (sos csdp) |
102 by sos |
103 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" |
103 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" |
104 by (sos csdp) |
104 by sos |
105 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" |
105 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" |
106 by (sos csdp) |
106 by sos |
107 lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c" |
107 lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c" |
108 by (sos csdp) |
108 by sos |
109 lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x" |
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 csdp) |
110 by sos |
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)" |
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 csdp) |
112 by sos |
113 |
113 |
114 |
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?*) |
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 |
116 |
117 lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x" |
117 lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x" |
118 by (sos csdp) |
118 by sos |
119 |
119 |
120 lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)" |
120 lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)" |
121 by (sos csdp) |
121 by sos |
122 |
122 |
123 lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)" |
123 lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)" |
124 by (sos csdp) |
124 by sos |
125 |
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" |
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 csdp) |
127 by sos |
128 |
128 |
129 end |
129 end |
130 |
130 |