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1c3ffd8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 | import Mathlib
set_option linter.unusedVariables.analyzeTactics true
theorem imo_1965_p2
(x y z : ℝ)
(a : ℕ → ℝ)
(h₀ : 0 < a 0 ∧ 0 < a 4 ∧ 0 < a 8)
(h₁ : a 1 < 0 ∧ a 2 < 0)
(h₂ : a 3 < 0 ∧ a 5 < 0)
(h₃ : a 6 < 0 ∧ a 7 < 0)
(h₄ : 0 < a 0 + a 1 + a 2)
(h₅ : 0 < a 3 + a 4 + a 5)
(h₆ : 0 < a 6 + a 7 + a 8)
(h₇ : a 0 * x + a 1 * y + a 2 * z = 0)
(h₈ : a 3 * x + a 4 * y + a 5 * z = 0)
(h₉ : a 6 * x + a 7 * y + a 8 * z = 0) :
x = 0 ∧ y = 0 ∧ z = 0 := by
by_cases hx0: x = 0
. rw [hx0] at h₇
constructor
. exact hx0
. rw [hx0] at h₈ h₉
simp at h₇ h₈ h₉
by_cases hy0: y = 0
. constructor
. exact hy0
. rw [hy0] at h₇
simp at h₇
. cases' h₇ with h₇₀ h₇₁
. exfalso
linarith
. exact h₇₁
. by_cases hyn: y < 0
. have g1: 0 < a 1 * y := by exact mul_pos_of_neg_of_neg h₁.1 hyn
have g2: a 1 * y = -a 2 * z := by linarith
rw [g2] at g1
have g3: a 2 *z < 0 := by linarith
have hzp: 0 < z := by exact pos_of_mul_neg_right g3 (le_of_lt h₁.2)
exfalso
have g4: a 4 * y < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 hyn
have g5: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzp
linarith
. push_neg at hy0 hyn
have hyp: 0 < y := by exact lt_of_le_of_ne hyn hy0.symm
exfalso
have g1: a 1 * y < 0 := by exact mul_neg_of_neg_of_pos h₁.1 hyp
have g2: 0 < z * a 2 := by linarith
have hzp: z < 0 := by exact neg_of_mul_pos_left g2 (le_of_lt h₁.2)
have g3: 0 < a 4 * y := by exact mul_pos h₀.2.1 hyp
have g4: 0 < a 5 * z := by exact mul_pos_of_neg_of_neg h₂.2 hzp
linarith
. exfalso
push_neg at hx0
by_cases hxp: 0 < x
. by_cases hy0: y = 0
. rw [hy0] at h₇ h₈ h₉
simp at h₇ h₈ h₉
have g1: 0 < a 0 * x := by exact mul_pos h₀.1 hxp
have g2: a 2 * z < 0 := by linarith
have hzn: 0 < z := by exact pos_of_mul_neg_right g2 (le_of_lt h₁.2)
have g3: a 3 * x < 0 := by exact mul_neg_of_neg_of_pos h₂.1 hxp
have g4: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzn
linarith
. push_neg at hy0
by_cases hyp: 0 < y
. have g1: a 6 * x < 0 := by exact mul_neg_of_neg_of_pos h₃.1 hxp
have g2: a 7 * y < 0 := by exact mul_neg_of_neg_of_pos h₃.2 hyp
have g3: 0 < z * a 8 := by linarith
have hzp: 0 < z := by exact pos_of_mul_pos_left g3 (le_of_lt h₀.2.2)
------ here we consider all the possible relationships between x, y, z
by_cases rxy: x ≤ y
. by_cases ryz: y ≤ z
-- x <= y <= z
. have g2: 0 < (a 6 + a 7 + a 8) * y := by exact mul_pos h₆ hyp
have g3: 0 ≤ a 6 * (x-y) := by
exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₃.1) (by linarith)-- exact mul_nonneg (le_of_lt h₃.1) (by linarith),},
have g4: 0 ≤ a 8 * (z-y) := by exact mul_nonneg (le_of_lt h₀.2.2) (by linarith)
linarith
push_neg at ryz
by_cases rxz: x ≤ z
-- x <= z < y
. have g2: 0 < (a 3 + a 4 + a 5) * y := by exact mul_pos h₅ hyp
have g3: 0 ≤ a 3 * (x-y) := by
exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₂.1) (by linarith)
have g4: 0 < a 5 * (z-y) := by
exact mul_pos_of_neg_of_neg h₂.2 (by linarith)
linarith
push_neg at rxz -- z < x <= y
have g2: 0 < (a 3 + a 4 + a 5) * y := by exact mul_pos h₅ hyp
have g3: 0 ≤ a 3 * (x-y) := by
exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₂.1) (by linarith)
have g4: 0 < a 5 * (z-y) := by
exact mul_pos_of_neg_of_neg h₂.2 (by linarith)
linarith
push_neg at rxy
by_cases rzy: z ≤ y
-- z <= y < x
. have g2: 0 < (a 0 + a 1 + a 2) * y := by exact mul_pos h₄ hyp
have g3: 0 < a 0 * (x-y) := by exact mul_pos h₀.1 (by linarith)
have g4: 0 ≤ a 2 * (z-y) := by
exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₁.2) (by linarith)
linarith
. push_neg at rzy
by_cases rzx: z ≤ x
-- y < z <= x
. have g2: 0 < (a 0 + a 1 + a 2) * z := by exact mul_pos h₄ hzp
have g3: 0 ≤ a 0 * (x-z) := by exact mul_nonneg (le_of_lt h₀.1) (by linarith)
have g4: 0 < a 1 * (y-z) := by exact mul_pos_of_neg_of_neg h₁.1 (by linarith)
linarith
. push_neg at rzx
-- y < x < z
have g2: 0 < (a 6 + a 7 + a 8) * z := by exact mul_pos h₆ hzp
have g3: 0 < a 6 * (x-z) := by exact mul_pos_of_neg_of_neg h₃.1 (by linarith)
have g4: 0 < a 7 * (y-z) := by exact mul_pos_of_neg_of_neg h₃.2 (by linarith)
linarith
-------- new world where y < 0 and 0 < x
. push_neg at hyp
have hyn: y < 0 := by exact lt_of_le_of_ne hyp hy0
-- show from a 0 that 0 < z
have g1: 0 < a 0 * x := by exact mul_pos h₀.1 hxp
have g2: 0 < a 1 * y := by exact mul_pos_of_neg_of_neg h₁.1 hyn
have g3: a 2 * z < 0 := by linarith
have hzp: 0 < z := by exact pos_of_mul_neg_right g3 (le_of_lt h₁.2)
-- then show from a 3 that's not possible
have g4: a 3 * x < 0 := by exact mul_neg_of_neg_of_pos h₂.1 hxp
have g5: a 4 * y < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 hyn
have g6: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzp
linarith
. push_neg at hxp
have hxn: x < 0 := by exact lt_of_le_of_ne hxp hx0
by_cases hyp: 0 ≤ y
. have g1: a 0 * x < 0 := by exact mul_neg_of_pos_of_neg h₀.1 hxn
have g2: a 1 * y ≤ 0 := by
refine mul_nonpos_iff.mpr ?_
right
constructor
. exact le_of_lt h₁.1
. exact hyp
have g3: 0 < z * a 2 := by linarith
have hzn: z < 0 := by exact neg_of_mul_pos_left g3 (le_of_lt h₁.2)
-- demonstrate the contradiction
have g4: 0 < a 3 * x := by exact mul_pos_of_neg_of_neg h₂.1 hxn
have g5: 0 ≤ a 4 * y := by exact mul_nonneg (le_of_lt h₀.2.1) hyp
have g6: 0 < a 5 * z := by exact mul_pos_of_neg_of_neg h₂.2 hzn
linarith
. push_neg at hyp
-- have hyn: y < 0, {exact lt_of_le_of_ne hyp hy0,},
have g1: 0 < a 6 * x := by exact mul_pos_of_neg_of_neg h₃.1 hxn
have g2: 0 < a 7 * y := by exact mul_pos_of_neg_of_neg h₃.2 hyp
have g3: z * a 8 < 0 := by linarith
have hzp: z < 0 := by exact neg_of_mul_neg_left g3 (le_of_lt h₀.2.2)
-- we have x,y,z < 0 -- we will examine all the orders they can have
by_cases rxy: x ≤ y
. by_cases ryz: y ≤ z
-- x <= y <= z
. have g2: (a 0 + a 1 + a 2) * y < 0 := by exact mul_neg_of_pos_of_neg h₄ hyp
have g3: a 0 * (x-y) ≤ 0 := by
exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.1) (by linarith)
have g4: a 2 * (z-y) ≤ 0 := by
exact mul_nonpos_of_nonpos_of_nonneg (le_of_lt h₁.2) (by linarith)
linarith
. push_neg at ryz
by_cases rxz: x ≤ z
-- x <= z < y
. have g2: (a 0 + a 1 + a 2) * z < 0 := by exact mul_neg_of_pos_of_neg h₄ hzp
have g3: a 0 * (x-z) ≤ 0 := by
exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.1) (by linarith)
have g4: a 1 * (y-z) < 0 := by
exact mul_neg_of_neg_of_pos h₁.1 (by linarith)
linarith
. push_neg at rxz -- z < x <= y
have g2: (a 6 + a 7 + a 8) * z < 0 := by exact mul_neg_of_pos_of_neg h₆ hzp
have g3: a 6 * (x-z) < 0 := by exact mul_neg_of_neg_of_pos h₃.1 (by linarith)
have g4: a 7 * (y-z) < 0 := by exact mul_neg_of_neg_of_pos h₃.2 (by linarith)
linarith
. push_neg at rxy
by_cases rzy: z ≤ y
-- z <= y < x
. have g2: (a 6 + a 7 + a 8) * y < 0 := by exact mul_neg_of_pos_of_neg h₆ hyp
have g3: a 6 * (x-y) < 0 := by exact mul_neg_of_neg_of_pos h₃.1 (by linarith)
have g4: a 8 * (z-y) ≤ 0 := by
exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.2.2) (by linarith)
linarith
. push_neg at rzy
by_cases rzx: z ≤ x
-- y < z <= x
. have g2: (a 3 + a 4 + a 5) * z < 0 := by exact mul_neg_of_pos_of_neg h₅ hzp
have g3: a 3 * (x-z) ≤ 0 := by
exact mul_nonpos_of_nonpos_of_nonneg (le_of_lt h₂.1) (by linarith)
have g4: a 4 * (y-z) < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 (by linarith)
linarith
. push_neg at rzx
-- y < x < z
have g2: (a 3 + a 4 + a 5) * y < 0 := by exact mul_neg_of_pos_of_neg h₅ hyp
have g3: a 3 * (x-y) < 0 := by exact mul_neg_of_neg_of_pos h₂.1 (by linarith)
have g4: a 5 * (z-y) < 0 := by exact mul_neg_of_neg_of_pos h₂.2 (by linarith)
linarith
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