| 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 | |