import Mathlib set_option linter.unusedVariables.analyzeTactics true open Real lemma imo_1965_p2_1 (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) (hx0 : x = 0) : x = 0 ∧ y = 0 ∧ z = 0 := by constructor . exact hx0 . rw [hx0] at h₇ 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 lemma imo_1965_p2_2 (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) (hx0 : x = 0) : y = 0 ∧ z = 0 := by rw [hx0] 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 lemma imo_1965_p2_3 (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) (hx0: x = 0) (hy0 : y = 0) : y = 0 ∧ z = 0 := by rw [hx0] at h₇ h₈ h₉ constructor . exact hy0 . rw [hy0] at h₇ simp at h₇ . cases' h₇ with h₇₀ h₇₁ . exfalso linarith . exact h₇₁ lemma imo_1965_p2_4 -- (x : ℝ) (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) -- (hx0 : x = 0) (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) (h₉ : a 7 * y + a 8 * z = 0) (hy0 : y = 0) : z = 0 := by rw [hy0] at h₇ simp at h₇ . cases' h₇ with h₇₀ h₇₁ . exfalso linarith . exact h₇₁ lemma imo_1965_p2_5 -- (x : ℝ) (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) -- (hx0 : x = 0) (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) -- (h₉ : a 7 * y + a 8 * z = 0) -- (hy0 : ¬y = 0) (hyn : y < 0) : y = 0 ∧ z = 0 := by 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 lemma imo_1965_p2_6 -- (x : ℝ) (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) -- (hx0 : x = 0) (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) -- (h₉ : a 7 * y + a 8 * z = 0) -- (hy0 : ¬y = 0) (hyn : y < 0) : False := by 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) 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 lemma imo_1965_p2_7 -- (x : ℝ) (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) -- (hx0 : x = 0) -- (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) -- (h₉ : a 7 * y + a 8 * z = 0) -- (hy0 : ¬y = 0) (hyn : y < 0) -- (g1 : 0 < -a 2 * z) -- (g2 : a 1 * y = -a 2 * z) -- (g3 : a 2 * z < 0) (hzp : 0 < z) : False := by 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 lemma imo_1965_p2_8 -- (x : ℝ) (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) -- (hx0 : x = 0) (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) -- (h₉ : a 7 * y + a 8 * z = 0) -- (hy0 : y ≠ 0) (hyp : 0 < y) : y = 0 ∧ z = 0 := by 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 lemma imo_1965_p2_9 -- (x : ℝ) (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) -- (hx0 : x = 0) (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) -- (h₉ : a 7 * y + a 8 * z = 0) -- (hy0 : y ≠ 0) (hyp : 0 < y) : False := by 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 lemma imo_1965_p2_10 -- (x : ℝ) (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) -- (hx0 : x = 0) -- (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) -- (h₉ : a 7 * y + a 8 * z = 0) -- (hy0 : y ≠ 0) -- (hyn : 0 ≤ y) (hyp : 0 < y) -- (g1 : a 1 * y < 0) (g2 : 0 < z * a 2) : False := by 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 lemma imo_1965_p2_11 -- (x : ℝ) (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) -- (hx0 : x = 0) -- (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) -- (h₉ : a 7 * y + a 8 * z = 0) -- (hy0 : y ≠ 0) -- (hyn : 0 ≤ y) (hyp : 0 < y) -- (g1 : a 1 * y < 0) -- (g2 : 0 < z * a 2) (hzp : z < 0) : False := by 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 lemma imo_1965_p2_12 -- (x : ℝ) (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) -- (hx0 : x = 0) -- (h₇ : a 1 * y + a 2 * z = 0) (h₈ : a 4 * y + a 5 * z = 0) -- (h₉ : a 7 * y + a 8 * z = 0) -- (hy0 : y ≠ 0) -- (hyn : 0 ≤ y) -- (hyp : 0 < y) -- (g1 : a 1 * y < 0) -- (g2 : 0 < z * a 2) (hzp : z < 0) (g3 : 0 < a 4 * y) : False := by have g4: 0 < a 5 * z := by exact mul_pos_of_neg_of_neg h₂.2 hzp linarith lemma imo_1965_p2_13 (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) (hx0 : ¬x = 0) : x = 0 ∧ y = 0 ∧ z = 0 := by 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) 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 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 lemma imo_1965_p2_14 (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) (hx0 : x ≠ 0) : False := by 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) 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 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 lemma imo_1965_p2_15 (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) (hx0 : x ≠ 0) (hxp : 0 < x) : False := by 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) 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 lemma imo_1965_p2_16 (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) (hx0 : x ≠ 0) (hxp : 0 < x) (hy0 : y = 0) : False := by 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 lemma imo_1965_p2_17 -- (y : ℝ) (x 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) -- (hx0 : x ≠ 0) (hxp : 0 < x) -- (hy0 : y = 0) (h₇ : a 0 * x + a 2 * z = 0) (h₈ : a 3 * x + a 5 * z = 0) : -- (h₉ : a 6 * x + a 8 * z = 0) : False := by 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 lemma imo_1965_p2_18 -- (y : ℝ) (x 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) -- (hx0 : x ≠ 0) (hxp : 0 < x) -- (hy0 : y = 0) (h₇ : a 0 * x + a 2 * z = 0) : -- (h₈ : a 3 * x + a 5 * z = 0) -- (h₉ : a 6 * x + a 8 * z = 0) : 0 < z := by have g1: 0 < a 0 * x := by exact mul_pos h₀.1 hxp have g2: a 2 * z < 0 := by linarith exact pos_of_mul_neg_right g2 (le_of_lt h₁.2) lemma imo_1965_p2_19 -- (x y : ℝ) (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y = 0) -- (h₇ : a 0 * x + a 2 * z = 0) -- (h₈ : a 3 * x + a 5 * z = 0) -- (h₉ : a 6 * x + a 8 * z = 0) -- (g1 : 0 < a 0 * x) (g2 : a 2 * z < 0) : 0 < z := by refine pos_of_mul_neg_right g2 ?_ exact le_of_lt h₁.2 lemma imo_1965_p2_20 (x z : ℝ) -- (y : 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) -- (hx0 : x ≠ 0) (hxp : 0 < x) -- (hy0 : y = 0) -- (h₇ : a 0 * x + a 2 * z = 0) (h₈ : a 3 * x + a 5 * z = 0) -- (h₉ : a 6 * x + a 8 * z = 0) -- (g1 : 0 < a 0 * x) -- (g2 : a 2 * z < 0) (hzn : 0 < z) : False := by 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 lemma imo_1965_p2_21 -- (y : ℝ) (x 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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y = 0) -- (h₇ : a 0 * x + a 2 * z = 0) (h₈ : a 3 * x + a 5 * z = 0) -- (h₉ : a 6 * x + a 8 * z = 0) -- (g1 : 0 < a 0 * x) -- (g2 : a 2 * z < 0) (hzn : 0 < z) (g3 : a 3 * x < 0) : -- (g4 : a 5 * z < 0) : False := by have g4: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzn linarith lemma imo_1965_p2_22 (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) -- (hx0 : x ≠ 0) (hxp : 0 < x) (hy0 : ¬y = 0) : False := by 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) 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 lemma imo_1965_p2_23 (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) -- (hx0 : x ≠ 0) (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) : False := by 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) 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 lemma imo_1965_p2_24 (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) -- (hx0 : x ≠ 0) (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) : -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) : 0 < z := by 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 refine pos_of_mul_pos_left g3 ?_ exact le_of_lt h₀.2.2 lemma imo_1965_p2_25 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) (hzp : 0 < z) : False := by 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) 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 lemma imo_1965_p2_26 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) (rxy : x ≤ y) : False := by 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) 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 -- z < x <= y . push_neg at rxz 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 lemma imo_1965_p2_27 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) (rxy : x ≤ y) (ryz : y ≤ z) : False := by 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) have g4: 0 ≤ a 8 * (z-y) := by exact mul_nonneg (le_of_lt h₀.2.2) (by linarith) linarith lemma imo_1965_p2_28 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g11 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) (rxy : x ≤ y) (ryz : y ≤ z) (g1 : (a 6 + a 7 + a 8) * y + a 6 * (x - y) + a 8 * (z - y) = 0) : False := by 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) have g4: 0 ≤ a 8 * (z-y) := by exact mul_nonneg (le_of_lt h₀.2.2) (by linarith) linarith lemma imo_1965_p2_29 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : 0 < y) -- (g11 : a 6 * x < 0) -- (g21 : a 7 * y < 0) -- (g31 : 0 < z * a 8) -- (hzp : 0 < z) -- (rxy : x ≤ y) (ryz : y ≤ z) (g1 : (a 6 + a 7 + a 8) * y + a 6 * (x - y) + a 8 * (z - y) = 0) (g2 : 0 < (a 6 + a 7 + a 8) * y) (g3 : 0 ≤ a 6 * (x - y)) : False := by have g4: 0 ≤ a 8 * (z-y) := by exact mul_nonneg (le_of_lt h₀.2.2) (by linarith) linarith lemma imo_1965_p2_30 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) (rxy : x ≤ y) (ryz : z < y) : False := by 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 -- z < x <= y . push_neg at rxz 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 lemma imo_1965_p2_31 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) -- (rxy : x ≤ y) (ryz : z < y) (rxz : x ≤ z) : False := by 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 lemma imo_1965_p2_32 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g11 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) -- (rxy : x ≤ y) (ryz : z < y) (rxz : x ≤ z) (g1 : (a 3 + a 4 + a 5) * y + a 3 * (x - y) + a 5 * (z - y) = 0) : False := by 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 lemma imo_1965_p2_33 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) (rxy : x ≤ y) (ryz : z < y) : -- (rxz : z < x) : False := by 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 lemma imo_1965_p2_34 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g11 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) (rxy : x ≤ y) (ryz : z < y) -- (rxz : z < x) (g1 : (a 3 + a 4 + a 5) * y + a 3 * (x - y) + a 5 * (z - y) = 0) : False := by 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 lemma imo_1965_p2_35 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) (hzp : 0 < z) (rxy : ¬x ≤ y) : False := by 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 lemma imo_1965_p2_36 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) (rxy : y < x) (rzy : z ≤ y) : False := by 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 lemma imo_1965_p2_37 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) (hyp : 0 < y) -- (g11 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) -- (hzp : 0 < z) (rxy : y < x) (rzy : z ≤ y) (g1 : (a 0 + a 1 + a 2) * y + a 0 * (x - y) + a 2 * (z - y) = 0) : False := by 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 lemma imo_1965_p2_38 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) (hzp : 0 < z) -- (rxy : y < x) (rzy : y < z) : False := by 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 lemma imo_1965_p2_39 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) (hzp : 0 < z) -- (rxy : y < x) (rzy : y < z) (rzx : z ≤ x) : False := by 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 lemma imo_1965_p2_40 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : 0 < y) -- (g11 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) (hzp : 0 < z) -- (rxy : y < x) (rzy : y < z) (rzx : z ≤ x) (g1 : (a 0 + a 1 + a 2) * z + a 0 * (x-z) + a 1 * (y-z) = 0) : False := by 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 lemma imo_1965_p2_41 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : 0 < y) -- (g1 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) (hzp : 0 < z) -- (rxy : y < x) (rzy : y < z) (rzx : x < z) : False := by 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 lemma imo_1965_p2_42 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : 0 < y) -- (g11 : a 6 * x < 0) -- (g2 : a 7 * y < 0) -- (g3 : 0 < z * a 8) (hzp : 0 < z) -- (rxy : y < x) (rzy : y < z) (rzx : x < z) (g1 : (a 6 + a 7 + a 8) * z + a 6 * (x - z) + a 7 * (y - z) = 0) : False := by 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 lemma imo_1965_p2_43 (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) -- (hx0 : x ≠ 0) (hxp : 0 < x) (hy0 : y ≠ 0) (hyp : y ≤ 0) : False := by 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 lemma imo_1965_p2_44 (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) -- (hx0 : x ≠ 0) (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : y ≤ 0) (hyn : y < 0) : -- (g1 : 0 < a 0 * x) -- (g2 : 0 < a 1 * y) -- (g3 : a 2 * z < 0) : 0 < z := by 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 exact pos_of_mul_neg_right g3 (le_of_lt h₁.2) lemma imo_1965_p2_45 (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) -- (hx0 : x ≠ 0) (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : y ≤ 0) (hyn : y < 0) -- (g1 : 0 < a 0 * x) -- (g2 : 0 < a 1 * y) -- (g3 : a 2 * z < 0) (hzp : 0 < z) : False := by 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 lemma imo_1965_p2_46 (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) -- (hx0 : x ≠ 0) -- (hxp : 0 < x) -- (hy0 : y ≠ 0) -- (hyp : y ≤ 0) -- (hyn : y < 0) -- (g1 : 0 < a 0 * x) -- (g2 : 0 < a 1 * y) -- (g3 : a 2 * z < 0) (hzp : 0 < z) (g4 : a 3 * x < 0) (g5 : a 4 * y < 0) : False := by have g6: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzp linarith lemma imo_1965_p2_47 (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) (hx0 : x ≠ 0) (hxp : x ≤ 0) : False := by 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 lemma imo_1965_p2_48 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) (hxn : x < 0) (hyp : 0 ≤ y) : False := by 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 lemma imo_1965_p2_49 -- (x z : ℝ) (y : 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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : 0 ≤ y) : -- (g1 : a 0 * x < 0) : a 1 * y ≤ 0 := by refine mul_nonpos_iff.mpr ?_ right constructor . exact le_of_lt h₁.1 . exact hyp lemma imo_1965_p2_50 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) (hxn : x < 0) (hyp : 0 ≤ y) : -- g1 : a 0 * x < 0 -- g2 : a 1 * y ≤ 0 -- g3 : 0 < z * a 2 z < 0 := by 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 exact neg_of_mul_pos_left g3 (le_of_lt h₁.2) lemma imo_1965_p2_51 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : 0 ≤ y) (g1 : a 0 * x < 0) (g2 : a 1 * y ≤ 0) : -- g3 : 0 < z * a 2 z < 0 := by have g3: 0 < z * a 2 := by linarith exact neg_of_mul_pos_left g3 (le_of_lt h₁.2) lemma imo_1965_p2_52 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) (hxn : x < 0) (hyp : 0 ≤ y) -- (g1 : a 0 * x < 0) -- (g2 : a 1 * y ≤ 0) -- (g3 : 0 < z * a 2) (hzn : z < 0) : False := by 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 lemma imo_1965_p2_53 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : 0 ≤ y) -- (g1 : a 0 * x < 0) -- (g2 : a 1 * y ≤ 0) -- (g3 : 0 < z * a 2) (hzn : z < 0) (g4 : 0 < a 3 * x) : False := by 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 lemma imo_1965_p2_54 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) (hxn : x < 0) (hyp : y < 0) : False := by 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 lemma imo_1965_p2_55 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) (hxn : x < 0) (hyp : y < 0) : z < 0 := by 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 exact neg_of_mul_neg_left g3 (le_of_lt h₀.2.2) lemma imo_1965_p2_56 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : y < 0) (g1 : 0 < a 6 * x) (g2 : 0 < a 7 * y) : z < 0 := by have g3: z * a 8 < 0 := by linarith exact neg_of_mul_neg_left g3 (le_of_lt h₀.2.2) lemma imo_1965_p2_57 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) : False := by 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 lemma imo_1965_p2_58 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) (rxy : x ≤ y) : False := by 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 lemma imo_1965_p2_59 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) -- (hzp : z < 0) (rxy : x ≤ y) (ryz : y ≤ z) : False := by 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 lemma imo_1965_p2_60 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g11 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) -- (hzp : z < 0) (rxy : x ≤ y) (ryz : y ≤ z) (g1 : (a 0 + a 1 + a 2) * y + a 0 * (x - y) + a 2 * (z - y) = 0) : False := by 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 lemma imo_1965_p2_61 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) (ryz : z < y) : False := by 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 lemma imo_1965_p2_62 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) -- (rxy : x ≤ y) (ryz : z < y) (rxz : x ≤ z) : False := by 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 lemma imo_1965_p2_63 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : y < 0) -- (g11 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) -- (rxy : x ≤ y) (ryz : z < y) (rxz : x ≤ z) (g1 : (a 0 + a 1 + a 2) * z + a 0 * (x - z) + a 1 * (y - z) = 0) : False := by 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 lemma imo_1965_p2_64 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) (rxy : x ≤ y) -- (ryz : z < y) (rxz : z < x) : False := by 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 lemma imo_1965_p2_65 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : y < 0) -- (g11 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) -- (rxy : x ≤ y) (ryz : z < y) (rxz : z < x) (g1 : (a 6 + a 7 + a 8) * z + a 6 * (x - z) + a 7 * (y - z) = 0) : False := by 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 lemma imo_1965_p2_66 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) (rxy : y < x) : False := by 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 lemma imo_1965_p2_67 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) -- (hzp : z < 0) (rxy : y < x) (rzy : z ≤ y) : False := by 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 lemma imo_1965_p2_68 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g11 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) -- (hzp : z < 0) (rxy : y < x) (rzy : z ≤ y) (g1 : (a 6 + a 7 + a 8) * y + a 6 * (x - y) + a 8 * (z - y) = 0) : False := by 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 lemma imo_1965_p2_69 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) (rxy : y < x) (rzy : y < z) : False := by 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 lemma imo_1965_p2_70 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) -- (rxy : y < x) (rzy : y < z) (rzx : z ≤ x) : False := by 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 lemma imo_1965_p2_71 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) -- (hyp : y < 0) -- (g11 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) (hzp : z < 0) -- (rxy : y < x) (rzy : y < z) (rzx : z ≤ x) (g1 : (a 3 + a 4 + a 5) * z + a 3 * (x - z) + a 4 * (y - z) = 0) : False := by 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 lemma imo_1965_p2_72 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g1 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) -- (hzp : z < 0) (rxy : y < x) (rzy : y < z) : -- (rzx : x < z) : False := by 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 lemma imo_1965_p2_73 (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) -- (hx0 : x ≠ 0) -- (hxp : x ≤ 0) -- (hxn : x < 0) (hyp : y < 0) -- (g11 : 0 < a 6 * x) -- (g2 : 0 < a 7 * y) -- (g3 : z * a 8 < 0) -- (hzp : z < 0) (rxy : y < x) (rzy : y < z) -- (rzx : x < z) (g1 : (a 3 + a 4 + a 5) * y + a 3 * (x - y) + a 5 * (z - y) = 0) : False := by 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