diff --git "a/Lemmas/imo_1965_p2_lemmas.lean" "b/Lemmas/imo_1965_p2_lemmas.lean" new file mode 100644--- /dev/null +++ "b/Lemmas/imo_1965_p2_lemmas.lean" @@ -0,0 +1,2900 @@ +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