import Mathlib
set_option linter.unusedVariables.analyzeTactics true


theorem imo_1965_p2
  (x y z : ℝ)
  (a : ℕ → ℝ)
  (h₀ : 0 < a 0 ∧ 0 < a 4 ∧ 0 < a 8)
  (h₁ : a 1 < 0 ∧ a 2 < 0)
  (h₂ : a 3 < 0 ∧ a 5 < 0)
  (h₃ : a 6 < 0 ∧ a 7 < 0)
  (h₄ : 0 < a 0 + a 1 + a 2)
  (h₅ : 0 < a 3 + a 4 + a 5)
  (h₆ : 0 < a 6 + a 7 + a 8)
  (h₇ : a 0 * x + a 1 * y + a 2 * z = 0)
  (h₈ : a 3 * x + a 4 * y + a 5 * z = 0)
  (h₉ : a 6 * x + a 7 * y + a 8 * z = 0) :
  x = 0 ∧ y = 0 ∧ z = 0 := by
  by_cases hx0: x = 0
  . rw [hx0] at h₇
    constructor
    . exact hx0
    . rw [hx0] at h₈ h₉
      simp at h₇ h₈ h₉
      by_cases hy0: y = 0
      . constructor
        . exact hy0
        . rw [hy0] at h₇
          simp at h₇
          . cases' h₇ with h₇₀ h₇₁
            . exfalso
              linarith
            . exact h₇₁
      . by_cases hyn: y < 0
        . have g1: 0 < a 1 * y := by exact mul_pos_of_neg_of_neg h₁.1 hyn
          have g2: a 1 * y = -a 2 * z := by linarith
          rw [g2] at g1
          have g3: a 2 *z < 0 := by linarith
          have hzp: 0 < z := by exact pos_of_mul_neg_right g3 (le_of_lt h₁.2)
          exfalso
          have g4: a 4 * y < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 hyn
          have g5: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzp
          linarith
        . push_neg at hy0 hyn
          have hyp: 0 < y := by exact lt_of_le_of_ne hyn hy0.symm
          exfalso
          have g1: a 1 * y < 0 := by exact mul_neg_of_neg_of_pos h₁.1 hyp
          have g2: 0 < z * a 2 := by linarith
          have hzp: z < 0 := by exact neg_of_mul_pos_left g2 (le_of_lt h₁.2)
          have g3: 0 < a 4 * y := by exact mul_pos h₀.2.1 hyp
          have g4: 0 < a 5 * z := by exact mul_pos_of_neg_of_neg h₂.2 hzp
          linarith
  . exfalso
    push_neg at hx0
    by_cases hxp: 0 < x
    . by_cases hy0: y = 0
      . rw [hy0] at h₇ h₈ h₉
        simp at h₇ h₈ h₉
        have g1: 0 < a 0 * x := by exact mul_pos h₀.1 hxp
        have g2: a 2 * z < 0 := by linarith
        have hzn: 0 < z := by exact pos_of_mul_neg_right g2 (le_of_lt h₁.2)
        have g3: a 3 * x < 0 := by exact mul_neg_of_neg_of_pos h₂.1 hxp
        have g4: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzn
        linarith
      . push_neg at hy0
        by_cases hyp: 0 < y
        . have g1: a 6 * x < 0 := by exact mul_neg_of_neg_of_pos h₃.1 hxp
          have g2: a 7 * y < 0 := by exact mul_neg_of_neg_of_pos h₃.2 hyp
          have g3: 0 < z * a 8 := by linarith
          have hzp: 0 < z := by exact pos_of_mul_pos_left g3 (le_of_lt h₀.2.2)
          ------ here we consider all the possible relationships between x, y, z
          by_cases rxy: x ≤ y
          . by_cases ryz: y ≤ z
              -- x <= y <= z
            . have g2: 0 < (a 6 + a 7 + a 8) * y := by exact mul_pos h₆ hyp
              have g3: 0 ≤ a 6 * (x-y) := by
                exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₃.1) (by linarith)-- exact mul_nonneg (le_of_lt h₃.1) (by linarith),},
              have g4: 0 ≤ a 8 * (z-y) := by exact mul_nonneg (le_of_lt h₀.2.2) (by linarith)
              linarith
            push_neg at ryz
            by_cases rxz: x ≤ z
              -- x <= z < y
            . have g2: 0 < (a 3 + a 4 + a 5) * y := by exact mul_pos h₅ hyp
              have g3: 0 ≤ a 3 * (x-y) := by
                exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₂.1) (by linarith)
              have g4: 0 < a 5 * (z-y) := by
                exact mul_pos_of_neg_of_neg h₂.2 (by linarith)
              linarith
            push_neg at rxz -- z < x <= y
            have g2: 0 < (a 3 + a 4 + a 5) * y := by exact mul_pos h₅ hyp
            have g3: 0 ≤ a 3 * (x-y) := by
              exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₂.1) (by linarith)
            have g4: 0 < a 5 * (z-y) := by
              exact mul_pos_of_neg_of_neg h₂.2 (by linarith)
            linarith
          push_neg at rxy
          by_cases rzy: z ≤ y
            -- z <= y < x
          . have g2: 0 < (a 0 + a 1 + a 2) * y := by exact mul_pos h₄ hyp
            have g3: 0 < a 0 * (x-y) := by exact mul_pos h₀.1 (by linarith)
            have g4: 0 ≤ a 2 * (z-y) := by
              exact mul_nonneg_of_nonpos_of_nonpos (le_of_lt h₁.2) (by linarith)
            linarith
          . push_neg at rzy
            by_cases rzx: z ≤ x
              -- y < z <= x
            . have g2: 0 < (a 0 + a 1 + a 2) * z := by exact mul_pos h₄ hzp
              have g3: 0 ≤ a 0 * (x-z) := by exact mul_nonneg (le_of_lt h₀.1) (by linarith)
              have g4: 0 < a 1 * (y-z) := by exact mul_pos_of_neg_of_neg h₁.1 (by linarith)
              linarith
            . push_neg at rzx
                -- y < x < z
              have g2: 0 < (a 6 + a 7 + a 8) * z := by exact mul_pos h₆ hzp
              have g3: 0 < a 6 * (x-z) := by exact mul_pos_of_neg_of_neg h₃.1 (by linarith)
              have g4: 0 < a 7 * (y-z) := by exact mul_pos_of_neg_of_neg h₃.2 (by linarith)
              linarith
        -------- new world where y < 0 and 0 < x
        . push_neg at hyp
          have hyn: y < 0 :=  by exact lt_of_le_of_ne hyp hy0
          -- show from a 0 that 0 < z
          have g1: 0 < a 0 * x := by exact mul_pos h₀.1 hxp
          have g2: 0 < a 1 * y := by exact mul_pos_of_neg_of_neg h₁.1 hyn
          have g3: a 2 * z < 0 := by linarith
          have hzp: 0 < z := by exact pos_of_mul_neg_right g3 (le_of_lt h₁.2)
          -- then show from a 3 that's not possible
          have g4: a 3 * x < 0 := by exact mul_neg_of_neg_of_pos h₂.1 hxp
          have g5: a 4 * y < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 hyn
          have g6: a 5 * z < 0 := by exact mul_neg_of_neg_of_pos h₂.2 hzp
          linarith
    . push_neg at hxp
      have hxn: x < 0 := by exact lt_of_le_of_ne hxp hx0
      by_cases hyp: 0 ≤ y
      . have g1: a 0 * x < 0 := by exact mul_neg_of_pos_of_neg h₀.1 hxn
        have g2: a 1 * y ≤ 0 := by
          refine mul_nonpos_iff.mpr ?_
          right
          constructor
          . exact le_of_lt h₁.1
          . exact hyp
        have g3: 0 < z * a 2 := by linarith
        have hzn: z < 0 := by exact neg_of_mul_pos_left g3 (le_of_lt h₁.2)
        -- demonstrate the contradiction
        have g4: 0 < a 3 * x := by exact mul_pos_of_neg_of_neg h₂.1 hxn
        have g5: 0 ≤ a 4 * y := by exact mul_nonneg (le_of_lt h₀.2.1) hyp
        have g6: 0 < a 5 * z := by exact mul_pos_of_neg_of_neg h₂.2 hzn
        linarith
      . push_neg at hyp
        -- have hyn: y < 0, {exact lt_of_le_of_ne hyp hy0,},
        have g1: 0 < a 6 * x := by exact mul_pos_of_neg_of_neg h₃.1 hxn
        have g2: 0 < a 7 * y := by exact mul_pos_of_neg_of_neg h₃.2 hyp
        have g3: z * a 8 < 0 := by linarith
        have hzp: z < 0 := by exact neg_of_mul_neg_left g3 (le_of_lt h₀.2.2)
          -- we have x,y,z < 0 -- we will examine all the orders they can have
        by_cases rxy: x ≤ y
        . by_cases ryz: y ≤ z
            -- x <= y <= z
          . have g2: (a 0 + a 1 + a 2) * y < 0 := by exact mul_neg_of_pos_of_neg h₄ hyp
            have g3: a 0 * (x-y) ≤ 0 := by
              exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.1) (by linarith)
            have g4: a 2 * (z-y) ≤ 0 := by
              exact mul_nonpos_of_nonpos_of_nonneg (le_of_lt h₁.2) (by linarith)
            linarith
          . push_neg at ryz
            by_cases rxz: x ≤ z
              -- x <= z < y
            . have g2: (a 0 + a 1 + a 2) * z < 0 := by exact mul_neg_of_pos_of_neg h₄ hzp
              have g3: a 0 * (x-z) ≤ 0 := by
                exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.1) (by linarith)
              have g4: a 1 * (y-z) < 0 := by
                exact mul_neg_of_neg_of_pos h₁.1 (by linarith)
              linarith
            . push_neg at rxz -- z < x <= y
              have g2: (a 6 + a 7 + a 8) * z < 0 := by exact mul_neg_of_pos_of_neg h₆ hzp
              have g3: a 6 * (x-z) < 0 := by exact mul_neg_of_neg_of_pos h₃.1 (by linarith)
              have g4: a 7 * (y-z) < 0 := by exact mul_neg_of_neg_of_pos h₃.2 (by linarith)
              linarith
        . push_neg at rxy
          by_cases rzy: z ≤ y
            -- z <= y < x
          . have g2: (a 6 + a 7 + a 8) * y < 0 := by exact mul_neg_of_pos_of_neg h₆ hyp
            have g3: a 6 * (x-y) < 0 := by exact mul_neg_of_neg_of_pos h₃.1 (by linarith)
            have g4: a 8 * (z-y) ≤ 0 := by
              exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt h₀.2.2) (by linarith)
            linarith
          . push_neg at rzy
            by_cases rzx: z ≤ x
              -- y < z <= x
            . have g2: (a 3 + a 4 + a 5) * z < 0 := by exact mul_neg_of_pos_of_neg h₅ hzp
              have g3: a 3 * (x-z) ≤ 0 := by
                exact mul_nonpos_of_nonpos_of_nonneg (le_of_lt h₂.1) (by linarith)
              have g4: a 4 * (y-z) < 0 := by exact mul_neg_of_pos_of_neg h₀.2.1 (by linarith)
              linarith
            . push_neg at rzx
              -- y < x < z
              have g2: (a 3 + a 4 + a 5) * y < 0 := by exact mul_neg_of_pos_of_neg h₅ hyp
              have g3: a 3 * (x-y) < 0 := by exact mul_neg_of_neg_of_pos h₂.1 (by linarith)
              have g4: a 5 * (z-y) < 0 := by exact mul_neg_of_neg_of_pos h₂.2 (by linarith)
              linarith