import Mathlib
set_option linter.unusedVariables.analyzeTactics true


open Real



theorem imo_1962_p2_1
  (x : ℝ)
  -- (h₀ : 0 ≤ 3 - x)
  -- (h₁ : 0 ≤ x + 1)
  (h₂ : 1 / 2 < Real.sqrt (x - 3) - Real.sqrt (x + 1)) :
  -1 ≤ x := by
  refine neg_le_iff_add_nonneg.mpr ?_
  contrapose! h₂
  have h₃: x - 3 < 0 := by linarith [h₂]
  have h₄: Real.sqrt (x + 1) = 0 := by
    refine Real.sqrt_eq_zero'.mpr ?_
    exact le_of_lt h₂
  have h₅: Real.sqrt (x -3) = 0 := by
    refine Real.sqrt_eq_zero'.mpr ?_
    exact le_of_lt h₃
  rw [h₄, h₅, sub_zero]
  refine div_nonneg ?_ ?_
  all_goals try linarith


theorem imo_1962_p2_2
  (x : ℝ)
  (h₀ : 0 ≤ 3 - x)
  (h₁ : 0 ≤ x + 1)
  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
  (2 * √(3 - x) * √(x + 1)) ^ 2 < (4 - 1 / 4) ^ 2 := by
  refine' pow_lt_pow_left₀ _ _ (by norm_num)
  . refine lt_tsub_iff_left.mpr ?_
    refine lt_tsub_iff_right.mp ?_
    suffices g₀: 4 - 2 * sqrt (3 - x) * sqrt (x + 1) = (sqrt (3 - x) - sqrt (x + 1)) ^ 2
    . rw [g₀]
      have g₁:  (1:ℝ) / 4 = (1/2)^2 := by norm_num
      rw [g₁]
      exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num)
    rw [sub_sq]
    rw [sq_sqrt h₀, sq_sqrt h₁]
    ring_nf
  . refine' mul_nonneg _ _
    . refine mul_nonneg (by linarith) ?_
      exact sqrt_nonneg (3 - x)
    . exact sqrt_nonneg (x + 1)


theorem imo_1962_p2_3
  (x : ℝ)
  (h₀ : 0 ≤ 3 - x)
  (h₁ : 0 ≤ x + 1)
  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
  2 * √(3 - x) * √(x + 1) < 4 - 1 / 4 := by
  refine lt_tsub_iff_left.mpr ?refine'_1.a
  refine lt_tsub_iff_right.mp ?refine'_1.a.a
  suffices g₀: 4 - 2 * sqrt (3 - x) * sqrt (x + 1) = (sqrt (3 - x) - sqrt (x + 1)) ^ 2
  . rw [g₀]
    have g₁:  (1:ℝ) / 4 = (1/2)^2 := by norm_num
    rw [g₁]
    exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num)
  rw [sub_sq]
  rw [sq_sqrt h₀, sq_sqrt h₁]
  ring_nf


theorem imo_1962_p2_4
  (x : ℝ) :
  -- (h₀ : 0 ≤ 3 - x)
  -- (h₁ : 0 ≤ x + 1)
  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
  0 ≤ 2 * √(3 - x) * √(x + 1) := by
  refine' mul_nonneg ?_ ?_
  . refine mul_nonneg (by linarith) ?_
    exact sqrt_nonneg (3 - x)
  . exact sqrt_nonneg (x + 1)



theorem imo_1962_p2_5
  (x : ℝ)
  (h₀ : 0 ≤ 3 - x)
  (h₁ : 0 ≤ x + 1) :
  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
  4 - 2 * √(3 - x) * √(x + 1) = (√(3 - x) - √(x + 1)) ^ 2 := by
  rw [sub_sq]
  rw [sq_sqrt h₀, sq_sqrt h₁]
  ring_nf


theorem imo_1962_p2_6
  (x : ℝ)
  -- (h₀ : 0 ≤ 3 - x)
  -- (h₁ : 0 ≤ x + 1)
  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
  (h₃: 4 - 2 * √(3 - x) * √(x + 1) = (√(3 - x) - √(x + 1)) ^ 2) :
  1 / 4 < 4 - 2 * √(3 - x) * √(x + 1) := by
  rw [h₃]
  have g₁:  (1:ℝ) / 4 = (1/2) ^ 2 := by norm_num
  rw [g₁]
  exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num)


theorem imo_1962_p2_7
  (x : ℝ)
  (h₀ : 0 ≤ 3 - x)
  (h₁ : 0 ≤ x + 1)
  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
  (h₃: (2 *sqrt (3 - x) * sqrt (x + 1)) ^ 2 < (4 - 1 / 4) ^ 2) :
  4 * (x + 1) * (3 - x) < 225 / 16 := by
  norm_num at h₃
  suffices g₀: 4 * (x + 1) * (3 - x) = (2 * sqrt (3 - x) * sqrt (x + 1)) ^ 2
  . exact Eq.trans_lt g₀ h₃
  . rw [mul_pow, mul_pow, sq_sqrt h₀, sq_sqrt h₁]
    norm_num
    exact mul_right_comm 4 (x + 1) (3 - x)


theorem imo_1962_p2_8
  (x : ℝ)
  (h₀ : 0 ≤ 3 - x)
  (h₁ : 0 ≤ x + 1)
  (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
  x < 1 := by
  suffices g₀: x + 1 < 3 - x
  . linarith
  . rw [← sq_sqrt h₀, ← sq_sqrt h₁]
    refine' pow_lt_pow_left₀ ?_ ?_ (by norm_num)
    . linarith
    . exact sqrt_nonneg (x + 1)


theorem imo_1962_p2_9
  (x : ℝ)
  -- (h₀ : 0 ≤ 3 - x)
  -- (h₁ : 0 ≤ x + 1)
  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
  (h₄: 4 * (x + 1) * (3 - x) < 225 / 16) :
  x < 1 - sqrt 31 / 8 ∨ 1 + sqrt 31 / 8 < x := by
  ring_nf at h₄
  have g₀: 0 < x * x + -2 * x + 33 / 64 := by linarith
  let a:ℝ := sqrt 31 / 8
  have g₁: x * x + -2 * x + 33 / 64 = (x - (1 + a)) * (x - (1 - a)) := by
    simp
    ring_nf
    norm_num
    linarith
  rw [g₁] at g₀
  by_cases g₂: (x - (1 - a)) < 0
  . left
    exact sub_neg.mp g₂
  . push_neg at g₂
    right
    have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left g₀ g₂
    exact sub_pos.mp g₃


theorem imo_1962_p2_10
  (x : ℝ)
  -- (h₀ : 0 ≤ 3 - x)
  -- (h₁ : 0 ≤ x + 1)
  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
  (h₄: x < 1)
  (h₅: x < 1 - sqrt 31 / 8 ∨ 1 + sqrt 31 / 8 < x) :
  x < 1 - Real.sqrt 31 / 8 := by
  cases h₅ with
  | inl h₅ => exact h₅
  | inr h₅ => linarith


theorem imo_1962_p2_11
  (x a : ℝ)
  (ha: a = √31 / 8)
  -- (h₀ : 0 ≤ 3 - x)
  -- (h₁ : 0 ≤ x + 1)
  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
  (h₃: 0 < (x - (1 + a)) * (x - (1 - a))) :
  x < 1 - √31 / 8 ∨ 1 + √31 / 8 < x := by
  by_cases g₂: (x - (1 - a)) < 0
  . left
    rw [ha] at g₂
    exact sub_neg.mp g₂
  . push_neg at g₂
    right
    have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left h₃ g₂
    rw [ha] at g₃
    exact sub_pos.mp g₃


theorem imo_1962_p2_12
  (x a : ℝ)
  (ha: a = 0.5)
  -- (h₀ : 0 ≤ 3 - x)
  -- (h₁ : 0 ≤ x + 1)
  -- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1))
  (h₃: 0 < (x - (1 + a)) * (x - (1 - a))) :
  x < 1 - 0.5 ∨ 1 + 0.5 < x := by
  by_cases g₂: (x - (1 - a)) < 0
  . left
    rw [ha] at g₂
    exact sub_neg.mp g₂
  . push_neg at g₂
    right
    have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left h₃ g₂
    rw [ha] at g₃
    exact sub_pos.mp g₃


theorem imo_1962_p2_13
  (x a : ℝ)
  (ha: a = √31 / 8) :
  -- h₀ : 0 ≤ 3 - x
  -- h₁ : 0 ≤ x + 1
  -- h₄ : 12 + (x * 8 - x ^ 2 * 4) < 225 / 16
  -- g₀ : 0 < x * x + -2 * x + 33 / 64
  x ^ 2 - 2 * x + 33 / 64 = (x - (1 + a)) * (x - (1 - a)) := by
  rw [ha]
  ring_nf
  norm_num
  linarith

theorem imo_1962_p2_14
  (x : ℝ)
  -- (h₀ : 0 ≤ 3 - x)
  -- (h₁ : 0 ≤ x + 1)
  (h₄ : 12 + (x * 8 - x ^ 2 * 4) < 225 / 16) :
  0 < x * x + -2 * x + 33 / 64 := by
  ring_nf at h₄
  linarith