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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
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