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import Mathlib
open Real
theorem imo_1960_p2
(x : ℝ)
(h₀ : 0 ≤ 1 + 2 * x)
(h₁ : (1 - Real.sqrt (1 + 2 * x))^2 ≠ 0)
(h₂ : (4 * x^2) / (1 - Real.sqrt (1 + 2*x))^2 < 2*x + 9) :
-(1 / 2) ≤ x ∧ x < 45 / 8 := by
apply And.intro
. linarith
. have h₃: 4 * x ^ 2 < (2 * x + 9) * (1 - sqrt (1 + 2 * x) ) ^ 2 := by
refine' (div_lt_iff₀ _).mp h₂
refine Ne.lt_of_le (id (Ne.symm h₁)) ?_
exact sq_nonneg (1 - sqrt (1 + 2 * x))
have h₄: (1 - sqrt (1 + 2 * x)) ^ 2 = (2 + 2 * x) - 2 * sqrt (1 + 2 * x) := by
ring_nf at *
rw [Real.sq_sqrt h₀]
ring_nf
have h₅: (2 * x + 9) ^ 2 * (sqrt (1 + 2 * x)) ^ 2 < (11 * x + 9) ^ 2 := by
rw [← mul_pow]
refine' pow_lt_pow_left₀ _ _ (by norm_num)
rw [h₄] at h₃
simp_all only [ne_eq, zero_lt_two]
. linarith
. refine' mul_nonneg _ _
linarith
exact sqrt_nonneg (1 + 2 * x)
have h₆: 8 * x^3 < 45 * x^2 := by
rw [Real.sq_sqrt h₀] at h₅
ring_nf at h₅
linarith
have h₇₁: 0 ≤ x^2 := by exact sq_nonneg x
have h₇: 8 * x < 45 := by
refine' lt_of_mul_lt_mul_right ?_ h₇₁
ring_nf at *
exact h₆
linarith
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