| 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 | |