| import Mathlib | |
| set_option linter.unusedVariables.analyzeTactics true | |
| open Real | |
| theorem imo_1968_p5_1 | |
| (a : ℝ) | |
| (f : ℝ → ℝ) | |
| (h₀ : 0 < a) | |
| (h₁ : ∀ x, f (x + a) = 1 / 2 + Real.sqrt (f x - (f x)^2)) | |
| (h₂ : ∀ x, 1 / 2 ≤ f x ∧ f x ≤ 1) : | |
| ∃ b > 0, ∀ x, f (x + b) = f x := by | |
| use (2 * a) | |
| constructor | |
| . refine mul_pos (by norm_num) h₀ | |
| . intro x | |
| have h₃: f (x + a) = 1 / 2 + Real.sqrt (f x - (f x)^2) := by | |
| exact h₁ x | |
| have h₄: f (x + 2 * a) = 1 / 2 + Real.sqrt (f (x + a) - (f (x + a)^2)) := by | |
| rw [two_mul, ← add_assoc] | |
| exact h₁ (x + a) | |
| have h₅: f (x + a) - (f (x + a) ^ 2) = (f x - 1 / 2) ^ 2 := by | |
| have h₅₁: 0 ≤ f x - (f x)^2 := by | |
| refine sub_nonneg_of_le ?_ | |
| rw [pow_two] | |
| nth_rw 3 [← mul_one (f x)] | |
| refine (mul_le_mul_left ?_).mpr ?_ | |
| . linarith [h₂ x] | |
| . exact (h₂ x).2 | |
| rw [h₃, add_sq, sub_sq, sq_sqrt h₅₁] | |
| ring_nf | |
| rw [h₅, sqrt_sq ?_] at h₄ | |
| . linarith | |
| . have h₆: 1 / 2 ≤ f x := by | |
| exact (h₂ x).1 | |
| linarith [h₆] | |