| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem367 | |
| def binaryFusion (u v : ℝ) : ℝ := (u + v) / (1 + u * v) | |
| def binaryFusionLoss (u v : ℝ) : ℝ := | |
| u ^ 2 * v ^ 2 * (2 - u ^ 2 - v ^ 2) / (1 - u ^ 2 * v ^ 2) | |
| theorem binaryFusion_abs_mul_lt_one {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : |u * v| < 1 := by | |
| rw [abs_mul] | |
| calc | |
| |u| * |v| ≤ |u| * 1 := mul_le_mul_of_nonneg_left hv.le (abs_nonneg u) | |
| _ < 1 := by simpa using hu | |
| theorem binaryFusion_denominator_pos {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : 0 < 1 + u * v := by | |
| have h := (abs_lt.mp (binaryFusion_abs_mul_lt_one hu hv)).1 | |
| linarith | |
| theorem binaryFusionLoss_denominator_pos {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : 0 < 1 - u ^ 2 * v ^ 2 := by | |
| have h := (sq_lt_one_iff_abs_lt_one (u * v)).mpr | |
| (binaryFusion_abs_mul_lt_one hu hv) | |
| nlinarith | |
| theorem one_sub_binaryFusion_sq {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : | |
| 1 - binaryFusion u v ^ 2 = | |
| (1 - u ^ 2) * (1 - v ^ 2) / (1 + u * v) ^ 2 := by | |
| have hd := ne_of_gt (binaryFusion_denominator_pos hu hv) | |
| unfold binaryFusion | |
| field_simp | |
| ring | |
| theorem abs_binaryFusion_lt_one {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : |binaryFusion u v| < 1 := by | |
| apply (sq_lt_one_iff_abs_lt_one _).mp | |
| have hu' : 0 < 1 - u ^ 2 := by | |
| have := (sq_lt_one_iff_abs_lt_one u).mpr hu | |
| linarith | |
| have hv' : 0 < 1 - v ^ 2 := by | |
| have := (sq_lt_one_iff_abs_lt_one v).mpr hv | |
| linarith | |
| have hd := binaryFusion_denominator_pos hu hv | |
| have hpos : 0 < 1 - binaryFusion u v ^ 2 := by | |
| rw [one_sub_binaryFusion_sq hu hv] | |
| positivity | |
| linarith | |
| theorem binaryFusionLoss_nonneg {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : 0 ≤ binaryFusionLoss u v := by | |
| have hu' := (sq_lt_one_iff_abs_lt_one u).mpr hu | |
| have hv' := (sq_lt_one_iff_abs_lt_one v).mpr hv | |
| have hf : 0 ≤ 2 - u ^ 2 - v ^ 2 := by linarith | |
| have hd := binaryFusionLoss_denominator_pos hu hv | |
| unfold binaryFusionLoss | |
| positivity | |
| theorem binaryFusion_symmetric_square {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : | |
| ((u + v) ^ 2 / (1 + u * v) + (u - v) ^ 2 / (1 - u * v)) / 2 = | |
| u ^ 2 + v ^ 2 - binaryFusionLoss u v := by | |
| have hp := ne_of_gt (binaryFusion_denominator_pos hu hv) | |
| have hm : 1 - u * v ≠ 0 := by | |
| have := (abs_lt.mp (binaryFusion_abs_mul_lt_one hu hv)).2 | |
| linarith | |
| have hq := ne_of_gt (binaryFusionLoss_denominator_pos hu hv) | |
| unfold binaryFusionLoss | |
| field_simp | |
| ring | |
| theorem binaryFusion_symmetric_square_le {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : | |
| ((u + v) ^ 2 / (1 + u * v) + (u - v) ^ 2 / (1 - u * v)) / 2 ≤ | |
| u ^ 2 + v ^ 2 := by | |
| rw [binaryFusion_symmetric_square hu hv] | |
| exact sub_le_self _ (binaryFusionLoss_nonneg hu hv) | |
| theorem binaryFusionLoss_lower_bound {u v theta : ℝ} | |
| (hu : |u| ≤ theta) (hv : |v| ≤ theta) (htheta : theta < 1) : | |
| 2 * (1 - theta ^ 2) * u ^ 2 * v ^ 2 ≤ binaryFusionLoss u v := by | |
| have ht0 : 0 ≤ theta := le_trans (abs_nonneg u) hu | |
| have hu' : u ^ 2 ≤ theta ^ 2 := by | |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg u) ht0).mpr hu | |
| have hv' : v ^ 2 ≤ theta ^ 2 := by | |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg v) ht0).mpr hv | |
| have ht2 : theta ^ 2 < 1 := (sq_lt_one_iff₀ ht0).mpr htheta | |
| have hp : 0 ≤ u ^ 2 * v ^ 2 := mul_nonneg (sq_nonneg u) (sq_nonneg v) | |
| have hf : 0 ≤ 2 - u ^ 2 - v ^ 2 := by linarith | |
| have hd := binaryFusionLoss_denominator_pos (lt_of_le_of_lt hu htheta) | |
| (lt_of_le_of_lt hv htheta) | |
| have hfactor : 2 * (1 - theta ^ 2) ≤ 2 - u ^ 2 - v ^ 2 := by linarith | |
| have hnum : 0 ≤ u ^ 2 * v ^ 2 * (2 - u ^ 2 - v ^ 2) := mul_nonneg hp hf | |
| unfold binaryFusionLoss | |
| calc | |
| 2 * (1 - theta ^ 2) * u ^ 2 * v ^ 2 = | |
| (u ^ 2 * v ^ 2) * (2 * (1 - theta ^ 2)) := by ring | |
| _ ≤ u ^ 2 * v ^ 2 * (2 - u ^ 2 - v ^ 2) := | |
| mul_le_mul_of_nonneg_left hfactor hp | |
| _ ≤ u ^ 2 * v ^ 2 * (2 - u ^ 2 - v ^ 2) / (1 - u ^ 2 * v ^ 2) := by | |
| apply (le_div_iff₀ hd).mpr | |
| exact mul_le_of_le_one_right hnum (by linarith) | |
| theorem binaryFusion_integrand_le_two {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : (u + v) ^ 2 / (1 + u * v) ≤ 2 := by | |
| apply (div_le_iff₀ (binaryFusion_denominator_pos hu hv)).mpr | |
| have hu' := (sq_lt_one_iff_abs_lt_one u).mpr hu | |
| have hv' := (sq_lt_one_iff_abs_lt_one v).mpr hv | |
| nlinarith | |
| theorem binaryFusionLoss_le_sum_sq {u v : ℝ} | |
| (hu : |u| < 1) (hv : |v| < 1) : binaryFusionLoss u v ≤ u ^ 2 + v ^ 2 := by | |
| have hp : 0 ≤ (u + v) ^ 2 / (1 + u * v) := | |
| div_nonneg (sq_nonneg _) (binaryFusion_denominator_pos hu hv).le | |
| have hmden : 0 < 1 - u * v := by | |
| have := (abs_lt.mp (binaryFusion_abs_mul_lt_one hu hv)).2 | |
| linarith | |
| have hm : 0 ≤ (u - v) ^ 2 / (1 - u * v) := | |
| div_nonneg (sq_nonneg _) hmden.le | |
| have hid := binaryFusion_symmetric_square hu hv | |
| linarith | |
| end Problem367 | |
| end | |
| end OAI | |