openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingDerivativeWeight.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingWeightAddition | |
| namespace OAI | |
| /-! # One physical derivative costs one high-frequency Sobolev order uniformly in radius -/ | |
| namespace DefocusingNLS | |
| private theorem derivative_density_bound (a k L x : ℝ) (ha : 0 < a) (ha1 : a < 1) | |
| (hk : 8 < k) (hL : 1 ≤ L) (hx : 0 ≤ x) : | |
| ((L ^ (-2 : ℝ) + x ^ 2) ^ (6 - a) + x ^ (2 * k)) * x ^ 2 ≤ | |
| (2 ^ (6 - a) + 1) ^ 2 * | |
| ((L ^ (-2 : ℝ) + x ^ 2) ^ (6 - a) + x ^ (2 * (k + 1))) := by | |
| by_cases hz : x = 0 | |
| · subst x | |
| simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, add_zero, mul_zero] | |
| positivity | |
| have hxp : 0 < x := lt_of_le_of_ne hx (Ne.symm hz) | |
| have hp (s : ℝ) : x ^ (2 * s) * x ^ 2 = x ^ (2 * (s + 1)) := by | |
| rw [← Real.rpow_natCast, ← Real.rpow_add hxp] | |
| congr 1 | |
| ring | |
| let A := (L ^ (-2 : ℝ) + x ^ 2) ^ (6 - a) | |
| let H := x ^ (2 * (k + 1)) | |
| let C : ℝ := 2 ^ (6 - a) + 1 | |
| have hA : 0 ≤ A := Real.rpow_nonneg (by positivity) _ | |
| have hH : 0 ≤ H := Real.rpow_nonneg hx _ | |
| have hC : 1 ≤ C := le_add_of_nonneg_left (by positivity) | |
| have hlow : A * x ^ 2 ≤ C * (A + H) - H := by | |
| by_cases hx1 : x ≤ 1 | |
| · have hx2 : x ^ 2 ≤ 1 := by nlinarith | |
| have hl := mul_le_mul_of_nonneg_left hx2 hA | |
| have he : 0 ≤ (C - 1) * (A + H) := mul_nonneg (by linarith) (add_nonneg hA hH) | |
| nlinarith | |
| · have hx1' : 1 ≤ x := le_of_lt (lt_of_not_ge hx1) | |
| have hLpow : L ^ (-2 : ℝ) ≤ 1 := | |
| Real.rpow_le_one_of_one_le_of_nonpos hL (by norm_num) | |
| have hbase : L ^ (-2 : ℝ) + x ^ 2 ≤ 2 * x ^ 2 := by nlinarith | |
| have hAp : A ≤ 2 ^ (6 - a) * x ^ (2 * (6 - a)) := by | |
| calc | |
| A ≤ (2 * x ^ 2) ^ (6 - a) := Real.rpow_le_rpow (by positivity) hbase (by linarith) | |
| _ = _ := by | |
| rw [Real.mul_rpow (by norm_num) (sq_nonneg x), | |
| ← Real.rpow_natCast, ← Real.rpow_mul hx] | |
| norm_num | |
| have he : x ^ (2 * (6 - a)) * x ^ 2 ≤ H := by | |
| rw [hp] | |
| exact Real.rpow_le_rpow_of_exponent_le hx1' (by linarith) | |
| have hb : A * x ^ 2 ≤ 2 ^ (6 - a) * H := by | |
| calc | |
| _ ≤ (2 ^ (6 - a) * x ^ (2 * (6 - a))) * x ^ 2 := | |
| mul_le_mul_of_nonneg_right hAp (sq_nonneg x) | |
| _ = 2 ^ (6 - a) * (x ^ (2 * (6 - a)) * x ^ 2) := by ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_left he (by positivity) | |
| have hh : 0 ≤ C * A := mul_nonneg (by linarith) hA | |
| dsimp [C] at * | |
| nlinarith | |
| have hCsq : C ≤ C ^ 2 := by nlinarith | |
| calc | |
| _ = A * x ^ 2 + H := by rw [add_mul, hp] | |
| _ ≤ C * (A + H) := by linarith | |
| _ ≤ C ^ 2 * (A + H) := mul_le_mul_of_nonneg_right hCsq (add_nonneg hA hH) | |
| _ = _ := rfl | |
| theorem expandingSobolevWeight_derivative_le (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) | |
| (hk : 8 < k) (hL : 1 ≤ L) (n : frequencyLattice) : | |
| expandingSobolevWeight a k L n * (‖n‖ / L) ≤ | |
| (2 ^ (6 - a) + 1) * expandingSobolevWeight a (k + 1) L n := by | |
| have hLp : 0 < L := by linarith | |
| have hb := mul_le_mul_of_nonneg_left | |
| (derivative_density_bound a k L (‖n‖ / L) ha ha1 hk hL (by positivity)) | |
| (show 0 ≤ (2 * Real.pi * L) ^ (12 : ℕ) by positivity) | |
| have hw (j : ℝ) : expandingSobolevWeight a j L n ^ 2 = | |
| (2 * Real.pi * L) ^ (12 : ℕ) * | |
| ((L ^ (-2 : ℝ) + (‖n‖ / L) ^ 2) ^ (6 - a) + (‖n‖ / L) ^ (2 * j)) := by | |
| rw [expandingSobolevWeight_sq a j L hL, expandingSobolevWeightSq_original a j L hL] | |
| have hs : (expandingSobolevWeight a k L n * (‖n‖ / L)) ^ 2 ≤ | |
| ((2 ^ (6 - a) + 1) * expandingSobolevWeight a (k + 1) L n) ^ 2 := by | |
| rw [mul_pow, mul_pow, hw k, hw (k + 1)] | |
| nlinarith only [hb] | |
| exact (sq_le_sq₀ (mul_nonneg (expandingSobolevWeight_pos a k L hL n).le (by positivity)) (by | |
| exact mul_nonneg (by positivity) (expandingSobolevWeight_pos a (k + 1) L hL n).le)).mp hs | |
| end DefocusingNLS | |
| end OAI | |