openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingEnergyInner.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingEnergyScaling | |
| import Mathlib.Analysis.InnerProductSpace.Calculus | |
| namespace OAI | |
| /-! # Polarization of the exact expanding-torus energy identities -/ | |
| namespace DefocusingNLS | |
| theorem expandingEnergy_inner (a k L : ℝ) (hL : 1 ≤ L) (f g : FourierL2) : | |
| inner ℝ f g = inner ℝ (expandingLowEnergy a k L hL f) (expandingLowEnergy a k L hL g) + | |
| inner ℝ (expandingHighEnergy a k L hL f) (expandingHighEnergy a k L hL g) := by | |
| have h := expandingEnergy_norm_sq a k L hL (f + g) | |
| simp only [map_add] at h | |
| rw [norm_add_sq_real, norm_add_sq_real, norm_add_sq_real, | |
| expandingEnergy_norm_sq a k L hL f, expandingEnergy_norm_sq a k L hL g] at h | |
| linarith | |
| theorem expandingLowEnergy_free_inner (a b k L s : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (hs : 0 ≤ s) (f g : FourierL2) : | |
| inner ℝ (expandingLowEnergy a k (expandingRadius L s) | |
| (hL.trans (expandingRadius_ge L s hL hs)) (expandingFreeStep a b k L s ha hk hL hs f)) | |
| (expandingLowEnergy a k (expandingRadius L s) | |
| (hL.trans (expandingRadius_ge L s hL hs)) (expandingFreeStep a b k L s ha hk hL hs g)) = | |
| Real.exp (-a * s) * inner ℝ (expandingLowEnergy a k L hL f) (expandingLowEnergy a k L hL g) := by | |
| have h := expandingLowEnergy_free a b k L s ha hk hL hs (f + g) | |
| simp only [map_add] at h | |
| rw [norm_add_sq_real, norm_add_sq_real, | |
| expandingLowEnergy_free a b k L s ha hk hL hs f, | |
| expandingLowEnergy_free a b k L s ha hk hL hs g] at h | |
| nlinarith [h] | |
| theorem expandingHighEnergy_free_inner (a b k L s : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (hs : 0 ≤ s) (f g : FourierL2) : | |
| inner ℝ (expandingHighEnergy a k (expandingRadius L s) | |
| (hL.trans (expandingRadius_ge L s hL hs)) (expandingFreeStep a b k L s ha hk hL hs f)) | |
| (expandingHighEnergy a k (expandingRadius L s) | |
| (hL.trans (expandingRadius_ge L s hL hs)) (expandingFreeStep a b k L s ha hk hL hs g)) = | |
| Real.exp ((6 - 2 * a - k) * s) * | |
| inner ℝ (expandingHighEnergy a k L hL f) (expandingHighEnergy a k L hL g) := by | |
| have h := expandingHighEnergy_free a b k L s ha hk hL hs (f + g) | |
| simp only [map_add] at h | |
| rw [norm_add_sq_real, norm_add_sq_real, | |
| expandingHighEnergy_free a b k L s ha hk hL hs f, | |
| expandingHighEnergy_free a b k L s ha hk hL hs g] at h | |
| nlinarith [h] | |
| end DefocusingNLS | |
| end OAI | |