openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingCircularMoment.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingPolynomialMoment | |
| namespace OAI | |
| /-! # Quantitative moments of both circular coefficients -/ | |
| namespace DefocusingNLS | |
| attribute [local irreducible] expandingPower expandingProduct expandingFourierCoefficient | |
| expandingCircularCoefficient expandingAnticircularCoefficient fourierConjugate | |
| variable (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (N m : ℕ) | |
| theorem expandingCircular_moment (q : FourierL2) | |
| (hq : Summable (latticeMomentMass L N (expandingFourierCoefficient a k L q))) : | |
| let M := ∑' n, latticeMomentMass L N (expandingFourierCoefficient a k L q) n | |
| Summable (latticeMomentMass L N (expandingFourierCoefficient a k L | |
| (expandingCircularCoefficient a k L ha ha1 hk hL m q))) ∧ | |
| (∑' n, latticeMomentMass L N (expandingFourierCoefficient a k L | |
| (expandingCircularCoefficient a k L ha ha1 hk hL m q)) n) ≤ | |
| (m + 1 : ℕ) * (M ^ m * M ^ m) := by | |
| intro M | |
| have hconj := expandingConjugate_moment a k L N q hq | |
| have hp := expandingPower_moment a k L ha ha1 hk hL N q hq m | |
| have hp' := expandingPower_moment a k L ha ha1 hk hL N (fourierConjugate q) hconj.1 m | |
| rw [hconj.2] at hp' | |
| have hprod := expandingProduct_moment a k L ha ha1 hk hL N | |
| (expandingPower a k L ha ha1 hk hL q m) | |
| (expandingPower a k L ha ha1 hk hL (fourierConjugate q) m) hp.1 hp'.1 | |
| have hc := expandingScalar_moment a k L N ((m + 1 : ℕ) : ℂ) | |
| (expandingProduct a k L ha ha1 hk hL | |
| (expandingPower a k L ha ha1 hk hL q m) | |
| (expandingPower a k L ha ha1 hk hL (fourierConjugate q) m)) hprod.1 | |
| rw [expandingCircularCoefficient] | |
| refine ⟨hc.1, ?_⟩ | |
| rw [hc.2] | |
| have hM : 0 ≤ M := tsum_nonneg (fun n => mul_nonneg | |
| (le_trans zero_le_one (latticeMomentWeight_one_le L hL N n)) (norm_nonneg _)) | |
| calc | |
| _ ≤ ‖((m + 1 : ℕ) : ℂ)‖ * (M ^ m * M ^ m) := by | |
| apply mul_le_mul_of_nonneg_left (hprod.2.trans ?_) (norm_nonneg _) | |
| exact mul_le_mul hp.2 hp'.2 | |
| (tsum_nonneg (fun n => mul_nonneg (le_trans zero_le_one (latticeMomentWeight_one_le L hL N n)) (norm_nonneg _))) | |
| (pow_nonneg hM m) | |
| _ = _ := by rw [Complex.norm_natCast] | |
| theorem expandingAnticircular_moment (q : FourierL2) | |
| (hq : Summable (latticeMomentMass L N (expandingFourierCoefficient a k L q))) : | |
| let M := ∑' n, latticeMomentMass L N (expandingFourierCoefficient a k L q) n | |
| Summable (latticeMomentMass L N (expandingFourierCoefficient a k L | |
| (expandingAnticircularCoefficient a k L ha ha1 hk hL m q))) ∧ | |
| (∑' n, latticeMomentMass L N (expandingFourierCoefficient a k L | |
| (expandingAnticircularCoefficient a k L ha ha1 hk hL m q)) n) ≤ | |
| (m : ℝ) * (M ^ (m + 1) * M ^ (m - 1)) := by | |
| intro M | |
| have hconj := expandingConjugate_moment a k L N q hq | |
| have hp := expandingPower_moment a k L ha ha1 hk hL N q hq (m + 1) | |
| have hp' := expandingPower_moment a k L ha ha1 hk hL N (fourierConjugate q) hconj.1 (m - 1) | |
| rw [hconj.2] at hp' | |
| have hprod := expandingProduct_moment a k L ha ha1 hk hL N | |
| (expandingPower a k L ha ha1 hk hL q (m + 1)) | |
| (expandingPower a k L ha ha1 hk hL (fourierConjugate q) (m - 1)) hp.1 hp'.1 | |
| have hc := expandingScalar_moment a k L N (m : ℂ) | |
| (expandingProduct a k L ha ha1 hk hL | |
| (expandingPower a k L ha ha1 hk hL q (m + 1)) | |
| (expandingPower a k L ha ha1 hk hL (fourierConjugate q) (m - 1))) hprod.1 | |
| rw [expandingAnticircularCoefficient] | |
| refine ⟨hc.1, ?_⟩ | |
| rw [hc.2] | |
| have hM : 0 ≤ M := tsum_nonneg (fun n => mul_nonneg | |
| (le_trans zero_le_one (latticeMomentWeight_one_le L hL N n)) (norm_nonneg _)) | |
| calc | |
| _ ≤ ‖(m : ℂ)‖ * (M ^ (m + 1) * M ^ (m - 1)) := by | |
| apply mul_le_mul_of_nonneg_left (hprod.2.trans ?_) (norm_nonneg _) | |
| exact mul_le_mul hp.2 hp'.2 | |
| (tsum_nonneg (fun n => mul_nonneg (le_trans zero_le_one (latticeMomentWeight_one_le L hL N n)) (norm_nonneg _))) | |
| (pow_nonneg hM (m + 1)) | |
| _ = _ := by rw [Complex.norm_natCast] | |
| end DefocusingNLS | |
| end OAI | |