openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingEnergyObservation.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingNonprincipalObservation | |
| namespace OAI | |
| /-! # Dissipative energy with a compact physical observation on large tori -/ | |
| open Filter Topology MeasureTheory | |
| namespace DefocusingNLS | |
| local notation "T" => UnitAddTorus (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| noncomputable local instance energyObservationMeasure : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ | |
| local instance energyObservationProbability : IsProbabilityMeasure (volume : Measure UnitAddCircle) := | |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) | |
| theorem exists_expandingEnergy_observation (a Q : ℝ) (N : ℕ) | |
| (ha : 0 < a) (ha1 : a < 1) (hN : 8 < ((N + 1 : ℕ) : ℝ)) (hQ : 0 ≤ Q) | |
| (q : Radius → FourierL2) (hqb : ∀ L, ‖q L‖ ≤ Q) | |
| (hq : ∀ (L : ℕ → Radius), Tendsto (fun n => (L n).1) atTop atTop → | |
| ExpandingCompactApproximation a (N + 1 : ℕ) ha1 hN | |
| (fun n => (L n).1) (fun n => (L n).2) (fun n => q (L n))) | |
| (m : ℕ) (hm : 0 < m) (P : ℝ) (hP : 0 ≤ P) | |
| (hQB : ∀ (L : Radius) (x : T), ‖expandingUnitTorusFunction a (N + 1 : ℕ) L.1 (q L) x‖ ^ (2 * m) ≤ P) | |
| (hgap : 0 < (N + 1 : ℕ) + 2 * a - 6 - 2 * ((2 * (m : ℝ) + 1) * P)) : | |
| ∃ c : ℝ, 0 < c ∧ ∃ n : ℕ, ∃ C : ℝ, 0 ≤ C ∧ | |
| ∀ (L : Radius), (n : ℝ) ≤ L.1 → ∀ f : FourierL2, | |
| -a * ‖expandingLowEnergy a (N + 1 : ℕ) L.1 L.2 f‖ ^ 2 + | |
| (6 - 2 * a - (N + 1 : ℕ)) * ‖expandingHighEnergy a (N + 1 : ℕ) L.1 L.2 f‖ ^ 2 + | |
| 2 * inner ℝ f (expandingLinearizedPotential a (N + 1 : ℕ) L.1 ha ha1 hN L.2 m (q L) f) ≤ | |
| -c * ‖f‖ ^ 2 + C * ‖expandingPhysicalBall a (N + 1 : ℕ) L.1 n ha ha1 hN L.2 f‖ ^ 2 := by | |
| let B := (2 * (m : ℝ) + 1) * P | |
| let g := (N + 1 : ℕ) + 2 * a - 6 - 2 * B | |
| let γ := min a g | |
| have hγ : 0 < γ := lt_min ha hgap | |
| obtain ⟨n, C, hC, herror⟩ := exists_expandingNonprincipal_observation a Q N ha ha1 hN hQ q hqb hq | |
| m hm P hP hQB (γ / 4) (by positivity) | |
| refine ⟨γ / 2, half_pos hγ, n, 2 * C, by positivity, ?_⟩ | |
| intro L hL f | |
| have he := herror L hL f | |
| have hp := expandingOrderedPrincipal_energy_le a L.1 (N + 1) m L.2 | |
| (expandingUnitTorusFunction a (N + 1 : ℕ) L.1 (q L)) | |
| (expandingUnitTorusFunction a (N + 1 : ℕ) L.1 (q L)).continuous P hP (hQB L) f | |
| have hnorm := expandingEnergy_norm_sq a (N + 1 : ℕ) L.1 L.2 f | |
| have hl := mul_nonneg (sub_nonneg.mpr (min_le_left a g)) | |
| (sq_nonneg ‖expandingLowEnergy a (N + 1 : ℕ) L.1 L.2 f‖) | |
| have hh := mul_nonneg (sub_nonneg.mpr (min_le_right a g)) | |
| (sq_nonneg ‖expandingHighEnergy a (N + 1 : ℕ) L.1 L.2 f‖) | |
| change 0 ≤ (a - γ) * _ at hl | |
| change 0 ≤ (g - γ) * _ at hh | |
| unfold expandingNonprincipalEnergy at he | |
| dsimp only [B, g] at hh | |
| nlinarith | |
| end DefocusingNLS | |
| end OAI | |