openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingBlowupClock.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingBlowupObservation | |
| namespace OAI | |
| /-! # The exact rescaled pointwise limit in physical time -/ | |
| open Filter Topology | |
| namespace DefocusingNLS | |
| theorem expanding_clock_denominator_pos_of_lt (L t : ℝ) (hL : 0 < L) | |
| (ht : t < L ^ (-2 : ℝ)) : 0 < 1 - L ^ 2 * t := by | |
| have hT : 0 < L ^ (-2 : ℝ) := Real.rpow_pos_of_pos hL _ | |
| have hc : L ^ (-2 : ℝ) * L ^ 2 = 1 := by | |
| rw [← Real.rpow_two, ← Real.rpow_add hL] | |
| norm_num | |
| have he : L ^ (-2 : ℝ) - t = L ^ (-2 : ℝ) * (1 - L ^ 2 * t) := by | |
| calc | |
| _ = L ^ (-2 : ℝ) - (L ^ (-2 : ℝ) * L ^ 2) * t := by rw [hc, one_mul] | |
| _ = _ := by ring | |
| have hp : 0 < L ^ (-2 : ℝ) - t := sub_pos.mpr ht | |
| rw [he] at hp | |
| exact (mul_pos_iff_of_pos_left hT).mp hp | |
| theorem expandingInverseClock_nonneg (L t : ℝ) (hL : 0 < L) | |
| (ht : t ∈ Set.Ico 0 (L ^ (-2 : ℝ))) : 0 ≤ expandingInverseClock L t := by | |
| unfold expandingInverseClock | |
| apply neg_nonneg.mpr | |
| exact Real.log_nonpos (expanding_clock_denominator_pos_of_lt L t hL ht.2).le | |
| (by nlinarith [mul_nonneg (sq_nonneg L) ht.1]) | |
| theorem expandingInverseClock_eq_similarityTime (L t : ℝ) (hL : 0 < L) | |
| (ht : t < L ^ (-2 : ℝ)) : | |
| expandingInverseClock L t = similarityTime (L ^ (-2 : ℝ)) t := by | |
| have hT : 0 < L ^ (-2 : ℝ) := Real.rpow_pos_of_pos hL _ | |
| have hc : L ^ (-2 : ℝ) * L ^ 2 = 1 := by | |
| rw [← Real.rpow_two, ← Real.rpow_add hL] | |
| norm_num | |
| have he : L ^ (-2 : ℝ) - t = L ^ (-2 : ℝ) * (1 - L ^ 2 * t) := by | |
| calc | |
| _ = L ^ (-2 : ℝ) - (L ^ (-2 : ℝ) * L ^ 2) * t := by rw [hc, one_mul] | |
| _ = _ := by ring | |
| have hb : 0 < 1 - L ^ 2 * t := | |
| (mul_pos_iff_of_pos_left hT).mp (he ▸ sub_pos.mpr ht) | |
| have hratio : L ^ (-2 : ℝ) / (L ^ (-2 : ℝ) - t) = (1 - L ^ 2 * t)⁻¹ := by | |
| rw [he] | |
| field_simp [hT.ne', hb.ne'] | |
| rw [expandingInverseClock, similarityTime, hratio, Real.log_inv] | |
| theorem expandingInverseClock_tendsto (L : ℝ) (hL : 0 < L) : | |
| Tendsto (expandingInverseClock L) (𝓝[<] (L ^ (-2 : ℝ))) atTop := by | |
| apply (similarityTime_tendsto (L ^ (-2 : ℝ)) (Real.rpow_pos_of_pos hL _)).congr' | |
| filter_upwards [self_mem_nhdsWithin] with t ht | |
| exact (expandingInverseClock_eq_similarityTime L t hL ht).symm | |
| theorem expanding_rescaled_amplitude (a b L s : ℝ) (hL : 0 < L) : | |
| (L ^ (-2 : ℝ) - expandingFreeTime L s) ^ a * | |
| ‖expandingPhysicalAmplitude a b L s‖ = 1 := by | |
| have he : L ^ (-2 : ℝ) - expandingFreeTime L s = L ^ (-2 : ℝ) * Real.exp (-s) := by | |
| unfold expandingFreeTime | |
| ring | |
| have hp : (L ^ (-2 : ℝ)) ^ a * L ^ (2 * a) = 1 := by | |
| rw [← Real.rpow_mul hL.le, ← Real.rpow_add hL] | |
| rw [show (-2 : ℝ) * a + 2 * a = 0 by ring, Real.rpow_zero] | |
| rw [he, expandingPhysicalAmplitude_norm a b L s hL, | |
| Real.mul_rpow (Real.rpow_nonneg hL.le _) (Real.exp_pos _).le, ← Real.exp_mul] | |
| calc | |
| _ = ((L ^ (-2 : ℝ)) ^ a * L ^ (2 * a)) * | |
| (Real.exp (-s * a) * Real.exp (a * s)) := by ring | |
| _ = 1 := by | |
| rw [hp, ← Real.exp_add, show -s * a + a * s = 0 by ring, Real.exp_zero, one_mul] | |
| theorem expanding_rescaled_observation (a b L t : ℝ) (hL : 0 < L) | |
| (ht : t < L ^ (-2 : ℝ)) (z : ℂ) : | |
| (L ^ (-2 : ℝ) - t) ^ a * | |
| ‖expandingPhysicalAmplitude a b L (expandingInverseClock L t) * z‖ = ‖z‖ := by | |
| have hd := expanding_clock_denominator_pos_of_lt L t hL ht | |
| have h := expanding_rescaled_amplitude a b L (expandingInverseClock L t) hL | |
| rw [expandingFreeTime_inverseClock L t hL hd] at h | |
| rw [norm_mul, ← mul_assoc, h, one_mul] | |
| /-- The rescaled modulus limit required in the main theorem follows from | |
| the similarity observation limit. -/ | |
| theorem expanding_rescaled_observation_limit (a b L : ℝ) (hL : 0 < L) | |
| (g : ℝ → ℂ) (c : ℂ) (hg : Tendsto g atTop (𝓝 c)) : | |
| Tendsto (fun t => (L ^ (-2 : ℝ) - t) ^ a * | |
| ‖expandingPhysicalAmplitude a b L (expandingInverseClock L t) * | |
| g (expandingInverseClock L t)‖) | |
| (𝓝[<] (L ^ (-2 : ℝ))) (𝓝 ‖c‖) := by | |
| have h := (hg.comp (expandingInverseClock_tendsto L hL)).norm | |
| apply h.congr' | |
| filter_upwards [self_mem_nhdsWithin] with t ht | |
| exact (expanding_rescaled_observation a b L t hL ht _).symm | |
| end DefocusingNLS | |
| end OAI | |