openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralActionGronwall.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralIntegralGronwall | |
| namespace OAI | |
| /-! Remove the same-channel action before applying Gronwall. This keeps the | |
| integral of the WKB residual, rather than the length of the whole shell. -/ | |
| open Set MeasureTheory | |
| namespace DefocusingNLS | |
| theorem spectral_action_gronwall (a b A : ℝ) (hab : a≤ b) (X g H : ℝ → ℝ) | |
| (hX : ContinuousOn X (Icc a b)) (hg : ContinuousOn g (Icc a b)) | |
| (hH : ContinuousOn H (Icc a b)) (hg0 : ∀ r ∈ Icc a b, 0≤ g r) | |
| (hbound : ∀ r ∈ Icc a b, | |
| X r≤ A*Real.exp (H r)+∫ t in a..r, g t*Real.exp (H r-H t)*X t) : | |
| ∀ r ∈ Icc a b, X r≤ A*Real.exp (H r+∫ t in a..r, g t) := by | |
| let Y := fun r => Real.exp (-H r)*X r | |
| have hY : ContinuousOn Y (Icc a b) := | |
| (Real.continuous_exp.comp_continuousOn hH.neg).mul hX | |
| have hprod (r t : ℝ) : | |
| Real.exp (-H r)*(g t*Real.exp (H r-H t)*X t)=g t*Y t := by | |
| have he : Real.exp (-H r)*Real.exp (H r-H t)=Real.exp (-H t) := by | |
| rw [← Real.exp_add] | |
| congr 1 | |
| ring | |
| dsimp only [Y] | |
| calc | |
| _ = g t*(Real.exp (-H r)*Real.exp (H r-H t))*X t := by ring | |
| _ = _ := by rw [he]; ring | |
| have hYbound (r : ℝ) (hr : r ∈ Icc a b) : Y r≤ A+∫ t in a..r, g t*Y t := by | |
| have hb := mul_le_mul_of_nonneg_left (hbound r hr) (Real.exp_pos (-H r)).le | |
| have he : Real.exp (-H r)*Real.exp (H r)=1 := by | |
| rw [← Real.exp_add,neg_add_cancel,Real.exp_zero] | |
| change Y r≤ _ at hb | |
| calc | |
| Y r ≤ Real.exp (-H r)*(A*Real.exp (H r)+∫ t in a..r, g t*Real.exp (H r-H t)*X t) := hb | |
| _ = A+∫ t in a..r, g t*Y t := by | |
| rw [mul_add,← mul_assoc, mul_comm (Real.exp (-H r)) A,mul_assoc,he,mul_one, | |
| ← intervalIntegral.integral_const_mul] | |
| congr 1 | |
| exact intervalIntegral.integral_congr (fun t _ => hprod r t) | |
| intro r hr | |
| have hh := spectral_integral_gronwall a b A hab Y g hY hg hg0 hYbound r hr | |
| have hm := mul_le_mul_of_nonneg_left hh (Real.exp_pos (H r)).le | |
| have he : Real.exp (H r)*Real.exp (-H r)=1 := by | |
| rw [← Real.exp_add,add_neg_cancel,Real.exp_zero] | |
| change Real.exp (H r)*(Real.exp (-H r)*X r)≤ _ at hm | |
| rw [← mul_assoc,he,one_mul] at hm | |
| calc | |
| X r ≤ Real.exp (H r)*(A*Real.exp (∫ t in a..r, g t)) := hm | |
| _ = _ := by rw [← mul_assoc,mul_comm (Real.exp (H r)) A,mul_assoc,← Real.exp_add] | |
| end DefocusingNLS | |
| end OAI | |