openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Localization /InverseObservationScaleBarrier.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.RadialBounds.InverseKernelMassBarrier | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open Filter | |
| open scoped Topology | |
| namespace CoulombAtom | |
| open CoulombAnalysis | |
| lemma inverse_observation_geometry_tendsto {ι : Type*} {l : Filter ι} | |
| {a t ell : ι → ℝ} {c E : ℝ} (hc : 0 < c) (hE : 0 ≤ E) | |
| (ha : ∀ᶠ i in l, 0 < a i) (ha0 : Tendsto a l (𝓝 0)) | |
| (ht : ∀ᶠ i in l, c*(a i)^(1+masterExponent) ≤ t i) | |
| (hell : ∀ᶠ i in l, 0 ≤ ell i ∧ ell i ≤ E*(a i)^(101/100:ℝ)) : | |
| Tendsto (fun i => (a i)^3*(t i)⁻¹^4*((a i)^(6/5:ℝ)+2*ell i)) l (𝓝 0) := by | |
| have he1 : 0 < (21/5:ℝ)-(1+masterExponent)*4 := by norm_num [masterExponent] | |
| have he2 : 0 < (401/100:ℝ)-(1+masterExponent)*4 := by norm_num [masterExponent] | |
| have hh : Tendsto (fun i => c⁻¹^4*(a i)^((21/5:ℝ)-(1+masterExponent)*4)+ | |
| (2*E*c⁻¹^4)*(a i)^((401/100:ℝ)-(1+masterExponent)*4)) l (𝓝 0) := by | |
| simpa only [mul_zero,add_zero] using | |
| ((ha0.rpow_const_nhds_zero he1).const_mul (c⁻¹^4)).add | |
| ((ha0.rpow_const_nhds_zero he2).const_mul (2*E*c⁻¹^4)) | |
| apply squeeze_zero' _ _ hh | |
| · filter_upwards [ha,ht,hell] with i hai hti helli | |
| have hti' := (mul_pos hc (Real.rpow_pos_of_pos hai (1+masterExponent))).trans_le hti | |
| have hli := helli.1 | |
| positivity | |
| · filter_upwards [ha,ht,hell] with i hai hti helli | |
| have hti' := (mul_pos hc (Real.rpow_pos_of_pos hai (1+masterExponent))).trans_le hti | |
| have he1' : (a i)^3*(a i)^(6/5:ℝ) = (a i)^(21/5:ℝ) := by | |
| rw [←Real.rpow_natCast,←Real.rpow_add hai] | |
| norm_num | |
| have he2' : (a i)^3*(a i)^(101/100:ℝ) = (a i)^(401/100:ℝ) := by | |
| rw [←Real.rpow_natCast,←Real.rpow_add hai] | |
| norm_num | |
| calc | |
| _ ≤ (a i)^3*(t i)⁻¹^4*((a i)^(6/5:ℝ)+2*(E*(a i)^(101/100:ℝ))) := by | |
| gcongr | |
| exact helli.2 | |
| _ = ((a i)^3*(a i)^(6/5:ℝ))*(t i)⁻¹^4+ | |
| (2*E)*((a i)^3*(a i)^(101/100:ℝ))*(t i)⁻¹^4 := by ring | |
| _ = (a i)^(21/5:ℝ)*(t i)⁻¹^4+(2*E)*((a i)^(401/100:ℝ)*(t i)⁻¹^4) := by | |
| rw [he1',he2']; ring | |
| _ ≤ c⁻¹^4*(a i)^((21/5:ℝ)-(1+masterExponent)*4)+ | |
| (2*E)*(c⁻¹^4*(a i)^((401/100:ℝ)-(1+masterExponent)*4)) := | |
| add_le_add (rpow_inv_pow_bound hai hc hti 4) | |
| (mul_le_mul_of_nonneg_left (rpow_inv_pow_bound hai hc hti 4) (by positivity)) | |
| _ = _ := by ring | |
| theorem inverse_observation_error_tendsto {ι : Type*} {l : Filter ι} | |
| {a t ell Q : ι → ℝ} {c E L q : ℝ} (hc : 0 < c) (hE : 0 ≤ E) | |
| (ha : ∀ᶠ i in l, 0 < a i) (ha0 : Tendsto a l (𝓝 0)) | |
| (ht : ∀ᶠ i in l, c*(a i)^(1+masterExponent) ≤ t i) | |
| (hell : ∀ᶠ i in l, 0 ≤ ell i ∧ ell i ≤ E*(a i)^(101/100:ℝ)) | |
| (hQ : Tendsto (fun i => (a i)^3*Q i) l (𝓝 q)) : | |
| Tendsto (fun i => (a i)^6*(Q i*(L*(t i)⁻¹^4*Real.sqrt 3* | |
| ((a i)^(6/5:ℝ)+2*ell i)))) l (𝓝 0) := by | |
| have hh := (hQ.mul (inverse_observation_geometry_tendsto hc hE ha ha0 ht hell)).const_mul (L*Real.sqrt 3) | |
| simp only [mul_zero] at hh | |
| apply hh.congr' | |
| exact Eventually.of_forall (fun i => by ring) | |
| end CoulombAtom | |
| end | |
| end OAI | |