openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Localization /FiniteCellObservation.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.RadialBounds.ArbitraryCutExcess | |
| namespace OAI | |
| open MeasureTheory Set Metric | |
| noncomputable section | |
| namespace CoulombAtom | |
| namespace CoreObservationEnsemble | |
| def observeCell (E : CoreObservationEnsemble) (y : Space) : CoreObservationEnsemble := | |
| if h : y = 0 then E else | |
| E.observe (coreFirstRadialCut y (localCellRadius_pos h).le (localCellRadius_pos h)) | |
| (coreFirstRadialCut_partition y (localCellRadius_pos h).le (localCellRadius_pos h)) | |
| def observeCells (E : CoreObservationEnsemble) : List Space → CoreObservationEnsemble | |
| | [] => E | |
| | y::ys => (E.observeCell y).observeCells ys | |
| lemma observeCell_mass (E : CoreObservationEnsemble) (y : Space) : | |
| (E.observeCell y).mass = E.mass := by | |
| classical | |
| unfold observeCell | |
| split_ifs <;> simp only [observe_mass] | |
| lemma observeCell_countMoment (E : CoreObservationEnsemble) (y z : Space) (R : ℝ) : | |
| (E.observeCell y).countMoment z R ≤ E.countMoment z R := by | |
| classical | |
| unfold observeCell | |
| split_ifs | |
| · exact le_rfl | |
| · exact observe_countMoment .. | |
| lemma observeCells_mass (E : CoreObservationEnsemble) (ys : List Space) : | |
| (E.observeCells ys).mass = E.mass := by | |
| induction ys generalizing E with | |
| | nil => rfl | |
| | cons y ys ih => rw [observeCells,ih,observeCell_mass] | |
| lemma observeCells_countMoment (E : CoreObservationEnsemble) (ys : List Space) | |
| (z : Space) (R : ℝ) : | |
| (E.observeCells ys).countMoment z R ≤ E.countMoment z R := by | |
| induction ys generalizing E with | |
| | nil => exact le_rfl | |
| | cons y ys ih => exact (ih _).trans (observeCell_countMoment ..) | |
| theorem observeCells_excess (E : CoreObservationEnsemble) (ys : List Space) | |
| {Z lam C H : ℝ} (hmass : E.mass = 1) (hZ : 0 ≤ Z) (hlam : 0 < lam) | |
| (hC : 1 ≤ C) (hH : 0 ≤ H) | |
| (hb : ∀ y ∈ ys, y ≠ 0 ∧ ∃ m : ℝ, 1 ≤ m ∧ 1/(localCellRadius y)^3 ≤ m ∧ | |
| E.countMoment y (16*localCellRadius y) ≤ C*m^2 ∧ m^2/localCellRadius y ≤ H) : | |
| max ((E.observeCells ys).excess Z lam) 0+H ≤ | |
| (radialCutExcessConstant C+1)^ys.length*(max (E.excess Z lam) 0+H) := by | |
| classical | |
| induction ys generalizing E with | |
| | nil => simpa only [observeCells,List.length_nil,pow_zero,one_mul] using (le_refl (max (E.excess Z lam) 0+H)) | |
| | cons y ys ih => | |
| obtain ⟨hy,m,hm,h3,hB,hmH⟩ := hb y (List.mem_cons_self ..) | |
| have ha := localCellRadius_pos hy | |
| have hf := E.observe_excess_linear y ha ha.le (by linarith) | |
| (by unfold localCellRadius; nlinarith [norm_nonneg y]) hmass hZ hlam hm h3 hC hB | |
| have hp : max ((E.observeCell y).excess Z lam) 0 ≤ | |
| radialCutExcessConstant C*(max (E.excess Z lam) 0+H) := by | |
| rw [observeCell, dite_eq_right hy] | |
| apply max_le | |
| · exact (le_max_left _ _).trans (hf.trans | |
| (mul_le_mul_of_nonneg_left (add_le_add le_rfl hmH) | |
| (zero_le_one.trans (radialCutExcessConstant_one_le hC)))) | |
| · exact mul_nonneg (zero_le_one.trans (radialCutExcessConstant_one_le hC)) | |
| (add_nonneg (le_max_right _ _) hH) | |
| have hb' : ∀ z ∈ ys, z ≠ 0 ∧ ∃ m : ℝ, 1 ≤ m ∧ 1/(localCellRadius z)^3 ≤ m ∧ | |
| (E.observeCell y).countMoment z (16*localCellRadius z) ≤ C*m^2 ∧ | |
| m^2/localCellRadius z ≤ H := by | |
| intro z hz | |
| obtain ⟨hz0,m,hm,h3,hB,hH⟩ := hb z (List.mem_cons_of_mem y hz) | |
| exact ⟨hz0,m,hm,h3,(observeCell_countMoment ..).trans hB,hH⟩ | |
| have h0 : 0 ≤ max (E.excess Z lam) 0 := le_max_right _ _ | |
| have hK : 0 ≤ radialCutExcessConstant C+1 := by | |
| have := radialCutExcessConstant_one_le hC; linarith | |
| have hh : max ((E.observeCell y).excess Z lam) 0+H ≤ | |
| (radialCutExcessConstant C+1)*(max (E.excess Z lam) 0+H) := by | |
| nlinarith only [hp,h0] | |
| calc | |
| _ ≤ (radialCutExcessConstant C+1)^ys.length* | |
| (max ((E.observeCell y).excess Z lam) 0+H) := | |
| ih _ ((observeCell_mass E y).trans hmass) hb' | |
| _ ≤ (radialCutExcessConstant C+1)^ys.length* | |
| ((radialCutExcessConstant C+1)*(max (E.excess Z lam) 0+H)) := | |
| mul_le_mul_of_nonneg_left hh (pow_nonneg hK _) | |
| _ = _ := by simp only [List.length_cons,pow_succ]; ring | |
| end CoreObservationEnsemble | |
| end CoulombAtom | |
| end | |
| end OAI | |