| import Mathlib |
| import OAI.Analysis.CoulombIonization.RadialBounds.RadialPatchGeometry |
| import OAI.Analysis.CoulombIonization.Localization.FormRawLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
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| open MeasureTheory Set Filter |
| open scoped BigOperators |
|
|
| namespace CoulombAtom |
| open CoulombRadialAveraging |
|
|
| def spatialCutExpectation {N : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) (ψ : FormVector N) |
| (F : (Fin N → Fin 2) → Configuration N → ℝ) : ℝ := |
| ∑ c : Fin N → Fin 2, ∑ s : Spins N, ∫ x, |
| F c x*‖(multiplyForm (spatialProduct p hp c) ψ).value s x‖^2 |
|
|
| lemma spatialCutExpectation_raw {N : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) {ψ : FormVector N} (hψ : SobolevVector ψ) |
| {g : Configuration N → ℝ} (hg : Measurable g) {B : ℝ} (hB : ∀ x, ‖g x‖ ≤ B) : |
| spatialCutExpectation p hp ψ (fun _ => g) = ∫ x, g x ∂formRawLaw ψ := by |
| rw [formRawLaw_integral_eq_spin_sum hψ hg hB] |
| unfold spatialCutExpectation |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro s _ |
| have hi (c : Fin N → Fin 2) : Integrable (fun x => |
| g x*‖(multiplyForm (spatialProduct p hp c) ψ).value s x‖^2) := |
| (((hψ.multiply _).1 s).norm.integrable_sq).bdd_mul hg.aestronglyMeasurable (ae_of_all _ hB) |
| rw [←integral_finsetSum _ (fun c _ => hi c)] |
| apply integral_congr_ae (ae_of_all _ _) |
| intro x |
| rw [←Finset.mul_sum,finite_partition_value (spatialProduct p hp) (spatial_square_partition p hp)] |
|
|
| lemma spatialCutExpectation_le_raw {N : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) {ψ : FormVector N} (hψ : SobolevVector ψ) |
| {F : (Fin N → Fin 2) → Configuration N → ℝ} {g : Configuration N → ℝ} |
| (hF : ∀ c, Measurable (F c)) {A : ℝ} (hA : ∀ c x, ‖F c x‖ ≤ A) |
| (hg : Measurable g) {B : ℝ} (hB : ∀ x, ‖g x‖ ≤ B) |
| (hfg : ∀ c x, (spatialProduct p hp c).value x ≠ 0 → F c x ≤ g x) : |
| spatialCutExpectation p hp ψ F ≤ ∫ x, g x ∂formRawLaw ψ := by |
| rw [←spatialCutExpectation_raw p hp hψ hg hB] |
| apply Finset.sum_le_sum; intro c _ |
| apply Finset.sum_le_sum; intro s _ |
| apply integral_mono |
| ((((hψ.multiply _).1 s).norm.integrable_sq).bdd_mul (hF c).aestronglyMeasurable |
| (ae_of_all _ (hA c))) |
| ((((hψ.multiply _).1 s).norm.integrable_sq).bdd_mul hg.aestronglyMeasurable |
| (ae_of_all _ hB)) |
| intro x |
| by_cases hx : (spatialProduct p hp c).value x = 0 |
| · simp only [multiplyForm,hx,Complex.ofReal_zero,zero_mul,norm_zero,zero_pow (by decide : 2 ≠ 0),mul_zero,le_refl] |
| · exact mul_le_mul_of_nonneg_right (hfg c x hx) (sq_nonneg _) |
|
|
| lemma spatialCutExpectation_outMoment {N : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) (ψ : FormVector N) |
| (q : ℕ → ℝ) : |
| (∑ c : Fin N → Fin 2, q (cutOutNumber c)*formMass (orderedCutForm p hp ψ c)) = |
| spatialCutExpectation p hp ψ (fun c _ => q (cutOutNumber c)) := by |
| simp_rw [orderedCutForm_mass] |
| simp only [spatialCutExpectation,formMass,Finset.mul_sum,integral_const_mul] |
|
|
| lemma radialCut_out_distance {N : ℕ} (y : Space) {t b : ℝ} (ht : 0 ≤ t) (hb : 0 < b) |
| (c : Fin N → Fin 2) (x : Configuration N) |
| (hx : (spatialProduct (coreFirstRadialCut y ht hb) |
| (coreFirstRadialCut_partition y ht hb) c).value x ≠ 0) |
| (i : Fin N) (hi : c i ≠ 0) : ‖x i-y‖ < t+b := by |
| classical |
| have hc : c i = 1 := by omega |
| change (∏ j, (coreFirstRadialCut y ht hb (c j)).value (x j)) ≠ 0 at hx |
| have hf : (coreFirstRadialCut y ht hb (c i)).value (x i) ≠ 0 := |
| (Finset.prod_ne_zero_iff.mp hx) i (Finset.mem_univ i) |
| by_contra hh |
| have hz := (radialCut_outer ht hb (le_of_not_gt hh)).1 |
| apply hf |
| simpa only [hc,coreFirstRadialCut,radialCut,Matrix.cons_val_one,Matrix.cons_val_zero] using hz |
|
|
| lemma radialCut_outNumber_le_rawBall {N : ℕ} (y : Space) {t b R : ℝ} |
| (ht : 0 ≤ t) (hb : 0 < b) (hR : t+b ≤ R) |
| (c : Fin N → Fin 2) (x : Configuration N) |
| (hx : (spatialProduct (coreFirstRadialCut y ht hb) |
| (coreFirstRadialCut_partition y ht hb) c).value x ≠ 0) : |
| (cutOutNumber c:ℝ) ≤ rawBallCount y R x := by |
| classical |
| rw [cutOutNumber_eq_sum] |
| apply Finset.sum_le_sum |
| intro i _ |
| by_cases hi : c i ≠ 0 |
| · have hd := lt_of_lt_of_le (radialCut_out_distance y ht hb c x hx i hi) hR |
| simp [hi,hd] |
| · simp only [hi,ite_false] |
| split_ifs <;> norm_num |
|
|
| def cutDeletedCount {N : ℕ} (y : Space) (t b : ℝ) |
| (c : Fin N → Fin 2) (x : Configuration N) : ℝ := |
| ∑ i, if c i ≠ 0 ∧ t-7*b ≤ ‖x i-y‖ then 1 else 0 |
|
|
| lemma cutDeletedCount_nonneg {N : ℕ} (y : Space) (t b : ℝ) |
| (c : Fin N → Fin 2) (x : Configuration N) : 0 ≤ cutDeletedCount y t b c x := by |
| apply Finset.sum_nonneg; intro i _; split_ifs <;> norm_num |
|
|
| lemma cutDeletedCount_le {N : ℕ} (y : Space) (t b : ℝ) |
| (c : Fin N → Fin 2) (x : Configuration N) : cutDeletedCount y t b c x ≤ N := by |
| calc |
| _ ≤ ∑ _ : Fin N, (1:ℝ) := Finset.sum_le_sum (fun _ _ => by split_ifs <;> norm_num) |
| _ = N := by simp |
|
|
| lemma cutDeletedCount_measurable {N : ℕ} (y : Space) (t b : ℝ) (c : Fin N → Fin 2) : |
| Measurable (cutDeletedCount y t b c) := by |
| apply Finset.measurable_sum; intro i _ |
| apply Measurable.ite _ measurable_const measurable_const |
| by_cases hc : c i ≠ 0 |
| · simpa [hc] using (measurableSet_le (measurable_const (a := t-7*b)) |
| (configuration_distance_measurable y i)) |
| · simp [hc] |
|
|
| lemma radialCut_deleted_le_rawStrip {N : ℕ} (y : Space) {t b : ℝ} |
| (ht : 0 ≤ t) (hb : 0 < b) (c : Fin N → Fin 2) (x : Configuration N) |
| (hx : (spatialProduct (coreFirstRadialCut y ht hb) |
| (coreFirstRadialCut_partition y ht hb) c).value x ≠ 0) : |
| cutDeletedCount y t b c x ≤ rawStripCount y t b x := by |
| apply Finset.sum_le_sum |
| intro i _ |
| by_cases hh : c i ≠ 0 ∧ t-7*b ≤ ‖x i-y‖ |
| · have hd := radialCut_out_distance y ht hb c x hx i hh.1 |
| have hs : ‖x i-y‖-b ≤ t ∧ t ≤ ‖x i-y‖+7*b := ⟨by linarith,by linarith [hh.2]⟩ |
| simp [hh,slidingBit,hs] |
| · simp only [hh,ite_false] |
| exact slidingBit_nonneg _ _ _ _ |
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| end CoulombAtom |
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| end |
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| end OAI |
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