openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Variational /ExpectedTruncatedControl.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Localization.MasterPotential | |
| import OAI.Analysis.CoulombIonization.Variational.PotentialRecovery | |
| namespace OAI | |
| noncomputable section | |
| namespace CoulombAtom | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| section Work_ExpectedTruncatedControl_scope | |
| open MeasureTheory Set Metric | |
| open scoped BigOperators ENNReal NNReal ContDiff | |
| open CoulombNeumann CoulombAnalysis | |
| local instance : Fact ((5/3:ℝ≥0∞) ≠ (⊤ : ℝ≥0∞)) := ⟨by finiteness⟩ | |
| local instance : Fact ((5/2:ℝ≥0∞) ≠ (⊤ : ℝ≥0∞)) := ⟨by finiteness⟩ | |
| local instance (R : ℝ) : MeasurableSpace (TFField R) := borel _ | |
| local instance (R : ℝ) : BorelSpace (TFField R) := ⟨rfl⟩ | |
| local instance (R : ℝ) : MeasurableSpace (TFLp (ballMeasure R)) := borel _ | |
| local instance (R : ℝ) : BorelSpace (TFLp (ballMeasure R)) := ⟨rfl⟩ | |
| def weightedPatchTestAbs {N M : ℕ} (ψ : FormVector (N+M)) (s : Spins M) | |
| (Z lam : ℝ) (y : Space) (R : ℝ) {b : ℝ} (hb : 0 < b) | |
| (S : Configuration M → Finset (Fin M)) (g : TFSpace → ℝ) | |
| (u : Configuration M) : ℝ := | |
| formMass (coreSlice ψ s u)* | |
| |∫ x, g x*(retainedPatchLp hb (S u) u y R- | |
| tfPatchMinimizer R tfKinetic tfKinetic_pos (conditionalPatchField ψ s Z lam y R u)) x | |
| ∂ballMeasure R| | |
| lemma patchTestError_aestronglyMeasurable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (s : Spins M) (Z lam : ℝ) (y : Space) (R : ℝ) | |
| {b : ℝ} (hb : 0 < b) (S : Configuration M → Finset (Fin M)) | |
| (hS : ∀ i, MeasurableSet {u | i ∈ S u}) {g : TFSpace → ℝ} | |
| (hm : MemLp g (5/2) (ballMeasure R)) : | |
| AEStronglyMeasurable (fun u => | |
| ∫ x, g x*(retainedPatchLp hb (S u) u y R- | |
| tfPatchMinimizer R tfKinetic tfKinetic_pos (conditionalPatchField ψ s Z lam y R u)) x | |
| ∂ballMeasure R) := by | |
| have hs : Measurable (fun u : Configuration M => retainedPatchLp hb (S u) u y R) := | |
| retainedPatchLp_measurable hb S id hS measurable_id y R | |
| have hr := conditionalPatchMinimizer_aemeasurable hψ s Z lam y R | |
| exact ((patch_test_continuous_of_memLp R hm).measurable.comp_aemeasurable | |
| (hs.aemeasurable.sub hr)).aestronglyMeasurable | |
| lemma weightedPatchTruncatedAbs_integrable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (s : Spins M) (Z lam : ℝ) (y : Space) | |
| {R r : ℝ} (hr : 0 < r) (hR : r < R) | |
| {b : ℝ} (hb : 0 < b) (S : Configuration M → Finset (Fin M)) | |
| (hS : ∀ i, MeasurableSet {u | i ∈ S u}) | |
| (hi : Integrable (weightedPatchGap ψ s Z lam y R hb S)) : | |
| Integrable (weightedPatchTestAbs ψ s Z lam y R hb S (fun x => 1/max ‖x‖ r)) := | |
| (weighted_abs_integrable_and_bound (fun _ => formMass_nonneg _) | |
| (hψ.coreSlice_mass_integrable s) | |
| (patchTestError_aestronglyMeasurable hψ s Z lam y R hb S hS (truncated_kernel_memLp R hr)) | |
| (weightedPatchTruncatedSquare_integrable hψ s Z lam y hr hR hb S hS hi)).1 | |
| theorem radial_truncated_absolute_control {N : ℕ} {ψ : FormVector N} | |
| (hψ : SobolevFermion ψ) (y : Space) {t b : ℝ} (ht : 0 ≤ t) (hb : 0 < b) | |
| (htb : 7*b < t) (hy : t ≤ ‖y‖) {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam) | |
| {g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g) | |
| (hgn : ∫ z : Space, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1) | |
| {r : ℝ} (hr : 0 < r) (hR : r < t-4*b) : | |
| (∑ c : Fin N → Fin 2, ∑ s : Spins (cutOutNumber c), ∫ u, | |
| weightedPatchTestAbs (orderedCutForm (coreFirstRadialCut y ht hb) | |
| (coreFirstRadialCut_partition y ht hb) ψ c) s Z lam y (t-4*b) hb | |
| (radialPatchRetention N y t b c s) (fun x => 1/max ‖x‖ r) u) ≤ | |
| Real.sqrt (formMass ψ)*Real.sqrt ((2/r)* | |
| (∑ c : Fin N → Fin 2, ∑ s : Spins (cutOutNumber c), ∫ u, | |
| weightedPatchGap (orderedCutForm (coreFirstRadialCut y ht hb) | |
| (coreFirstRadialCut_partition y ht hb) ψ c) s Z lam y (t-4*b) hb | |
| (radialPatchRetention N y t b c s) u)) := by | |
| let p := coreFirstRadialCut y ht hb | |
| let hp := coreFirstRadialCut_partition y ht hb | |
| let I := (c : Fin N → Fin 2) × Spins (cutOutNumber c) | |
| let w := fun i : I => fun u : Configuration (cutOutNumber i.1) => | |
| formMass (coreSlice (orderedCutForm p hp ψ i.1) i.2 u) | |
| let f := fun i : I => fun u : Configuration (cutOutNumber i.1) => | |
| ∫ x, (1/max ‖x‖ r)*(retainedPatchLp hb (radialPatchRetention N y t b i.1 i.2 u) u y (t-4*b)- | |
| tfPatchMinimizer (t-4*b) tfKinetic tfKinetic_pos | |
| (conditionalPatchField (orderedCutForm p hp ψ i.1) i.2 Z lam y (t-4*b) u)) x | |
| ∂ballMeasure (t-4*b) | |
| have hχ (c : Fin N → Fin 2) : SobolevVector (orderedCutForm p hp ψ c) := | |
| orderedCutForm_sobolev p hp hψ.sobolevVector c | |
| have hgap (i : I) := radial_weightedPatchGap_integrable hψ y ht hb htb hy hZ hlam | |
| hg hcg hgn hrad hgs i.1 i.2 | |
| have hcs := finite_weighted_abs_bound (μ := fun i : I => (volume : Measure (Configuration (cutOutNumber i.1)))) | |
| (w := w) (f := f) (fun _ _ => formMass_nonneg _) | |
| (fun i => (hχ i.1).coreSlice_mass_integrable i.2) | |
| (fun i => patchTestError_aestronglyMeasurable (hχ i.1) i.2 Z lam y (t-4*b) hb _ | |
| (radialPatchRetention_measurable N y t b i.1 i.2) (truncated_kernel_memLp _ hr)) | |
| (fun i => weightedPatchTruncatedSquare_integrable (hχ i.1) i.2 Z lam y hr hR hb _ | |
| (radialPatchRetention_measurable N y t b i.1 i.2) (hgap i)) | |
| have hmass : (∑ i : I, ∫ u, w i u) = formMass ψ := by | |
| rw [Fintype.sum_sigma] | |
| change (∑ c, ∑ s, ∫ u, formMass (coreSlice (orderedCutForm p hp ψ c) s u)) = _ | |
| simp_rw [SobolevVector.integral_coreSlice_mass (hχ _)] | |
| exact orderedCutForm_mass_sum p hp hψ.sobolevVector | |
| rw [hmass,Fintype.sum_sigma,Fintype.sum_sigma] at hcs | |
| change (∑ c, ∑ s, ∫ u, weightedPatchTestAbs (orderedCutForm p hp ψ c) s Z lam y _ hb | |
| (radialPatchRetention N y t b c s) (fun x => 1/max ‖x‖ r) u) ≤ | |
| Real.sqrt (formMass ψ)*Real.sqrt | |
| (∑ c, ∑ s, ∫ u, weightedPatchTestSquare (orderedCutForm p hp ψ c) s Z lam y _ hb | |
| (radialPatchRetention N y t b c s) (fun x => 1/max ‖x‖ r) u) at hcs | |
| exact hcs.trans (mul_le_mul_of_nonneg_left | |
| (Real.sqrt_le_sqrt (radial_truncated_test_control hψ y ht hb htb hy hZ hlam hg hcg hgn hrad hgs hr hR)) | |
| (Real.sqrt_nonneg _)) | |
| end Work_ExpectedTruncatedControl_scope | |
| open MeasureTheory Filter Set Metric | |
| open scoped ENNReal ContDiff | |
| open CoulombAnalysis | |
| def masterSmearing (c₁ r₀ s : ℝ) (g ρ : Space → ℝ) (y : Space) : ℝ := | |
| ∫ x, ρ x*masterKernel c₁ r₀ s g x y | |
| lemma scaled_packet_density_compact {b : ℝ} (hb : 0 < b) {g : Space → ℝ} | |
| (hg : HasCompactSupport g) : HasCompactSupport (fun z => (scaledRealPacket b g z)^2) := by | |
| convert! (scaledRealPacket_compact hb hg).mul_right (f' := scaledRealPacket b g) using 1 | |
| funext z | |
| exact pow_two _ | |
| lemma masterKernel_pole_measurable {c₁ r₀ s : ℝ} (hc : 0 < c₁) (hr : 0 < r₀) | |
| (hs : 0 < s) {g : Space → ℝ} (hg : Continuous g) (y : Space) : | |
| Measurable (fun x => tfPotential (masterKernel c₁ r₀ s g x) y) := by | |
| have hm : Measurable (fun p : Space × Space => masterKernel c₁ r₀ s g p.1 p.2/‖y-p.2‖) := | |
| (masterKernel_joint_continuous hc hr hs hg).measurable.div (by fun_prop) | |
| exact hm.stronglyMeasurable.integral_prod_right.measurable | |
| lemma master_potential_integrable {c₁ r₀ s : ℝ} (hc : 0 < c₁) (hr : 0 < r₀) (hs : 0 < s) | |
| {g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g) | |
| (hgn : ∫ z, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1) | |
| {ρ : Space → ℝ} (hmρ : Measurable ρ) (hρ : Integrable ρ) (hp : MemLp ρ (5/3)) (y : Space) : | |
| Integrable (fun x => ρ x*tfPotential (masterKernel c₁ r₀ s g x) y) := by | |
| apply (tfPotential_integrable hρ.norm hp.norm y).mono' | |
| (hmρ.mul (masterKernel_pole_measurable hc hr hs hg.continuous y)).aestronglyMeasurable | |
| filter_upwards [ae_ne_point y] with x hx | |
| change ‖ρ x*tfPotential (masterKernel c₁ r₀ s g x) y‖ ≤ _ | |
| rw [norm_mul,Real.norm_eq_abs (tfPotential _ _), | |
| abs_of_nonneg (masterKernel_potential_nonneg c₁ r₀ s g x y)] | |
| have hh := mul_le_mul_of_nonneg_left | |
| (masterKernel_potential_le hc hr hs hg hcg hgn hrad hgs hx) (norm_nonneg (ρ x)) | |
| simpa only [mul_one_div] using hh | |
| lemma master_smearing_pole_integrable {c₁ r₀ s : ℝ} (hc : 0 < c₁) (hr : 0 < r₀) (hs : 0 < s) | |
| {g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g) | |
| (hgn : ∫ z, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1) | |
| {ρ : Space → ℝ} (hmρ : Measurable ρ) (hρ : Integrable ρ) (hp : MemLp ρ (5/3)) (y : Space) : | |
| Integrable (fun p : Space × Space => ρ p.1*masterKernel c₁ r₀ s g p.1 p.2/‖y-p.2‖) := by | |
| have hm : Measurable (fun p : Space × Space => ρ p.1*masterKernel c₁ r₀ s g p.1 p.2/‖y-p.2‖) := | |
| ((hmρ.comp measurable_fst).mul (masterKernel_joint_continuous hc hr hs hg.continuous).measurable).div (by fun_prop) | |
| apply (integrable_prod_iff hm.aestronglyMeasurable).2 | |
| refine ⟨Eventually.of_forall fun x => ?_,?_⟩ | |
| · have hη := scaled_packet_density_compact (masterWidth_pos hc hr hs x) hcg | |
| have hηc := (scaledRealPacket_smooth (masterWidth c₁ r₀ s x) hg).continuous.pow 2 | |
| have hh := (translated_pole_integrable (hηc.integrable_of_hasCompactSupport hη) | |
| (hηc.memLp_of_hasCompactSupport hη) y x).const_mul (ρ x) | |
| convert! hh using 1 | |
| funext z | |
| simp only [masterKernel,Pi.pow_apply,mul_div_assoc] | |
| · have hi := master_potential_integrable hc hr hs hg hcg hgn hrad hgs hmρ.norm hρ.norm hp.norm y | |
| convert! hi using 1 | |
| funext x | |
| simp only [norm_div,norm_mul,norm_norm, | |
| Real.norm_of_nonneg (masterKernel_nonneg c₁ r₀ s g x _),mul_div_assoc,integral_const_mul,tfPotential] | |
| lemma masterSmearing_potential {c₁ r₀ s : ℝ} (hc : 0 < c₁) (hr : 0 < r₀) (hs : 0 < s) | |
| {g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g) | |
| (hgn : ∫ z, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1) | |
| {ρ : Space → ℝ} (hmρ : Measurable ρ) (hρ : Integrable ρ) (hp : MemLp ρ (5/3)) (y : Space) : | |
| tfPotential (masterSmearing c₁ r₀ s g ρ) y = | |
| ∫ x, ρ x*tfPotential (masterKernel c₁ r₀ s g x) y := by | |
| change (∫ z, (∫ x, ρ x*masterKernel c₁ r₀ s g x z)/‖y-z‖) = _ | |
| calc | |
| _ = ∫ z, ∫ x, ρ x*masterKernel c₁ r₀ s g x z/‖y-z‖ := by simp only [integral_div] | |
| _ = ∫ x, ∫ z, ρ x*masterKernel c₁ r₀ s g x z/‖y-z‖ := | |
| integral_integral_swap (master_smearing_pole_integrable hc hr hs hg hcg hgn hrad hgs hmρ hρ hp y).swap | |
| _ = _ := by simp only [mul_div_assoc,integral_const_mul,tfPotential] | |
| theorem masterSmearing_potential_loss {c₁ r₀ s : ℝ} | |
| (hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹) | |
| (hr : 0 < r₀) (hs : 0 < s) (hs1 : s ≤ 1) | |
| {g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g) | |
| (hgn : ∫ z, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1) | |
| {ρ : Space → ℝ} (hmρ : Measurable ρ) (hρ : Integrable ρ) (hp : MemLp ρ (5/3)) | |
| (hn : ∀ x, 0 ≤ ρ x) (y : Space) : | |
| 0 ≤ tfPotential ρ y-tfPotential (masterSmearing c₁ r₀ s g ρ) y ∧ | |
| tfPotential ρ y-tfPotential (masterSmearing c₁ r₀ s g ρ) y ≤ | |
| ∫ x in ball y (2*masterWidth c₁ r₀ s y), ρ x/‖y-x‖ := by | |
| rw [masterSmearing_potential hc hr hs hg hcg hgn hrad hgs hmρ hρ hp y] | |
| have hraw := tfPotential_integrable hρ hp y | |
| have hms := master_potential_integrable hc hr hs hg hcg hgn hrad hgs hmρ hρ hp y | |
| change 0 ≤ (∫ x, ρ x/‖y-x‖)-(∫ x, ρ x*tfPotential (masterKernel c₁ r₀ s g x) y) ∧ | |
| (∫ x, ρ x/‖y-x‖)-(∫ x, ρ x*tfPotential (masterKernel c₁ r₀ s g x) y) ≤ _ | |
| rw [←integral_sub hraw hms] | |
| rw [←integral_indicator measurableSet_ball] | |
| have hpoint : ∀ᵐ x, 0 ≤ ρ x/‖y-x‖-ρ x*tfPotential (masterKernel c₁ r₀ s g x) y ∧ | |
| ρ x/‖y-x‖-ρ x*tfPotential (masterKernel c₁ r₀ s g x) y ≤ | |
| (ball y (2*masterWidth c₁ r₀ s y)).indicator (fun x => ρ x/‖y-x‖) x := by | |
| filter_upwards [ae_ne_point y] with x hx | |
| obtain ⟨h0,h1⟩ := masterKernel_potential_loss hc hcL hr hs hs1 hg hcg hgn hrad hgs hx | |
| have h0' := mul_nonneg (hn x) h0 | |
| have h1' := mul_le_mul_of_nonneg_left h1 (hn x) | |
| by_cases hxB : x ∈ ball y (2*masterWidth c₁ r₀ s y) | |
| · rw [indicator_of_mem hxB] at h1' ⊢ | |
| constructor <;> nlinarith [show ρ x*(1/‖y-x‖) = ρ x/‖y-x‖ by ring] | |
| · rw [indicator_of_notMem hxB] at h1' ⊢ | |
| constructor <;> nlinarith [show ρ x*(1/‖y-x‖) = ρ x/‖y-x‖ by ring] | |
| exact ⟨integral_nonneg_of_ae (hpoint.mono fun _ hx => hx.1), | |
| integral_mono_ae (hraw.sub hms) (hraw.indicator measurableSet_ball) (hpoint.mono fun _ hx => hx.2)⟩ | |
| end CoulombAtom | |
| end | |
| end OAI | |