openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Variational /IntegratedLipschitzControl.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.RadialBounds.PatchFailureMass | |
| import OAI.Analysis.CoulombIonization.ThomasFermi.CoulombLipschitzDuality | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Set Metric | |
| open scoped BigOperators ENNReal NNReal ContDiff | |
| namespace CoulombAtom | |
| 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⟩ | |
| lemma patch_test_continuous (R : ℝ) {g : TFSpace → ℝ} | |
| (hg : Continuous g) (hcg : HasCompactSupport g) : | |
| Continuous (fun f : TFLp (ballMeasure R) => ∫ x, g x*f x ∂ballMeasure R) := by | |
| have hm : MemLp g (5/2) (ballMeasure R) := | |
| (hg.memLp_of_hasCompactSupport hcg).mono_measure Measure.restrict_le_self | |
| have he : (fun f : TFLp (ballMeasure R) => ∫ x, g x*f x ∂ballMeasure R) = | |
| tfPatchPair R (hm.toLp g) := by | |
| funext f | |
| rw [tfPatchPair_apply] | |
| exact integral_congr_ae (hm.coeFn_toLp.mono fun x hx => by dsimp only; rw [hx]) | |
| rw [he] | |
| exact (tfPatchPair R (hm.toLp g)).continuous | |
| def weightedPatchTestSquare {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)^2 | |
| lemma weightedPatchTestSquare_nonneg {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) : 0 ≤ weightedPatchTestSquare ψ s Z lam y R hb S g u := | |
| mul_nonneg (formMass_nonneg _) (sq_nonneg _) | |
| lemma weightedPatchTestSquare_le {N M : ℕ} (ψ : FormVector (N+M)) (s : Spins M) | |
| (Z lam : ℝ) (y : Space) {R r : ℝ} (hr : 0 < r) (hR : 2*r < R) | |
| {b : ℝ} (hb : 0 < b) (S : Configuration M → Finset (Fin M)) | |
| {g : TFSpace → ℝ} {L : ℝ≥0} (hg : LipschitzWith L g) | |
| (hs : Function.support g ⊆ closedBall 0 r) (u : Configuration M) : | |
| weightedPatchTestSquare ψ s Z lam y R hb S g u ≤ | |
| (16*r^3*(L:ℝ)^2)*weightedPatchGap ψ s Z lam y R hb S u := by | |
| have hh := mul_le_mul_of_nonneg_left | |
| (tfPatchGap_lipschitz_control hr hR tfKinetic tfKinetic_pos | |
| (conditionalPatchField ψ s Z lam y R u) (retainedPatchLp_nonneg hb (S u) u y R) hg hs) | |
| (formMass_nonneg (coreSlice ψ s u)) | |
| simpa only [weightedPatchTestSquare,weightedPatchGap,mul_left_comm,mul_assoc] using hh | |
| lemma weightedPatchTestSquare_aemeasurable {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} | |
| (hg : Continuous g) (hcg : HasCompactSupport g) : | |
| AEMeasurable (weightedPatchTestSquare ψ s Z lam y R hb S g) := by | |
| have hm := (hψ.coreSlice_mass_integrable s).aestronglyMeasurable.aemeasurable | |
| 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 | |
| have hp := (patch_test_continuous R hg hcg).measurable.comp_aemeasurable | |
| (hs.aemeasurable.sub hr) | |
| exact hm.mul (hp.pow_const 2) | |
| lemma weightedPatchTestSquare_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 : 2*r < R) | |
| {b : ℝ} (hb : 0 < b) (S : Configuration M → Finset (Fin M)) | |
| (hS : ∀ i, MeasurableSet {u | i ∈ S u}) {g : TFSpace → ℝ} {L : ℝ≥0} | |
| (hg : LipschitzWith L g) (hs : Function.support g ⊆ closedBall 0 r) | |
| (hi : Integrable (weightedPatchGap ψ s Z lam y R hb S)) : | |
| Integrable (weightedPatchTestSquare ψ s Z lam y R hb S g) := by | |
| have hcg : HasCompactSupport g := | |
| (isCompact_closedBall (0 : TFSpace) r).of_isClosed_subset isClosed_closure | |
| (closure_minimal hs isClosed_closedBall) | |
| apply (hi.const_mul (16*r^3*(L:ℝ)^2)).mono' | |
| (weightedPatchTestSquare_aemeasurable hψ s Z lam y R hb S hS hg.continuous hcg).aestronglyMeasurable | |
| exact Filter.Eventually.of_forall fun u => by | |
| rw [Real.norm_eq_abs,abs_of_nonneg (weightedPatchTestSquare_nonneg ψ s Z lam y R hb S g u)] | |
| exact weightedPatchTestSquare_le ψ s Z lam y hr hR hb S hg hs u | |
| lemma integral_weightedPatchTestSquare_le {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 : 2*r < R) | |
| {b : ℝ} (hb : 0 < b) (S : Configuration M → Finset (Fin M)) | |
| (hS : ∀ i, MeasurableSet {u | i ∈ S u}) {g : TFSpace → ℝ} {L : ℝ≥0} | |
| (hg : LipschitzWith L g) (hs : Function.support g ⊆ closedBall 0 r) | |
| (hi : Integrable (weightedPatchGap ψ s Z lam y R hb S)) : | |
| (∫ u, weightedPatchTestSquare ψ s Z lam y R hb S g u) ≤ | |
| (16*r^3*(L:ℝ)^2)*(∫ u, weightedPatchGap ψ s Z lam y R hb S u) := by | |
| rw [←integral_const_mul] | |
| exact integral_mono (weightedPatchTestSquare_integrable hψ s Z lam y hr hR hb S hS hg hs hi) | |
| (hi.const_mul _) (weightedPatchTestSquare_le ψ s Z lam y hr hR hb S hg hs) | |
| theorem radial_lipschitz_test_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) | |
| {q : TFSpace → ℝ} {L : ℝ≥0} (hq : LipschitzWith L q) | |
| {r : ℝ} (hr : 0 < r) (hR : 2*r < t-4*b) | |
| (hs : Function.support q ⊆ closedBall 0 r) : | |
| (∑ c : Fin N → Fin 2, ∑ s : Spins (cutOutNumber c), ∫ u, | |
| weightedPatchTestSquare (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) q u) ≤ | |
| (16*r^3*(L:ℝ)^2)* | |
| (∑ 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 | |
| simp only [Finset.mul_sum] | |
| apply Finset.sum_le_sum; intro c _ | |
| apply Finset.sum_le_sum; intro s _ | |
| exact integral_weightedPatchTestSquare_le | |
| (orderedCutForm_sobolev (coreFirstRadialCut y ht hb) | |
| (coreFirstRadialCut_partition y ht hb) hψ.sobolevVector c) | |
| s Z lam y hr hR hb _ (radialPatchRetention_measurable N y t b c s) hq hs | |
| (radial_weightedPatchGap_integrable hψ y ht hb htb hy hZ hlam hg hcg hgn hrad hgs c s) | |
| end CoulombAtom | |
| end | |
| end OAI | |