openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Fermionic /QuantumEventMultiplier.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Fermionic.Multiplier2 | |
| import OAI.Analysis.CoulombIonization.Localization.ObservationIntegral | |
| import OAI.Analysis.CoulombIonization.Localization.ArraySymmetry | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Set Finset | |
| open scoped ENNReal NNReal BigOperators | |
| namespace CoulombAtom | |
| open CoulombObservation | |
| def quantumEventMultiplier {N : ℕ} {J : Type*} [Fintype J] | |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) | |
| (hsy : QuantumEventSymmetric s) (p : ℝ) : FermionLipschitzMultiplier N where | |
| value x := (Real.sqrt p)⁻¹ * Real.sqrt (quantumEventLikelihood b s x) | |
| constant := ‖(Real.sqrt p)⁻¹‖₊ * | |
| (sqrtLikelihoodLipschitzConstant (I := Fin N × Fin 3) b * ‖flattenConfiguration N‖₊) | |
| lipschitz := (lipschitzWith_smul ((Real.sqrt p)⁻¹)).comp | |
| ((sqrtArrayLikelihood_lipschitz b hs).comp (flattenConfiguration N).lipschitzWith) | |
| bound := ⟨(Real.sqrt p)⁻¹, fun x => by | |
| rw [abs_of_nonneg (by positivity)] | |
| calc | |
| _ ≤ (Real.sqrt p)⁻¹ * 1 := mul_le_mul_of_nonneg_left | |
| (by simpa using Real.sqrt_le_sqrt (quantumEventLikelihood_le_one b hs x)) | |
| (by positivity) | |
| _ = _ := mul_one _⟩ | |
| symmetric π x := by rw [quantumEventLikelihood_symmetric b hs hsy π x] | |
| @[simp] lemma quantumEventMultiplier_value {N : ℕ} {J : Type*} [Fintype J] | |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) | |
| (hsy : QuantumEventSymmetric s) (p : ℝ) (x : Configuration N) : | |
| (quantumEventMultiplier b hs hsy p).value x = | |
| (Real.sqrt p)⁻¹ * Real.sqrt (quantumEventLikelihood b s x) := rfl | |
| lemma quantumEventMultiplier_sq {N : ℕ} {J : Type*} [Fintype J] | |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) | |
| (hsy : QuantumEventSymmetric s) {p : ℝ} (hp : 0 < p) (x : Configuration N) : | |
| (quantumEventMultiplier b hs hsy p).value x ^ 2 = | |
| p⁻¹ * quantumEventLikelihood b s x := by | |
| rw [quantumEventMultiplier_value, mul_pow, inv_pow, Real.sq_sqrt hp.le, | |
| Real.sq_sqrt (quantumEventLikelihood_nonneg b hs x)] | |
| lemma quantumEventMultiplier_lineDeriv {N : ℕ} {J : Type*} [Fintype J] | |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) | |
| (hsy : QuantumEventSymmetric s) (p : ℝ) (x : Configuration N) (i : Fin N) (a : Fin 3) : | |
| lineDeriv ℝ (quantumEventMultiplier b hs hsy p).value x (direction i a) = | |
| (Real.sqrt p)⁻¹ * lineDeriv ℝ (fun y => Real.sqrt (arrayLikelihood b s y)) | |
| (flattenConfiguration N x) (Pi.single (i,a) 1) := by | |
| change deriv (fun t => (Real.sqrt p)⁻¹ * | |
| Real.sqrt (quantumEventLikelihood b s (x + t • direction i a))) 0 = _ | |
| rw [deriv_const_mul_field] | |
| exact congrArg ((Real.sqrt p)⁻¹ * ·) | |
| (lineDeriv_flattenConfiguration (fun y => Real.sqrt (arrayLikelihood b s y)) x i a) | |
| lemma quantumEventMultiplier_fisher {N : ℕ} {J : Type*} [Fintype J] | |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) | |
| (hsy : QuantumEventSymmetric s) {p : ℝ} (hp : 0 < p) (x : Configuration N) : | |
| (∑ q : Fin N × Fin 3, | |
| (lineDeriv ℝ (quantumEventMultiplier b hs hsy p).value x (direction q.1 q.2))^2) = | |
| p⁻¹ * arrayFisher b s (flattenConfiguration N x) := by | |
| simp only [quantumEventMultiplier_lineDeriv, mul_pow, inv_pow, Real.sq_sqrt hp.le, | |
| arrayFisher, mul_sum] | |
| lemma quantumEventMultiplier_fisher_integral {N : ℕ} {J : Type*} [Fintype J] | |
| (F : fermionGraph N) (hn : ‖fermionGraphValue N F‖^2 = 1) | |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) | |
| (hsy : QuantumEventSymmetric s) (hp : 0 < quantumEventProbability F b s) : | |
| (∫ x, ∑ q : Fin N × Fin 3, | |
| (lineDeriv ℝ (quantumEventMultiplier b hs hsy (quantumEventProbability F b s)).value x | |
| (direction q.1 q.2))^2 ∂graphRawLaw F) ≤ | |
| observationFisherConstant * (∑ j, (b j)^2) * | |
| (Real.log (Real.exp 1/quantumEventProbability F b s))^5 := by | |
| let := flatRawLaw_probability F hn | |
| simp only [quantumEventMultiplier_fisher b hs hsy hp] | |
| rw [integral_const_mul, ← integral_map (flattenConfiguration N).measurable.aemeasurable | |
| (arrayFisher_measurable b hs).aestronglyMeasurable] | |
| have hp' := hp | |
| rw [quantumEventProbability_eq_flat F b hs] at hp' | |
| have hh := arrayFisher_integral_log b hs (flatRawLaw F) hp' | |
| rw [← quantumEventProbability_eq_flat F b hs] at hh | |
| calc | |
| _ ≤ (quantumEventProbability F b s)⁻¹ * | |
| (observationFisherConstant * (∑ j, (b j)^2) * | |
| (Real.log (Real.exp 1/quantumEventProbability F b s))^5 * | |
| quantumEventProbability F b s) := mul_le_mul_of_nonneg_left hh (by positivity) | |
| _ = _ := by field_simp | |
| end CoulombAtom | |
| end | |
| end OAI | |