Datasets:

roseking
/

openAI-math-dataset

import OAI.Probability.FreeIsing.Gaussian.PiIntegrationByParts
import OAI.Probability.FreeIsing.Posterior.GaussianGradient
import OAI.Probability.FreeIsing.Posterior.GaussianTesting

namespace OAI

noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal

namespace Problem367.FinitePosterior

variable {S V : Type*} [Fintype S] [Nonempty S] [Fintype V] [DecidableEq V]

theorem integral_fieldSquareGradient_mul_noise
    (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
    (A : ℝ) (hbound : ∀ s, |observable s| ≤ A)
    (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1)
    (signal : V → ℝ) (t : ℝ) (v : V) :
    (∫ z, fieldSquareGradient energy spin observable
      (gaussianObservationCurve signal z t) v * z v
      ∂Measure.pi (fun _ : V => gaussianReal 0 1)) =
    Real.sqrt t * ∫ z, fieldSquareHessianDiagonal energy spin observable
      (gaussianObservationCurve signal z t) v
      ∂Measure.pi (fun _ : V => gaussianReal 0 1) := by
  have h := Problem367.gaussian_pi_integral_mul_affine_of_bounded_deriv
    (fun j => t * signal j) (fun _ => Real.sqrt t) v
    (continuous_fieldSquareGradient energy spin observable v)
    (continuous_fieldSquareHessianDiagonal energy spin observable v)
    (fun z r => hasDerivAt_fieldSquareGradient_update energy spin observable z v r
      (fun s => hcoords s v))
    (fun z => abs_fieldSquareGradient_le energy spin observable z v A
      (fun s => hcoords s v) hbound)
    (fun z => abs_fieldSquareHessianDiagonal_le energy spin observable z v A
      (fun s => hcoords s v) hbound)
  simpa only [gaussianObservationCurve, Pi.add_def, Pi.smul_def, smul_eq_mul,
    mul_comm] using h

section Planted

variable [MeasurableSpace S] [MeasurableSingletonClass S]

def finiteStandardObservation (spin : S → V → ℝ) (t : ℝ)
    (x : S × (V → ℝ)) : V → ℝ := gaussianObservationCurve (spin x.1) x.2 t

omit [Nonempty S] [Fintype V] [DecidableEq V] in
lemma measurable_finiteStandardObservation (spin : S → V → ℝ) (t : ℝ) :
    Measurable (finiteStandardObservation spin t) := by
  unfold finiteStandardObservation gaussianObservationCurve
  exact ((measurable_of_finite spin).comp measurable_fst).const_smul t |>.add
    (measurable_snd.const_smul (Real.sqrt t))

omit [DecidableEq V] in
private theorem integrable_bounded_test
    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
    (X : Ω → V → ℝ) (hX : Measurable X) (f : (V → ℝ) → ℝ)
    (hf : Continuous f) (C : ℝ) (hbound : ∀ z, |f z| ≤ C) :
    Integrable (fun x => f (X x)) μ := by
  apply Integrable.of_bound (hf.measurable.comp hX).aestronglyMeasurable C
  exact Filter.Eventually.of_forall fun x => by
    simpa only [Real.norm_eq_abs, Function.comp_def] using hbound (X x)

omit [Fintype S] [Nonempty S] [DecidableEq V] [MeasurableSingletonClass S] in

theorem integrable_finiteStandardNoise (μ : Measure S) [IsFiniteMeasure μ] (v : V) :
    Integrable (fun x : S × (V → ℝ) => x.2 v)
      (μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) := by
  have h := (measurePreserving_eval (fun _ : V => gaussianReal 0 1) v).integrable_comp_of_integrable ((memLp_id_gaussianReal 1).integrable (by norm_num))
  exact h.comp_snd μ

omit [DecidableEq V] in
theorem integrable_finiteStandardGradient (μ : Measure S) [IsFiniteMeasure μ]
    (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
    (A : ℝ) (hbound : ∀ s, |observable s| ≤ A)
    (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
    Integrable (fun x => fieldSquareGradient energy spin observable
      (finiteStandardObservation spin t x) v)
      (μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :=
  integrable_bounded_test _ _ (measurable_finiteStandardObservation spin t) _
    (continuous_fieldSquareGradient energy spin observable v) (4 * A ^ 2)
    (fun z => abs_fieldSquareGradient_le energy spin observable z v A
      (fun s => hcoords s v) hbound)

theorem integrable_finiteStandardHessian (μ : Measure S) [IsFiniteMeasure μ]
    (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
    (A : ℝ) (hbound : ∀ s, |observable s| ≤ A)
    (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
    Integrable (fun x => fieldSquareHessianDiagonal energy spin observable
      (finiteStandardObservation spin t x) v)
      (μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :=
  integrable_bounded_test _ _ (measurable_finiteStandardObservation spin t) _
    (continuous_fieldSquareHessianDiagonal energy spin observable v) (16 * A ^ 2)
    (fun z => abs_fieldSquareHessianDiagonal_le energy spin observable z v A
      (fun s => hcoords s v) hbound)

omit [DecidableEq V] in
theorem integrable_finiteStandardGradient_mul_noise
    (μ : Measure S) [IsFiniteMeasure μ]
    (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
    (A : ℝ) (hbound : ∀ s, |observable s| ≤ A)
    (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
    Integrable (fun x => fieldSquareGradient energy spin observable
      (finiteStandardObservation spin t x) v * x.2 v)
      (μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) := by
  have hg := integrable_finiteStandardGradient μ energy spin observable A hbound hcoords t v
  have h := (integrable_finiteStandardNoise μ v).mul_bdd hg.aestronglyMeasurable
    (Filter.Eventually.of_forall fun x => by
      simpa only [Real.norm_eq_abs, Function.comp_def] using abs_fieldSquareGradient_le energy spin observable
        (finiteStandardObservation spin t x) v A (fun s => hcoords s v) hbound)
  simpa only [mul_comm] using h

theorem integral_finiteStandardGradient_mul_noise
    (μ : Measure S) [IsFiniteMeasure μ]
    (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
    (A : ℝ) (hbound : ∀ s, |observable s| ≤ A)
    (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
    (∫ x, fieldSquareGradient energy spin observable
      (finiteStandardObservation spin t x) v * x.2 v
      ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) =
    Real.sqrt t * ∫ x, fieldSquareHessianDiagonal energy spin observable
      (finiteStandardObservation spin t x) v
      ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)) := by
  rw [integral_prod _ (integrable_finiteStandardGradient_mul_noise
    μ energy spin observable A hbound hcoords t v),
    integral_prod _ (integrable_finiteStandardHessian
    μ energy spin observable A hbound hcoords t v), ← integral_const_mul]
  apply integral_congr_ae
  exact Filter.Eventually.of_forall fun s =>
    integral_fieldSquareGradient_mul_noise energy spin observable A hbound hcoords (spin s) t v

theorem hasDerivAt_finiteStandardEnergy_of_posterior_test
    (μ : Measure S) [IsProbabilityMeasure μ]
    (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
    (A : ℝ) (hbound : ∀ s, |observable s| ≤ A)
    (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1)
    {t : ℝ} (ht : 0 < t)
    (hposterior : ∀ v,
      (∫ x, fieldSquareGradient energy spin observable
        (finiteStandardObservation spin t x) v * spin x.1 v
        ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) =
      ∫ x, fieldSquareGradient energy spin observable
        (finiteStandardObservation spin t x) v *
          fieldMean energy spin (fun s => spin s v) (finiteStandardObservation spin t x)
        ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :
    HasDerivAt
      (fun r => ∫ x, fieldMean energy spin observable (finiteStandardObservation spin r x) ^ 2
        ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)))
      (∫ x, ∑ v, fieldCovariance energy spin observable
        (finiteStandardObservation spin t x) v ^ 2
        ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) t := by
  let ρ := μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))
  let Y := finiteStandardObservation spin t
  let g := fun x v => fieldSquareGradient energy spin observable (Y x) v
  let H := fun x v => fieldSquareHessianDiagonal energy spin observable (Y x) v
  let p := fun x v => fieldMean energy spin (fun s => spin s v) (Y x)
  let c := 1 / (2 * Real.sqrt t)
  have hy : Measurable Y := measurable_finiteStandardObservation spin t
  have hg (v : V) : Integrable (fun x => g x v) ρ :=
    integrable_finiteStandardGradient μ energy spin observable A hbound hcoords t v
  have hH (v : V) : Integrable (fun x => H x v) ρ :=
    integrable_finiteStandardHessian μ energy spin observable A hbound hcoords t v
  have hn (v : V) : Integrable (fun x => g x v * x.2 v) ρ :=
    integrable_finiteStandardGradient_mul_noise μ energy spin observable A hbound hcoords t v
  have hs (v : V) : Integrable (fun x => g x v * spin x.1 v) ρ := by
    apply (hg v).mul_bdd
      (((measurable_of_finite (fun s => spin s v)).comp measurable_fst).aestronglyMeasurable)
    exact Filter.Eventually.of_forall fun x => by
      simpa only [Real.norm_eq_abs, Function.comp_def] using
        (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) x.1).le
  have hp (v : V) : Integrable (fun x => g x v * p x v) ρ := by
    apply (hg v).mul_bdd
      (((continuous_fieldMean energy spin (fun s => spin s v)).measurable.comp hy).aestronglyMeasurable)
    exact Filter.Eventually.of_forall fun x => by
      simpa only [Real.norm_eq_abs, Function.comp_def] using abs_fieldMean_le energy spin (fun s => spin s v)
        (Y x) 1 (fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le)
  have hvEq (v : V) :
      (fun x : S × (V → ℝ) => g x v * gaussianObservationVelocity (spin x.1) x.2 t v) =
        (fun x => g x v * spin x.1 v + c * (g x v * x.2 v)) := by
    funext x
    change g x v * (spin x.1 v + c * x.2 v) = _
    ring
  have hv (v : V) : Integrable
      (fun x => g x v * gaussianObservationVelocity (spin x.1) x.2 t v) ρ := by
    rw [hvEq]
    exact (hs v).add ((hn v).const_mul c)
  have hcancel (x : S × (V → ℝ)) (v : V) :
      g x v * p x v + (1 / 2 : ℝ) * H x v =
        fieldCovariance energy spin observable (Y x) v ^ 2 := by
    dsimp [g, H, p, fieldSquareGradient, fieldSquareHessianDiagonal]
    ring
  have hc (v : V) : Integrable
      (fun x => fieldCovariance energy spin observable (Y x) v ^ 2) ρ := by
    exact ((hp v).add ((hH v).const_mul (1 / 2 : ℝ))).congr
      (Filter.Eventually.of_forall fun x => hcancel x v)
  have hstep (v : V) :
      (∫ x, g x v * gaussianObservationVelocity (spin x.1) x.2 t v ∂ρ) =
        ∫ x, fieldCovariance energy spin observable (Y x) v ^ 2 ∂ρ := by
    rw [hvEq, integral_add (hs v) ((hn v).const_mul c), integral_const_mul]
    have hnoise : (∫ x, g x v * x.2 v ∂ρ) = Real.sqrt t * ∫ x, H x v ∂ρ :=
      integral_finiteStandardGradient_mul_noise μ energy spin observable A hbound hcoords t v
    rw [hnoise]
    have hcoef : c * Real.sqrt t = (1 / 2 : ℝ) := by
      dsimp [c]
      field_simp [ne_of_gt (Real.sqrt_pos.mpr ht)]
    rw [← mul_assoc, hcoef, hposterior v]
    rw [← integral_const_mul, ← integral_add (hp v) ((hH v).const_mul (1 / 2 : ℝ))]
    exact integral_congr_ae (Filter.Eventually.of_forall fun x => hcancel x v)
  have hd := hasDerivAt_integral_fieldMean_square_observation ρ energy spin observable A
    hbound hcoords (fun x : S × (V → ℝ) => spin x.1) Prod.snd
    ((measurable_of_finite spin).comp measurable_fst) measurable_snd
    (fun x v => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) x.1).le)
    (fun v => integrable_finiteStandardNoise μ v) ht
  apply hd.congr_deriv
  change (∫ x, ∑ v, g x v * gaussianObservationVelocity (spin x.1) x.2 t v ∂ρ) = _
  rw [integral_finsetSum _ (fun v _ => hv v), integral_finsetSum _ (fun v _ => hc v)]
  exact Finset.sum_congr rfl fun v _ => hstep v

theorem hasDerivAt_finiteStandardEnergy [StandardBorelSpace S]
    (μ : Measure S) [IsProbabilityMeasure μ] (energy : S → ℝ)
    (hprior : ∀ s, μ.real {s} = Real.exp (energy s) / ∑ a, Real.exp (energy a))
    (spin : S → V → ℝ) (observable : S → ℝ)
    (A : ℝ) (hbound : ∀ s, |observable s| ≤ A)
    (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1)
    {t : ℝ} (ht : 0 < t) :
    HasDerivAt
      (fun r => ∫ x, fieldMean energy spin observable (finiteStandardObservation spin r x) ^ 2
        ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)))
      (∫ x, ∑ v, fieldCovariance energy spin observable
        (finiteStandardObservation spin t x) v ^ 2
        ∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) t := by
  apply hasDerivAt_finiteStandardEnergy_of_posterior_test μ energy spin observable A hbound hcoords ht
  intro v
  have hsq : ∀ s i, spin s i ^ 2 = 1 := by
    intro s i
    rcases hcoords s i with h | h <;> simp [h]
  exact Problem367.finiteStandardGaussianPlanted_integral_test_fieldMean μ energy hprior t ht.le
    spin hsq (fun s => spin s v) (fun z => fieldSquareGradient energy spin observable z v)
    (continuous_fieldSquareGradient energy spin observable v).measurable (4 * A ^ 2)
    (fun z => by simpa only [Real.norm_eq_abs] using
      abs_fieldSquareGradient_le energy spin observable z v A (fun s => hcoords s v) hbound)

end Planted
end Problem367.FinitePosterior

end

end OAI

	import OAI.Probability.FreeIsing.Gaussian.PiIntegrationByParts
	import OAI.Probability.FreeIsing.Posterior.GaussianGradient
	import OAI.Probability.FreeIsing.Posterior.GaussianTesting

	namespace OAI

	noncomputable section
	open MeasureTheory ProbabilityTheory
	open scoped NNReal

	namespace Problem367.FinitePosterior

	variable {S V : Type*} [Fintype S] [Nonempty S] [Fintype V] [DecidableEq V]

	theorem integral_fieldSquareGradient_mul_noise
	(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
	(A : ℝ) (hbound : ∀ s, \|observable s\| ≤ A)
	(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1)
	(signal : V → ℝ) (t : ℝ) (v : V) :
	(∫ z, fieldSquareGradient energy spin observable
	(gaussianObservationCurve signal z t) v * z v
	∂Measure.pi (fun _ : V => gaussianReal 0 1)) =
	Real.sqrt t * ∫ z, fieldSquareHessianDiagonal energy spin observable
	(gaussianObservationCurve signal z t) v
	∂Measure.pi (fun _ : V => gaussianReal 0 1) := by
	have h := Problem367.gaussian_pi_integral_mul_affine_of_bounded_deriv
	(fun j => t * signal j) (fun _ => Real.sqrt t) v
	(continuous_fieldSquareGradient energy spin observable v)
	(continuous_fieldSquareHessianDiagonal energy spin observable v)
	(fun z r => hasDerivAt_fieldSquareGradient_update energy spin observable z v r
	(fun s => hcoords s v))
	(fun z => abs_fieldSquareGradient_le energy spin observable z v A
	(fun s => hcoords s v) hbound)
	(fun z => abs_fieldSquareHessianDiagonal_le energy spin observable z v A
	(fun s => hcoords s v) hbound)
	simpa only [gaussianObservationCurve, Pi.add_def, Pi.smul_def, smul_eq_mul,
	mul_comm] using h

	section Planted

	variable [MeasurableSpace S] [MeasurableSingletonClass S]

	def finiteStandardObservation (spin : S → V → ℝ) (t : ℝ)
	(x : S × (V → ℝ)) : V → ℝ := gaussianObservationCurve (spin x.1) x.2 t

	omit [Nonempty S] [Fintype V] [DecidableEq V] in
	lemma measurable_finiteStandardObservation (spin : S → V → ℝ) (t : ℝ) :
	Measurable (finiteStandardObservation spin t) := by
	unfold finiteStandardObservation gaussianObservationCurve
	exact ((measurable_of_finite spin).comp measurable_fst).const_smul t \|>.add
	(measurable_snd.const_smul (Real.sqrt t))

	omit [DecidableEq V] in
	private theorem integrable_bounded_test
	{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
	(X : Ω → V → ℝ) (hX : Measurable X) (f : (V → ℝ) → ℝ)
	(hf : Continuous f) (C : ℝ) (hbound : ∀ z, \|f z\| ≤ C) :
	Integrable (fun x => f (X x)) μ := by
	apply Integrable.of_bound (hf.measurable.comp hX).aestronglyMeasurable C
	exact Filter.Eventually.of_forall fun x => by
	simpa only [Real.norm_eq_abs, Function.comp_def] using hbound (X x)

	omit [Fintype S] [Nonempty S] [DecidableEq V] [MeasurableSingletonClass S] in

	theorem integrable_finiteStandardNoise (μ : Measure S) [IsFiniteMeasure μ] (v : V) :
	Integrable (fun x : S × (V → ℝ) => x.2 v)
	(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) := by
	have h := (measurePreserving_eval (fun _ : V => gaussianReal 0 1) v).integrable_comp_of_integrable ((memLp_id_gaussianReal 1).integrable (by norm_num))
	exact h.comp_snd μ

	omit [DecidableEq V] in
	theorem integrable_finiteStandardGradient (μ : Measure S) [IsFiniteMeasure μ]
	(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
	(A : ℝ) (hbound : ∀ s, \|observable s\| ≤ A)
	(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
	Integrable (fun x => fieldSquareGradient energy spin observable
	(finiteStandardObservation spin t x) v)
	(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :=
	integrable_bounded_test _ _ (measurable_finiteStandardObservation spin t) _
	(continuous_fieldSquareGradient energy spin observable v) (4 * A ^ 2)
	(fun z => abs_fieldSquareGradient_le energy spin observable z v A
	(fun s => hcoords s v) hbound)

	theorem integrable_finiteStandardHessian (μ : Measure S) [IsFiniteMeasure μ]
	(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
	(A : ℝ) (hbound : ∀ s, \|observable s\| ≤ A)
	(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
	Integrable (fun x => fieldSquareHessianDiagonal energy spin observable
	(finiteStandardObservation spin t x) v)
	(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :=
	integrable_bounded_test _ _ (measurable_finiteStandardObservation spin t) _
	(continuous_fieldSquareHessianDiagonal energy spin observable v) (16 * A ^ 2)
	(fun z => abs_fieldSquareHessianDiagonal_le energy spin observable z v A
	(fun s => hcoords s v) hbound)

	omit [DecidableEq V] in
	theorem integrable_finiteStandardGradient_mul_noise
	(μ : Measure S) [IsFiniteMeasure μ]
	(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
	(A : ℝ) (hbound : ∀ s, \|observable s\| ≤ A)
	(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
	Integrable (fun x => fieldSquareGradient energy spin observable
	(finiteStandardObservation spin t x) v * x.2 v)
	(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) := by
	have hg := integrable_finiteStandardGradient μ energy spin observable A hbound hcoords t v
	have h := (integrable_finiteStandardNoise μ v).mul_bdd hg.aestronglyMeasurable
	(Filter.Eventually.of_forall fun x => by
	simpa only [Real.norm_eq_abs, Function.comp_def] using abs_fieldSquareGradient_le energy spin observable
	(finiteStandardObservation spin t x) v A (fun s => hcoords s v) hbound)
	simpa only [mul_comm] using h

	theorem integral_finiteStandardGradient_mul_noise
	(μ : Measure S) [IsFiniteMeasure μ]
	(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
	(A : ℝ) (hbound : ∀ s, \|observable s\| ≤ A)
	(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
	(∫ x, fieldSquareGradient energy spin observable
	(finiteStandardObservation spin t x) v * x.2 v
	∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) =
	Real.sqrt t * ∫ x, fieldSquareHessianDiagonal energy spin observable
	(finiteStandardObservation spin t x) v
	∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)) := by
	rw [integral_prod _ (integrable_finiteStandardGradient_mul_noise
	μ energy spin observable A hbound hcoords t v),
	integral_prod _ (integrable_finiteStandardHessian
	μ energy spin observable A hbound hcoords t v), ← integral_const_mul]
	apply integral_congr_ae
	exact Filter.Eventually.of_forall fun s =>
	integral_fieldSquareGradient_mul_noise energy spin observable A hbound hcoords (spin s) t v

	theorem hasDerivAt_finiteStandardEnergy_of_posterior_test
	(μ : Measure S) [IsProbabilityMeasure μ]
	(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
	(A : ℝ) (hbound : ∀ s, \|observable s\| ≤ A)
	(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1)
	{t : ℝ} (ht : 0 < t)
	(hposterior : ∀ v,
	(∫ x, fieldSquareGradient energy spin observable
	(finiteStandardObservation spin t x) v * spin x.1 v
	∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) =
	∫ x, fieldSquareGradient energy spin observable
	(finiteStandardObservation spin t x) v *
	fieldMean energy spin (fun s => spin s v) (finiteStandardObservation spin t x)
	∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :
	HasDerivAt
	(fun r => ∫ x, fieldMean energy spin observable (finiteStandardObservation spin r x) ^ 2
	∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)))
	(∫ x, ∑ v, fieldCovariance energy spin observable
	(finiteStandardObservation spin t x) v ^ 2
	∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) t := by
	let ρ := μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))
	let Y := finiteStandardObservation spin t
	let g := fun x v => fieldSquareGradient energy spin observable (Y x) v
	let H := fun x v => fieldSquareHessianDiagonal energy spin observable (Y x) v
	let p := fun x v => fieldMean energy spin (fun s => spin s v) (Y x)
	let c := 1 / (2 * Real.sqrt t)
	have hy : Measurable Y := measurable_finiteStandardObservation spin t
	have hg (v : V) : Integrable (fun x => g x v) ρ :=
	integrable_finiteStandardGradient μ energy spin observable A hbound hcoords t v
	have hH (v : V) : Integrable (fun x => H x v) ρ :=
	integrable_finiteStandardHessian μ energy spin observable A hbound hcoords t v
	have hn (v : V) : Integrable (fun x => g x v * x.2 v) ρ :=
	integrable_finiteStandardGradient_mul_noise μ energy spin observable A hbound hcoords t v
	have hs (v : V) : Integrable (fun x => g x v * spin x.1 v) ρ := by
	apply (hg v).mul_bdd
	(((measurable_of_finite (fun s => spin s v)).comp measurable_fst).aestronglyMeasurable)
	exact Filter.Eventually.of_forall fun x => by
	simpa only [Real.norm_eq_abs, Function.comp_def] using
	(abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) x.1).le
	have hp (v : V) : Integrable (fun x => g x v * p x v) ρ := by
	apply (hg v).mul_bdd
	(((continuous_fieldMean energy spin (fun s => spin s v)).measurable.comp hy).aestronglyMeasurable)
	exact Filter.Eventually.of_forall fun x => by
	simpa only [Real.norm_eq_abs, Function.comp_def] using abs_fieldMean_le energy spin (fun s => spin s v)
	(Y x) 1 (fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le)
	have hvEq (v : V) :
	(fun x : S × (V → ℝ) => g x v * gaussianObservationVelocity (spin x.1) x.2 t v) =
	(fun x => g x v * spin x.1 v + c * (g x v * x.2 v)) := by
	funext x
	change g x v * (spin x.1 v + c * x.2 v) = _
	ring
	have hv (v : V) : Integrable
	(fun x => g x v * gaussianObservationVelocity (spin x.1) x.2 t v) ρ := by
	rw [hvEq]
	exact (hs v).add ((hn v).const_mul c)
	have hcancel (x : S × (V → ℝ)) (v : V) :
	g x v * p x v + (1 / 2 : ℝ) * H x v =
	fieldCovariance energy spin observable (Y x) v ^ 2 := by
	dsimp [g, H, p, fieldSquareGradient, fieldSquareHessianDiagonal]
	ring
	have hc (v : V) : Integrable
	(fun x => fieldCovariance energy spin observable (Y x) v ^ 2) ρ := by
	exact ((hp v).add ((hH v).const_mul (1 / 2 : ℝ))).congr
	(Filter.Eventually.of_forall fun x => hcancel x v)
	have hstep (v : V) :
	(∫ x, g x v * gaussianObservationVelocity (spin x.1) x.2 t v ∂ρ) =
	∫ x, fieldCovariance energy spin observable (Y x) v ^ 2 ∂ρ := by
	rw [hvEq, integral_add (hs v) ((hn v).const_mul c), integral_const_mul]
	have hnoise : (∫ x, g x v * x.2 v ∂ρ) = Real.sqrt t * ∫ x, H x v ∂ρ :=
	integral_finiteStandardGradient_mul_noise μ energy spin observable A hbound hcoords t v
	rw [hnoise]
	have hcoef : c * Real.sqrt t = (1 / 2 : ℝ) := by
	dsimp [c]
	field_simp [ne_of_gt (Real.sqrt_pos.mpr ht)]
	rw [← mul_assoc, hcoef, hposterior v]
	rw [← integral_const_mul, ← integral_add (hp v) ((hH v).const_mul (1 / 2 : ℝ))]
	exact integral_congr_ae (Filter.Eventually.of_forall fun x => hcancel x v)
	have hd := hasDerivAt_integral_fieldMean_square_observation ρ energy spin observable A
	hbound hcoords (fun x : S × (V → ℝ) => spin x.1) Prod.snd
	((measurable_of_finite spin).comp measurable_fst) measurable_snd
	(fun x v => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) x.1).le)
	(fun v => integrable_finiteStandardNoise μ v) ht
	apply hd.congr_deriv
	change (∫ x, ∑ v, g x v * gaussianObservationVelocity (spin x.1) x.2 t v ∂ρ) = _
	rw [integral_finsetSum _ (fun v _ => hv v), integral_finsetSum _ (fun v _ => hc v)]
	exact Finset.sum_congr rfl fun v _ => hstep v

	theorem hasDerivAt_finiteStandardEnergy [StandardBorelSpace S]
	(μ : Measure S) [IsProbabilityMeasure μ] (energy : S → ℝ)
	(hprior : ∀ s, μ.real {s} = Real.exp (energy s) / ∑ a, Real.exp (energy a))
	(spin : S → V → ℝ) (observable : S → ℝ)
	(A : ℝ) (hbound : ∀ s, \|observable s\| ≤ A)
	(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1)
	{t : ℝ} (ht : 0 < t) :
	HasDerivAt
	(fun r => ∫ x, fieldMean energy spin observable (finiteStandardObservation spin r x) ^ 2
	∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)))
	(∫ x, ∑ v, fieldCovariance energy spin observable
	(finiteStandardObservation spin t x) v ^ 2
	∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) t := by
	apply hasDerivAt_finiteStandardEnergy_of_posterior_test μ energy spin observable A hbound hcoords ht
	intro v
	have hsq : ∀ s i, spin s i ^ 2 = 1 := by
	intro s i
	rcases hcoords s i with h \| h <;> simp [h]
	exact Problem367.finiteStandardGaussianPlanted_integral_test_fieldMean μ energy hprior t ht.le
	spin hsq (fun s => spin s v) (fun z => fieldSquareGradient energy spin observable z v)
	(continuous_fieldSquareGradient energy spin observable v).measurable (4 * A ^ 2)
	(fun z => by simpa only [Real.norm_eq_abs] using
	abs_fieldSquareGradient_le energy spin observable z v A (fun s => hcoords s v) hbound)

	end Planted
	end Problem367.FinitePosterior

	end

	end OAI