Datasets:

roseking
/

openAI-math-dataset

import Mathlib.MeasureTheory.Integral.Marginal
import OAI.Probability.FreeIsing.Gaussian.Attenuation
import OAI.Probability.FreeIsing.Trees.RootCovariance

namespace OAI

noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal

namespace Problem367.GaussianTreeCovariance

def covariance {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
    (r v : TreeVertex d) (h : TreeVertex d → ℝ) : ℝ :=
  LeafGibbs.covariance
    (fun eta => GibbsLeaf.energy beta h E (extendFiniteSpin E eta))
    (fun eta => spinSign (extendFiniteSpin E eta r))
    (fun eta => spinSign (extendFiniteSpin E eta v))

lemma measurable_covariance {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
    (r v : TreeVertex d) : Measurable (covariance beta E r v) := by
  classical
  unfold covariance LeafGibbs.covariance LeafGibbs.expectation GibbsLeaf.energy
  fun_prop

lemma covariance_congr {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
    (r v : TreeVertex d) {h k : TreeVertex d → ℝ}
    (heq : ∀ x ∈ E, h x = k x) : covariance beta E r v h = covariance beta E r v k := by
  have he : (fun eta => GibbsLeaf.energy beta h E (extendFiniteSpin E eta)) =
      (fun eta => GibbsLeaf.energy beta k E (extendFiniteSpin E eta)) := by
    funext eta
    unfold GibbsLeaf.energy
    congr 1
    exact Finset.sum_congr rfl (fun x hx => by rw [heq x hx])
  unfold covariance
  rw [he]

lemma covariance_bounds {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
    {E : Finset (TreeVertex d)} {r v : TreeVertex d}
    (hE : ConnectedFinset E) (hr : r ∈ E) (hv : v ∈ E)
    (hmin : ∀ x ∈ E, r.1.length ≤ x.1.length) (h : TreeVertex d → ℝ) :
    0 ≤ covariance beta E r v h ∧
      covariance beta E r v h ≤ Real.tanh beta ^ (v.1.length - r.1.length) := by
  simpa only [covariance, extendFiniteSpin, dite_eq_left hr, dite_eq_left hv] using
    tree_gibbs_root_covariance_bounds hb hE hr hmin h v hv

lemma covariance_erase {d : ℕ} {E : Finset (TreeVertex d)}
    {w p r v : TreeVertex d} (beta : ℝ) (h : TreeVertex d → ℝ)
    (hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w) (hv : v ∈ E.erase w)
    (hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w))) :
    covariance beta E r v h = covariance beta (E.erase w) r v
      (Function.update h p (h p + cavityMessage (Real.tanh beta) (h w))) := by
  simpa [covariance, extendFiniteSpin, (Finset.mem_erase.mp hr).2,
    (Finset.mem_erase.mp hv).2, hr, hv, finiteSpinEraseEquiv] using
    tree_gibbs_covariance_erase beta h hw hp hedges
      (fun eta => spinSign (eta ⟨r, hr⟩)) (fun eta => spinSign (eta ⟨v, hv⟩))

lemma covariance_leaf {d : ℕ} {E : Finset (TreeVertex d)}
    {w p r : TreeVertex d} (beta : ℝ) (h : TreeVertex d → ℝ)
    (hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w)
    (hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w))) :
    covariance beta E r w h =
      (Real.tanh beta * responseAttenuation (Real.tanh beta) (h w)) *
        covariance beta (E.erase w) r p
          (Function.update h p (h p + cavityMessage (Real.tanh beta) (h w))) := by
  have hleaf := tree_gibbs_covariance_leaf beta h hw hp hedges
    (fun eta => spinSign (eta ⟨r, hr⟩))
  have he : covariance beta E r w h =
      (Real.tanh beta * responseAttenuation (Real.tanh beta) (h w)) *
        covariance beta E r p h := by
    simpa [covariance, extendFiniteSpin, (Finset.mem_erase.mp hr).2,
      (Finset.mem_erase.mp hp).2, hw, finiteSpinEraseEquiv] using hleaf
  rw [he, covariance_erase beta h hw hp hr hp hedges]

lemma measurable_covariance_sq {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
    (r v : TreeVertex d) (h : TreeVertex d → ℝ) :
    Measurable (fun x : TreeVertex d → ℝ => ENNReal.ofReal
      (covariance beta E r v (fun u => h u + x u) ^ 2)) := by
  exact (((measurable_covariance beta E r v).comp
    (Measurable.of_eval (fun u => measurable_const.add (measurable_pi_apply u)))).pow_const 2).ennreal_ofReal

lemma covariance_updated_fields {d : ℕ} {E : Finset (TreeVertex d)}
    {w p : TreeVertex d} (beta : ℝ) (r v : TreeVertex d)
    (hp : p ∈ E.erase w) (h x : TreeVertex d → ℝ) (z : ℝ)
    (y : E.erase w → ℝ) :
    covariance beta (E.erase w) r v
      (Function.update (fun u => h u + Function.updateFinset (Function.update x w z) (E.erase w) y u)
        p (h p + Function.updateFinset (Function.update x w z) (E.erase w) y p +
          cavityMessage (Real.tanh beta)
            (h w + Function.updateFinset (Function.update x w z) (E.erase w) y w))) =
    covariance beta (E.erase w) r v
      (fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u +
        Function.updateFinset x (E.erase w) y u) := by
  classical
  apply covariance_congr
  intro u hu
  by_cases hup : u = p
  · subst u
    simp only [Function.update_self, Function.updateFinset, dite_eq_left hp,
      Finset.notMem_erase, dite_false, Function.update_self]
    ring
  · simp only [Function.update_of_ne hup, Function.updateFinset, dite_eq_left hu]

lemma lmarginal_leaf {d : ℕ} {E : Finset (TreeVertex d)}
    {w p r : TreeVertex d} (beta : ℝ) (h x : TreeVertex d → ℝ) (z : ℝ)
    (hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w)
    (hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w)))
    (μ : TreeVertex d → Measure ℝ) :
    lmarginal μ (E.erase w)
      (fun y => ENNReal.ofReal (covariance beta E r w (fun u => h u + y u) ^ 2))
      (Function.update x w z) =
    ENNReal.ofReal (Real.tanh beta ^ 2 * responseAttenuation (Real.tanh beta) (h w + z) ^ 2) *
      lmarginal μ (E.erase w)
        (fun y => ENNReal.ofReal (covariance beta (E.erase w) r p
          (fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u + y u) ^ 2)) x := by
  classical
  unfold lmarginal
  rw [← lintegral_const_mul' _ _ (ENNReal.ofReal_ne_top)]
  apply lintegral_congr
  intro y
  dsimp only
  rw [covariance_leaf beta _ hw hp hr hedges, covariance_updated_fields beta r p hp h x z y]
  simp only [Function.updateFinset, Finset.notMem_erase, dite_false, Function.update_self,
    mul_pow, ENNReal.ofReal_mul (mul_nonneg (sq_nonneg _) (sq_nonneg _))]
lemma lmarginal_nonleaf {d : ℕ} {E : Finset (TreeVertex d)}
    {w p r v : TreeVertex d} (beta : ℝ) (h x : TreeVertex d → ℝ) (z : ℝ)
    (hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w) (hv : v ∈ E.erase w)
    (hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w)))
    (μ : TreeVertex d → Measure ℝ) :
    lmarginal μ (E.erase w)
      (fun y => ENNReal.ofReal (covariance beta E r v (fun u => h u + y u) ^ 2))
      (Function.update x w z) =
      lmarginal μ (E.erase w)
        (fun y => ENNReal.ofReal (covariance beta (E.erase w) r v
          (fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u + y u) ^ 2)) x := by
  classical
  unfold lmarginal
  apply lintegral_congr
  intro y
  dsimp only
  rw [covariance_erase beta _ hw hp hr hv hedges, covariance_updated_fields beta r v hp h x z y]
theorem lmarginal_covariance_sq_le
    {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
    (μ : TreeVertex d → Measure ℝ) [∀ u, IsProbabilityMeasure (μ u)]
    (q : ℝ≥0∞)
    (hq : ∀ (u : TreeVertex d) (a : ℝ),
      (∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (a + z) ^ 2) ∂μ u) ≤ q)
    {E : Finset (TreeVertex d)} {r : TreeVertex d}
    (hE : ConnectedFinset E) (hr : r ∈ E)
    (hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
    (h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
    lmarginal μ E (fun y => ENNReal.ofReal
      (covariance beta E r v (fun u => h u + y u) ^ 2)) x ≤
      (ENNReal.ofReal (Real.tanh beta ^ 2) * q) ^ (v.1.length - r.1.length) := by
  classical
  revert hE hr hmin h x v
  refine Finset.strongInductionOn E ?_
  intro E ih hE hr hmin h x v hv
  by_cases hvr : v = r
  · subst v
    simp only [Nat.sub_self, pow_zero]
    unfold lmarginal
    calc
      _ ≤ ∫⁻ _ : E → ℝ, (1 : ℝ≥0∞) ∂Measure.pi (fun u : E => μ u) := by
        apply lintegral_mono
        intro y
        apply ENNReal.ofReal_le_one.mpr
        have hc := covariance_bounds hb hE hr hr hmin
          (fun u => h u + Function.updateFinset x E y u)
        simp only [Nat.sub_self, pow_zero] at hc
        nlinarith
      _ = 1 := by simp
  have hsub : ({r} : Finset (TreeVertex d)) ⊆ E := Finset.singleton_subset_iff.mpr hr
  have hne : ({r} : Finset (TreeVertex d)) ≠ E := by
    intro heq
    exact hvr (Finset.mem_singleton.mp (heq.symm ▸ hv))
  obtain ⟨w, p, hwE, _, hpE, hpw, hEr, hsubr, hedges⟩ :=
    (connectedFinset_singleton r).exists_prunable_vertex_of_shared_min
      hE hsub (Finset.mem_singleton_self r) hmin hne
  have hrEr : r ∈ E.erase w := hsubr (Finset.mem_singleton_self r)
  have hpEr : p ∈ E.erase w := Finset.mem_erase.mpr ⟨hpw, hpE⟩
  have hErE : E.erase w ⊆ E := Finset.erase_subset _ _
  have hmin' : ∀ u ∈ E.erase w, r.1.length ≤ u.1.length :=
    fun u hu => hmin u (hErE hu)
  rw [lmarginal_erase _ (measurable_covariance_sq beta E r v h) hwE x]
  by_cases hvw : v = w
  · subst v
    have hedge : (p, w) ∈ inducedEdges E := by
      rw [hedges]
      exact Finset.mem_insert_self _ _
    have hdata := Finset.mem_filter.mp hedge
    have hadj : treeAdj p w := hdata.2.1
    have hlower : p.1.length < w.1.length := hdata.2.2
    have hlen := treeAdj_length hadj
    have hdist : w.1.length - r.1.length = (p.1.length - r.1.length) + 1 := by
      have := hmin p hpE
      omega
    let B := ENNReal.ofReal (Real.tanh beta ^ 2) * q
    have hm : Measurable (fun z : ℝ => ENNReal.ofReal
        (responseAttenuation (Real.tanh beta) (h w + z) ^ 2)) := by
      simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
      fun_prop
    calc
      _ ≤ ∫⁻ z, ENNReal.ofReal (Real.tanh beta ^ 2 *
          responseAttenuation (Real.tanh beta) (h w + z) ^ 2) *
          B ^ (p.1.length - r.1.length) ∂μ w := by
        apply lintegral_mono
        intro z
        dsimp only
        rw [lmarginal_leaf beta h x z hwE hpEr hrEr hedges]
        exact mul_le_mul_right (ih (E.erase w) (Finset.erase_ssubset hwE)
          hEr hrEr hmin' _ x p hpEr) _
      _ = ENNReal.ofReal (Real.tanh beta ^ 2) *
          (∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (h w + z) ^ 2) ∂μ w) *
          B ^ (p.1.length - r.1.length) := by
        simp_rw [ENNReal.ofReal_mul (sq_nonneg (Real.tanh beta))]
        have hcm : Measurable (fun z : ℝ => ENNReal.ofReal (Real.tanh beta ^ 2) *
            ENNReal.ofReal (responseAttenuation (Real.tanh beta) (h w + z) ^ 2)) :=
          measurable_const.mul hm
        rw [lintegral_mul_const _ hcm, lintegral_const_mul _ hm]
      _ ≤ ENNReal.ofReal (Real.tanh beta ^ 2) * q * B ^ (p.1.length - r.1.length) := by
        gcongr
        exact hq w (h w)
      _ = _ := by
        rw [hdist]
        change B * B ^ (p.1.length - r.1.length) = B ^ ((p.1.length - r.1.length) + 1)
        exact (pow_succ' B _).symm
  · have hvEr : v ∈ E.erase w := Finset.mem_erase.mpr ⟨hvw, hv⟩
    calc
      _ ≤ ∫⁻ _ : ℝ, (ENNReal.ofReal (Real.tanh beta ^ 2) * q) ^
          (v.1.length - r.1.length) ∂μ w := by
        apply lintegral_mono
        intro z
        dsimp only
        rw [lmarginal_nonleaf beta h x z hwE hpEr hrEr hvEr hedges]
        exact ih (E.erase w) (Finset.erase_ssubset hwE) hEr hrEr hmin' _ x v hvEr
      _ = _ := by simp
def gaussianContraction (beta : ℝ) (δ : ℝ≥0) : ℝ :=
  1 - ((1 - Real.tanh beta ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
lemma gaussianContraction_pos (beta : ℝ) (δ : ℝ≥0) :
    0 < gaussianContraction beta δ := by
  have ht := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
  have hp := mul_nonneg (sq_nonneg (Real.tanh beta))
    (sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
  unfold gaussianContraction
  nlinarith
lemma gaussianContraction_lt_one (beta : ℝ) {δ : ℝ≥0} (hδ : 0 < δ) :
    gaussianContraction beta δ < 1 := by
  have ht : 0 < Real.tanh (gaussianSmallRadius δ) := by
    rw [Real.tanh_eq_sinh_div_cosh]
    exact div_pos (Real.sinh_pos_iff.mpr (gaussianSmallRadius_pos hδ)) (Real.cosh_pos _)
  have hg : 0 < (1 - Real.tanh beta ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2 :=
    mul_pos (sub_pos.mpr (Real.tanh_sq_lt_one beta)) (sq_pos_of_pos ht)
  unfold gaussianContraction
  linarith
lemma lintegral_gaussian_attenuation_le (beta m a : ℝ) {δ t : ℝ≥0}
    (hδ : 0 < δ) (ht : δ ≤ t) :
    (∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (a + z) ^ 2)
      ∂gaussianReal m t) ≤ ENNReal.ofReal (gaussianContraction beta δ) := by
  have hm : Measurable (fun z : ℝ => responseAttenuation (Real.tanh beta) (a + z) ^ 2) := by
    simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
    fun_prop
  have hint : Integrable (fun z : ℝ => responseAttenuation (Real.tanh beta) (a + z) ^ 2)
      (gaussianReal m t) := by
    apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
    exact Filter.Eventually.of_forall fun z => by
      rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
      exact GaussianAttenuation.attenuation_sq_le_one (Real.tanh_sq_lt_one beta).le _
  rw [← ofReal_integral_eq_lintegral_ofReal hint (Filter.Eventually.of_forall fun _ => sq_nonneg _)]
  apply ENNReal.ofReal_le_ofReal
  have hlaw : HasLaw (fun z : ℝ => a + z) (gaussianReal (m + a) t) (gaussianReal m t) := by
    refine ⟨(measurable_const.add measurable_id).aemeasurable, ?_⟩
    simpa only [add_comm a] using (gaussianReal_map_add_const (μ := m) (v := t) a)
  exact GaussianAttenuation.integral_attenuation_sq_le_of_hasLaw hlaw
    (Real.tanh_sq_lt_one beta) hδ ht

theorem lintegral_gaussian_covariance_sq_le
    {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
    (m : TreeVertex d → ℝ) (t : TreeVertex d → ℝ≥0)
    {δ : ℝ≥0} (hδ : 0 < δ) (ht : ∀ u, δ ≤ t u)
    {E : Finset (TreeVertex d)} {r : TreeVertex d}
    (hE : ConnectedFinset E) (hr : r ∈ E)
    (hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
    (h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
    (∫⁻ y : E → ℝ, ENNReal.ofReal (covariance beta E r v
      (fun u => h u + Function.updateFinset x E y u) ^ 2)
      ∂Measure.pi (fun u : E => gaussianReal (m u) (t u))) ≤
    ENNReal.ofReal ((Real.tanh beta ^ 2 * gaussianContraction beta δ) ^
      (v.1.length - r.1.length)) := by
  have he := lmarginal_covariance_sq_le hb (fun u => gaussianReal (m u) (t u))
    (ENNReal.ofReal (gaussianContraction beta δ))
    (fun u a => lintegral_gaussian_attenuation_le beta (m u) a hδ (ht u))
    hE hr hmin h x v hv
  simpa only [lmarginal, ← ENNReal.ofReal_mul (sq_nonneg (Real.tanh beta)),
    ← ENNReal.ofReal_pow (mul_nonneg (sq_nonneg _) (gaussianContraction_pos beta δ).le)] using he

lemma integrable_covariance_sq {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
    {X : Ω → TreeVertex d → ℝ} (hX : Measurable X)
    {E : Finset (TreeVertex d)} {r : TreeVertex d}
    (hE : ConnectedFinset E) (hr : r ∈ E)
    (hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
    (h : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
    Integrable (fun ω => covariance beta E r v (fun u => h u + X ω u) ^ 2) P := by
  have hm : Measurable (fun ω => covariance beta E r v (fun u => h u + X ω u) ^ 2) :=
    ((measurable_covariance beta E r v).comp
      (Measurable.of_eval (fun u => measurable_const.add
        ((measurable_pi_apply u).comp hX)))).pow_const 2
  apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
  refine Filter.Eventually.of_forall fun ω => ?_
  rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
  have hc := covariance_bounds hb hE hr hv hmin (fun u => h u + X ω u)
  have ht : 0 ≤ Real.tanh beta := by
    rw [Real.tanh_eq_sinh_div_cosh]
    exact div_nonneg (Real.sinh_nonneg_iff.mpr hb) (Real.cosh_pos beta).le
  have hp : Real.tanh beta ^ (v.1.length - r.1.length) ≤ 1 :=
    pow_le_one₀ ht (Real.tanh_lt_one beta).le
  nlinarith
theorem integral_gaussian_covariance_sq_le
    {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
    (m : TreeVertex d → ℝ) (t : TreeVertex d → ℝ≥0)
    {δ : ℝ≥0} (hδ : 0 < δ) (ht : ∀ u, δ ≤ t u)
    {E : Finset (TreeVertex d)} {r : TreeVertex d}
    (hE : ConnectedFinset E) (hr : r ∈ E)
    (hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
    (h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
    (∫ y : E → ℝ, covariance beta E r v
      (fun u => h u + Function.updateFinset x E y u) ^ 2
      ∂Measure.pi (fun u : E => gaussianReal (m u) (t u))) ≤
    (Real.tanh beta ^ 2 * gaussianContraction beta δ) ^
      (v.1.length - r.1.length) := by
  apply (ENNReal.ofReal_le_ofReal_iff
    (pow_nonneg (mul_nonneg (sq_nonneg _) (gaussianContraction_pos beta δ).le) _)).mp
  have hint := integrable_covariance_sq hb
    (P := Measure.pi (fun u : E => gaussianReal (m u) (t u)))
    (X := fun y => Function.updateFinset x E y) (by fun_prop)
    hE hr hmin h v hv
  rw [ofReal_integral_eq_lintegral_ofReal hint (Filter.Eventually.of_forall fun _ => sq_nonneg _)]
  exact lintegral_gaussian_covariance_sq_le hb m t hδ ht hE hr hmin h x v hv
end Problem367.GaussianTreeCovariance
end
end OAI

	import Mathlib.MeasureTheory.Integral.Marginal
	import OAI.Probability.FreeIsing.Gaussian.Attenuation
	import OAI.Probability.FreeIsing.Trees.RootCovariance

	namespace OAI

	noncomputable section
	open MeasureTheory ProbabilityTheory
	open scoped NNReal ENNReal

	namespace Problem367.GaussianTreeCovariance

	def covariance {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
	(r v : TreeVertex d) (h : TreeVertex d → ℝ) : ℝ :=
	LeafGibbs.covariance
	(fun eta => GibbsLeaf.energy beta h E (extendFiniteSpin E eta))
	(fun eta => spinSign (extendFiniteSpin E eta r))
	(fun eta => spinSign (extendFiniteSpin E eta v))

	lemma measurable_covariance {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
	(r v : TreeVertex d) : Measurable (covariance beta E r v) := by
	classical
	unfold covariance LeafGibbs.covariance LeafGibbs.expectation GibbsLeaf.energy
	fun_prop

	lemma covariance_congr {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
	(r v : TreeVertex d) {h k : TreeVertex d → ℝ}
	(heq : ∀ x ∈ E, h x = k x) : covariance beta E r v h = covariance beta E r v k := by
	have he : (fun eta => GibbsLeaf.energy beta h E (extendFiniteSpin E eta)) =
	(fun eta => GibbsLeaf.energy beta k E (extendFiniteSpin E eta)) := by
	funext eta
	unfold GibbsLeaf.energy
	congr 1
	exact Finset.sum_congr rfl (fun x hx => by rw [heq x hx])
	unfold covariance
	rw [he]

	lemma covariance_bounds {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
	{E : Finset (TreeVertex d)} {r v : TreeVertex d}
	(hE : ConnectedFinset E) (hr : r ∈ E) (hv : v ∈ E)
	(hmin : ∀ x ∈ E, r.1.length ≤ x.1.length) (h : TreeVertex d → ℝ) :
	0 ≤ covariance beta E r v h ∧
	covariance beta E r v h ≤ Real.tanh beta ^ (v.1.length - r.1.length) := by
	simpa only [covariance, extendFiniteSpin, dite_eq_left hr, dite_eq_left hv] using
	tree_gibbs_root_covariance_bounds hb hE hr hmin h v hv

	lemma covariance_erase {d : ℕ} {E : Finset (TreeVertex d)}
	{w p r v : TreeVertex d} (beta : ℝ) (h : TreeVertex d → ℝ)
	(hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w) (hv : v ∈ E.erase w)
	(hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w))) :
	covariance beta E r v h = covariance beta (E.erase w) r v
	(Function.update h p (h p + cavityMessage (Real.tanh beta) (h w))) := by
	simpa [covariance, extendFiniteSpin, (Finset.mem_erase.mp hr).2,
	(Finset.mem_erase.mp hv).2, hr, hv, finiteSpinEraseEquiv] using
	tree_gibbs_covariance_erase beta h hw hp hedges
	(fun eta => spinSign (eta ⟨r, hr⟩)) (fun eta => spinSign (eta ⟨v, hv⟩))

	lemma covariance_leaf {d : ℕ} {E : Finset (TreeVertex d)}
	{w p r : TreeVertex d} (beta : ℝ) (h : TreeVertex d → ℝ)
	(hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w)
	(hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w))) :
	covariance beta E r w h =
	(Real.tanh beta * responseAttenuation (Real.tanh beta) (h w)) *
	covariance beta (E.erase w) r p
	(Function.update h p (h p + cavityMessage (Real.tanh beta) (h w))) := by
	have hleaf := tree_gibbs_covariance_leaf beta h hw hp hedges
	(fun eta => spinSign (eta ⟨r, hr⟩))
	have he : covariance beta E r w h =
	(Real.tanh beta * responseAttenuation (Real.tanh beta) (h w)) *
	covariance beta E r p h := by
	simpa [covariance, extendFiniteSpin, (Finset.mem_erase.mp hr).2,
	(Finset.mem_erase.mp hp).2, hw, finiteSpinEraseEquiv] using hleaf
	rw [he, covariance_erase beta h hw hp hr hp hedges]

	lemma measurable_covariance_sq {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
	(r v : TreeVertex d) (h : TreeVertex d → ℝ) :
	Measurable (fun x : TreeVertex d → ℝ => ENNReal.ofReal
	(covariance beta E r v (fun u => h u + x u) ^ 2)) := by
	exact (((measurable_covariance beta E r v).comp
	(Measurable.of_eval (fun u => measurable_const.add (measurable_pi_apply u)))).pow_const 2).ennreal_ofReal

	lemma covariance_updated_fields {d : ℕ} {E : Finset (TreeVertex d)}
	{w p : TreeVertex d} (beta : ℝ) (r v : TreeVertex d)
	(hp : p ∈ E.erase w) (h x : TreeVertex d → ℝ) (z : ℝ)
	(y : E.erase w → ℝ) :
	covariance beta (E.erase w) r v
	(Function.update (fun u => h u + Function.updateFinset (Function.update x w z) (E.erase w) y u)
	p (h p + Function.updateFinset (Function.update x w z) (E.erase w) y p +
	cavityMessage (Real.tanh beta)
	(h w + Function.updateFinset (Function.update x w z) (E.erase w) y w))) =
	covariance beta (E.erase w) r v
	(fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u +
	Function.updateFinset x (E.erase w) y u) := by
	classical
	apply covariance_congr
	intro u hu
	by_cases hup : u = p
	· subst u
	simp only [Function.update_self, Function.updateFinset, dite_eq_left hp,
	Finset.notMem_erase, dite_false, Function.update_self]
	ring
	· simp only [Function.update_of_ne hup, Function.updateFinset, dite_eq_left hu]

	lemma lmarginal_leaf {d : ℕ} {E : Finset (TreeVertex d)}
	{w p r : TreeVertex d} (beta : ℝ) (h x : TreeVertex d → ℝ) (z : ℝ)
	(hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w)
	(hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w)))
	(μ : TreeVertex d → Measure ℝ) :
	lmarginal μ (E.erase w)
	(fun y => ENNReal.ofReal (covariance beta E r w (fun u => h u + y u) ^ 2))
	(Function.update x w z) =
	ENNReal.ofReal (Real.tanh beta ^ 2 * responseAttenuation (Real.tanh beta) (h w + z) ^ 2) *
	lmarginal μ (E.erase w)
	(fun y => ENNReal.ofReal (covariance beta (E.erase w) r p
	(fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u + y u) ^ 2)) x := by
	classical
	unfold lmarginal
	rw [← lintegral_const_mul' _ _ (ENNReal.ofReal_ne_top)]
	apply lintegral_congr
	intro y
	dsimp only
	rw [covariance_leaf beta _ hw hp hr hedges, covariance_updated_fields beta r p hp h x z y]
	simp only [Function.updateFinset, Finset.notMem_erase, dite_false, Function.update_self,
	mul_pow, ENNReal.ofReal_mul (mul_nonneg (sq_nonneg _) (sq_nonneg _))]

	lemma lmarginal_nonleaf {d : ℕ} {E : Finset (TreeVertex d)}
	{w p r v : TreeVertex d} (beta : ℝ) (h x : TreeVertex d → ℝ) (z : ℝ)
	(hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w) (hv : v ∈ E.erase w)
	(hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w)))
	(μ : TreeVertex d → Measure ℝ) :
	lmarginal μ (E.erase w)
	(fun y => ENNReal.ofReal (covariance beta E r v (fun u => h u + y u) ^ 2))
	(Function.update x w z) =
	lmarginal μ (E.erase w)
	(fun y => ENNReal.ofReal (covariance beta (E.erase w) r v
	(fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u + y u) ^ 2)) x := by
	classical
	unfold lmarginal
	apply lintegral_congr
	intro y
	dsimp only
	rw [covariance_erase beta _ hw hp hr hv hedges, covariance_updated_fields beta r v hp h x z y]

	theorem lmarginal_covariance_sq_le
	{d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
	(μ : TreeVertex d → Measure ℝ) [∀ u, IsProbabilityMeasure (μ u)]
	(q : ℝ≥0∞)
	(hq : ∀ (u : TreeVertex d) (a : ℝ),
	(∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (a + z) ^ 2) ∂μ u) ≤ q)
	{E : Finset (TreeVertex d)} {r : TreeVertex d}
	(hE : ConnectedFinset E) (hr : r ∈ E)
	(hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
	(h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
	lmarginal μ E (fun y => ENNReal.ofReal
	(covariance beta E r v (fun u => h u + y u) ^ 2)) x ≤
	(ENNReal.ofReal (Real.tanh beta ^ 2) * q) ^ (v.1.length - r.1.length) := by
	classical
	revert hE hr hmin h x v
	refine Finset.strongInductionOn E ?_
	intro E ih hE hr hmin h x v hv
	by_cases hvr : v = r
	· subst v
	simp only [Nat.sub_self, pow_zero]
	unfold lmarginal
	calc
	_ ≤ ∫⁻ _ : E → ℝ, (1 : ℝ≥0∞) ∂Measure.pi (fun u : E => μ u) := by
	apply lintegral_mono
	intro y
	apply ENNReal.ofReal_le_one.mpr
	have hc := covariance_bounds hb hE hr hr hmin
	(fun u => h u + Function.updateFinset x E y u)
	simp only [Nat.sub_self, pow_zero] at hc
	nlinarith
	_ = 1 := by simp
	have hsub : ({r} : Finset (TreeVertex d)) ⊆ E := Finset.singleton_subset_iff.mpr hr
	have hne : ({r} : Finset (TreeVertex d)) ≠ E := by
	intro heq
	exact hvr (Finset.mem_singleton.mp (heq.symm ▸ hv))
	obtain ⟨w, p, hwE, _, hpE, hpw, hEr, hsubr, hedges⟩ :=
	(connectedFinset_singleton r).exists_prunable_vertex_of_shared_min
	hE hsub (Finset.mem_singleton_self r) hmin hne
	have hrEr : r ∈ E.erase w := hsubr (Finset.mem_singleton_self r)
	have hpEr : p ∈ E.erase w := Finset.mem_erase.mpr ⟨hpw, hpE⟩
	have hErE : E.erase w ⊆ E := Finset.erase_subset _ _
	have hmin' : ∀ u ∈ E.erase w, r.1.length ≤ u.1.length :=
	fun u hu => hmin u (hErE hu)
	rw [lmarginal_erase _ (measurable_covariance_sq beta E r v h) hwE x]
	by_cases hvw : v = w
	· subst v
	have hedge : (p, w) ∈ inducedEdges E := by
	rw [hedges]
	exact Finset.mem_insert_self _ _
	have hdata := Finset.mem_filter.mp hedge
	have hadj : treeAdj p w := hdata.2.1
	have hlower : p.1.length < w.1.length := hdata.2.2
	have hlen := treeAdj_length hadj
	have hdist : w.1.length - r.1.length = (p.1.length - r.1.length) + 1 := by
	have := hmin p hpE
	omega
	let B := ENNReal.ofReal (Real.tanh beta ^ 2) * q
	have hm : Measurable (fun z : ℝ => ENNReal.ofReal
	(responseAttenuation (Real.tanh beta) (h w + z) ^ 2)) := by
	simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
	fun_prop
	calc
	_ ≤ ∫⁻ z, ENNReal.ofReal (Real.tanh beta ^ 2 *
	responseAttenuation (Real.tanh beta) (h w + z) ^ 2) *
	B ^ (p.1.length - r.1.length) ∂μ w := by
	apply lintegral_mono
	intro z
	dsimp only
	rw [lmarginal_leaf beta h x z hwE hpEr hrEr hedges]
	exact mul_le_mul_right (ih (E.erase w) (Finset.erase_ssubset hwE)
	hEr hrEr hmin' _ x p hpEr) _
	_ = ENNReal.ofReal (Real.tanh beta ^ 2) *
	(∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (h w + z) ^ 2) ∂μ w) *
	B ^ (p.1.length - r.1.length) := by
	simp_rw [ENNReal.ofReal_mul (sq_nonneg (Real.tanh beta))]
	have hcm : Measurable (fun z : ℝ => ENNReal.ofReal (Real.tanh beta ^ 2) *
	ENNReal.ofReal (responseAttenuation (Real.tanh beta) (h w + z) ^ 2)) :=
	measurable_const.mul hm
	rw [lintegral_mul_const _ hcm, lintegral_const_mul _ hm]
	_ ≤ ENNReal.ofReal (Real.tanh beta ^ 2) * q * B ^ (p.1.length - r.1.length) := by
	gcongr
	exact hq w (h w)
	_ = _ := by
	rw [hdist]
	change B * B ^ (p.1.length - r.1.length) = B ^ ((p.1.length - r.1.length) + 1)
	exact (pow_succ' B _).symm
	· have hvEr : v ∈ E.erase w := Finset.mem_erase.mpr ⟨hvw, hv⟩
	calc
	_ ≤ ∫⁻ _ : ℝ, (ENNReal.ofReal (Real.tanh beta ^ 2) * q) ^
	(v.1.length - r.1.length) ∂μ w := by
	apply lintegral_mono
	intro z
	dsimp only
	rw [lmarginal_nonleaf beta h x z hwE hpEr hrEr hvEr hedges]
	exact ih (E.erase w) (Finset.erase_ssubset hwE) hEr hrEr hmin' _ x v hvEr
	_ = _ := by simp

	def gaussianContraction (beta : ℝ) (δ : ℝ≥0) : ℝ :=
	1 - ((1 - Real.tanh beta ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2

	lemma gaussianContraction_pos (beta : ℝ) (δ : ℝ≥0) :
	0 < gaussianContraction beta δ := by
	have ht := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
	have hp := mul_nonneg (sq_nonneg (Real.tanh beta))
	(sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
	unfold gaussianContraction
	nlinarith

	lemma gaussianContraction_lt_one (beta : ℝ) {δ : ℝ≥0} (hδ : 0 < δ) :
	gaussianContraction beta δ < 1 := by
	have ht : 0 < Real.tanh (gaussianSmallRadius δ) := by
	rw [Real.tanh_eq_sinh_div_cosh]
	exact div_pos (Real.sinh_pos_iff.mpr (gaussianSmallRadius_pos hδ)) (Real.cosh_pos _)
	have hg : 0 < (1 - Real.tanh beta ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2 :=
	mul_pos (sub_pos.mpr (Real.tanh_sq_lt_one beta)) (sq_pos_of_pos ht)
	unfold gaussianContraction
	linarith

	lemma lintegral_gaussian_attenuation_le (beta m a : ℝ) {δ t : ℝ≥0}
	(hδ : 0 < δ) (ht : δ ≤ t) :
	(∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (a + z) ^ 2)
	∂gaussianReal m t) ≤ ENNReal.ofReal (gaussianContraction beta δ) := by
	have hm : Measurable (fun z : ℝ => responseAttenuation (Real.tanh beta) (a + z) ^ 2) := by
	simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
	fun_prop
	have hint : Integrable (fun z : ℝ => responseAttenuation (Real.tanh beta) (a + z) ^ 2)
	(gaussianReal m t) := by
	apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
	exact Filter.Eventually.of_forall fun z => by
	rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
	exact GaussianAttenuation.attenuation_sq_le_one (Real.tanh_sq_lt_one beta).le _
	rw [← ofReal_integral_eq_lintegral_ofReal hint (Filter.Eventually.of_forall fun _ => sq_nonneg _)]
	apply ENNReal.ofReal_le_ofReal
	have hlaw : HasLaw (fun z : ℝ => a + z) (gaussianReal (m + a) t) (gaussianReal m t) := by
	refine ⟨(measurable_const.add measurable_id).aemeasurable, ?_⟩
	simpa only [add_comm a] using (gaussianReal_map_add_const (μ := m) (v := t) a)
	exact GaussianAttenuation.integral_attenuation_sq_le_of_hasLaw hlaw
	(Real.tanh_sq_lt_one beta) hδ ht

	theorem lintegral_gaussian_covariance_sq_le
	{d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
	(m : TreeVertex d → ℝ) (t : TreeVertex d → ℝ≥0)
	{δ : ℝ≥0} (hδ : 0 < δ) (ht : ∀ u, δ ≤ t u)
	{E : Finset (TreeVertex d)} {r : TreeVertex d}
	(hE : ConnectedFinset E) (hr : r ∈ E)
	(hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
	(h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
	(∫⁻ y : E → ℝ, ENNReal.ofReal (covariance beta E r v
	(fun u => h u + Function.updateFinset x E y u) ^ 2)
	∂Measure.pi (fun u : E => gaussianReal (m u) (t u))) ≤
	ENNReal.ofReal ((Real.tanh beta ^ 2 * gaussianContraction beta δ) ^
	(v.1.length - r.1.length)) := by
	have he := lmarginal_covariance_sq_le hb (fun u => gaussianReal (m u) (t u))
	(ENNReal.ofReal (gaussianContraction beta δ))
	(fun u a => lintegral_gaussian_attenuation_le beta (m u) a hδ (ht u))
	hE hr hmin h x v hv
	simpa only [lmarginal, ← ENNReal.ofReal_mul (sq_nonneg (Real.tanh beta)),
	← ENNReal.ofReal_pow (mul_nonneg (sq_nonneg _) (gaussianContraction_pos beta δ).le)] using he

	lemma integrable_covariance_sq {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
	{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
	{X : Ω → TreeVertex d → ℝ} (hX : Measurable X)
	{E : Finset (TreeVertex d)} {r : TreeVertex d}
	(hE : ConnectedFinset E) (hr : r ∈ E)
	(hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
	(h : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
	Integrable (fun ω => covariance beta E r v (fun u => h u + X ω u) ^ 2) P := by
	have hm : Measurable (fun ω => covariance beta E r v (fun u => h u + X ω u) ^ 2) :=
	((measurable_covariance beta E r v).comp
	(Measurable.of_eval (fun u => measurable_const.add
	((measurable_pi_apply u).comp hX)))).pow_const 2
	apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
	refine Filter.Eventually.of_forall fun ω => ?_
	rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
	have hc := covariance_bounds hb hE hr hv hmin (fun u => h u + X ω u)
	have ht : 0 ≤ Real.tanh beta := by
	rw [Real.tanh_eq_sinh_div_cosh]
	exact div_nonneg (Real.sinh_nonneg_iff.mpr hb) (Real.cosh_pos beta).le
	have hp : Real.tanh beta ^ (v.1.length - r.1.length) ≤ 1 :=
	pow_le_one₀ ht (Real.tanh_lt_one beta).le
	nlinarith

	theorem integral_gaussian_covariance_sq_le
	{d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
	(m : TreeVertex d → ℝ) (t : TreeVertex d → ℝ≥0)
	{δ : ℝ≥0} (hδ : 0 < δ) (ht : ∀ u, δ ≤ t u)
	{E : Finset (TreeVertex d)} {r : TreeVertex d}
	(hE : ConnectedFinset E) (hr : r ∈ E)
	(hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
	(h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
	(∫ y : E → ℝ, covariance beta E r v
	(fun u => h u + Function.updateFinset x E y u) ^ 2
	∂Measure.pi (fun u : E => gaussianReal (m u) (t u))) ≤
	(Real.tanh beta ^ 2 * gaussianContraction beta δ) ^
	(v.1.length - r.1.length) := by
	apply (ENNReal.ofReal_le_ofReal_iff
	(pow_nonneg (mul_nonneg (sq_nonneg _) (gaussianContraction_pos beta δ).le) _)).mp
	have hint := integrable_covariance_sq hb
	(P := Measure.pi (fun u : E => gaussianReal (m u) (t u)))
	(X := fun y => Function.updateFinset x E y) (by fun_prop)
	hE hr hmin h v hv
	rw [ofReal_integral_eq_lintegral_ofReal hint (Filter.Eventually.of_forall fun _ => sq_nonneg _)]
	exact lintegral_gaussian_covariance_sq_le hb m t hδ ht hE hr hmin h x v hv

	end Problem367.GaussianTreeCovariance

	end

	end OAI