Datasets:

roseking
/

openAI-math-dataset

import OAI.Probability.FreeIsing.Estimates.InfiniteBlockIndependence
import OAI.Probability.FreeIsing.Gaussian.Attenuation

namespace OAI

noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal

namespace Problem367.GaussianAttenuation

def pathTail (i : ℕ) (x : ℕ → ℝ) : ℕ → ℝ := fun j => x (i + 1 + j)

def pathProduct (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
    (x : ℕ → ℝ) : ℝ :=
  ∏ i ∈ Finset.range n, responseAttenuation θ (x i + shift i (pathTail i x)) ^ 2

@[fun_prop] lemma measurable_pathTail (i : ℕ) : Measurable (pathTail i) := by
  exact Measurable.of_eval (fun index => measurable_pi_apply _)

lemma measurable_pathProduct (θ : ℝ) (n : ℕ)
    {shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
    Measurable (pathProduct θ n shift) := by
  unfold pathProduct
  apply Finset.measurable_fun_prod
  intro i hi
  have h := (measurable_pi_apply i).add ((hs i).comp (measurable_pathTail i))
  simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
  fun_prop

lemma pathProduct_nonneg (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
    (x : ℕ → ℝ) : 0 ≤ pathProduct θ n shift x := by
  exact Finset.prod_nonneg (fun i _ => sq_nonneg _)

lemma pathProduct_le_one {θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
    (shift : ℕ → (ℕ → ℝ) → ℝ) (x : ℕ → ℝ) : pathProduct θ n shift x ≤ 1 := by
  exact Finset.prod_le_one₀ (fun index _ => sq_nonneg _)
    (fun i _ => attenuation_sq_le_one hθ _)

lemma integrable_pathProduct_comp {Ω : Type*} [MeasurableSpace Ω]
    {P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ}
    (hX : ∀ i, Measurable (X i)) {θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
    {shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
    Integrable (fun ω => pathProduct θ n shift (fun i => X i ω)) P := by
  apply (integrable_const (1 : ℝ)).mono'
    ((measurable_pathProduct θ n hs).comp (Measurable.of_eval hX)).aestronglyMeasurable
  exact Filter.Eventually.of_forall fun ω => by
    dsimp only [Function.comp_def]
    rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
    exact pathProduct_le_one hθ _ _ _

lemma pathProduct_succ (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
    (x : ℕ → ℝ) :
    pathProduct θ (n + 1) shift x =
      responseAttenuation θ (x 0 + shift 0 (pathTail 0 x)) ^ 2 *
        pathProduct θ n (fun i => shift (i + 1)) (pathTail 0 x) := by
  unfold pathProduct
  rw [Finset.prod_range_succ', mul_comm]
  congr 1
  apply Finset.prod_congr rfl
  intro i hi
  have hh : pathTail 0 x i = x (i + 1) := by
    simp only [pathTail, Nat.zero_add, Nat.add_comm]
  have ht : pathTail i (pathTail 0 x) = pathTail (i + 1) x := by
    funext j
    simp only [pathTail]
    congr 1
    omega
  rw [hh, ht]

lemma head_indep_tail {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
    {X : ℕ → Ω → ℝ} (hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P) :
    IndepFun (X 0) (fun ω j => X (j + 1) ω) P := by
  have h := indepFun_of_disjoint_blocks hind hX ({0} : Set ℕ) (Set.Ioi 0)
    (by simp : Disjoint ({0} : Set ℕ) (Set.Ioi 0))
    (fun y => y ⟨0, by simp⟩)
    (fun y : Set.Ioi (0 : ℕ) → ℝ => fun j => y ⟨j + 1, show 0 < j + 1 from Nat.succ_pos j⟩)
    (measurable_pi_apply _) (by fun_prop)
  exact h

theorem integral_pathProduct_le {Ω : Type*} [MeasurableSpace Ω]
    {P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ}
    (hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P)
    {m : ℕ → ℝ} {v : ℕ → ℝ≥0}
    (hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
    {θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
    (hv : ∀ i, δ ≤ v i) (n : ℕ)
    {shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
    (∫ ω, pathProduct θ n shift (fun i => X i ω) ∂P) ≤
      (1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
  let q := 1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
  have hq : 0 ≤ q := by
    have ht := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
    have hp := mul_nonneg (sq_nonneg θ) (sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
    dsimp [q]
    nlinarith
  change (∫ ω, pathProduct θ n shift (fun i => X i ω) ∂P) ≤ q ^ n
  induction n generalizing X m v shift with
  | zero => simp [pathProduct]
  | succ n ih =>
      have hi := ih (fun i => hX (i + 1)) (hind.precomp (g := fun i : ℕ => i + 1) (by intro i j hij; exact Nat.add_right_cancel hij))
        (fun i => hlaw (i + 1)) (fun i => hv (i + 1)) (fun i => hs (i + 1))
      have he := integral_shifted_attenuation_sq_mul_le_of_indepFun (hlaw 0)
        (show AEMeasurable (fun ω j => X (j + 1) ω) P from
          (Measurable.of_eval (fun index => hX (index + 1))).aemeasurable)
        (head_indep_tail hX hind) (hs 0)
        (measurable_pathProduct θ n (fun i => hs (i + 1)))
        (integrable_pathProduct_comp (fun i => hX (i + 1)) hθ.le n (fun i => hs (i + 1)))
        (pathProduct_nonneg θ n (fun i => shift (i + 1))) hθ hδ (hv 0)
      calc
        (∫ ω, pathProduct θ (n + 1) shift (fun i => X i ω) ∂P) =
            ∫ ω, responseAttenuation θ (X 0 ω + shift 0 (fun j => X (j + 1) ω)) ^ 2 *
              pathProduct θ n (fun i => shift (i + 1)) (fun j => X (j + 1) ω) ∂P := by
          apply integral_congr_ae
          exact Filter.Eventually.of_forall fun ω => by
            have hz : pathTail 0 (fun i => X i ω) = (fun j => X (j + 1) ω) := by
              funext j
              simp only [pathTail, Nat.zero_add, Nat.add_comm]
            simpa only [hz] using pathProduct_succ θ n shift (fun i => X i ω)
        _ ≤ q * ∫ ω, pathProduct θ n (fun i => shift (i + 1))
            (fun j => X (j + 1) ω) ∂P := he
        _ ≤ q * q ^ n := mul_le_mul_of_nonneg_left hi hq
        _ = q ^ (n + 1) := (pow_succ' q n).symm

lemma measurable_pathProduct_parameter {Y : Type*} [MeasurableSpace Y]
    (θ : ℝ) (n : ℕ) {shift : Y → ℕ → (ℕ → ℝ) → ℝ}
    (hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
    Measurable (fun p : Y × (ℕ → ℝ) => pathProduct θ n (shift p.1) p.2) := by
  unfold pathProduct
  apply Finset.measurable_fun_prod
  intro i hi
  have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
    simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
    fun_prop
  exact ha.comp (((measurable_pi_apply i).comp measurable_snd).add
    ((hs i).comp (measurable_fst.prodMk ((measurable_pathTail i).comp measurable_snd))))

theorem integral_pathProduct_parameter_le
    {Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
    (ν : Measure Y) [IsProbabilityMeasure ν]
    {P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ}
    (hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P)
    {m : ℕ → ℝ} {v : ℕ → ℝ≥0}
    (hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
    {θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
    (hv : ∀ i, δ ≤ v i) (n : ℕ)
    {shift : Y → ℕ → (ℕ → ℝ) → ℝ}
    (hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
    (∫ p : Y × Ω, pathProduct θ n (shift p.1) (fun i => X i p.2) ∂ν.prod P) ≤
      (1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
  let F : Y × Ω → ℝ := fun p => pathProduct θ n (shift p.1) (fun i => X i p.2)
  have hFm : Measurable F := by
    dsimp only [F, pathProduct]
    apply Finset.measurable_fun_prod
    intro i hi
    have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
      simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
      fun_prop
    have hg : Measurable (fun p : Y × Ω => (p.1, pathTail i (fun j => X j p.2))) := by
      apply Measurable.prodMk measurable_fst
      apply Measurable.of_eval
      intro j
      exact (hX (i + 1 + j)).comp measurable_snd
    exact ha.comp (((hX i).comp measurable_snd).add ((hs i).comp hg))
  have hFi : Integrable F (ν.prod P) := by
    apply (integrable_const (1 : ℝ)).mono' hFm.aestronglyMeasurable
    exact Filter.Eventually.of_forall fun p => by
      change ‖pathProduct θ n (shift p.1) (fun i => X i p.2)‖ ≤ 1
      rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
      exact pathProduct_le_one hθ.le _ _ _
  let q := 1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
  change (∫ p, F p ∂ν.prod P) ≤ q ^ n
  rw [integral_prod F hFi]
  calc
    (∫ y, ∫ ω, F (y, ω) ∂P ∂ν) ≤ ∫ _y, q ^ n ∂ν := by
      apply integral_mono hFi.integral_prod_left (integrable_const _)
      intro y
      have hsy (i : ℕ) : Measurable (shift y i) := by
        exact (hs i).comp (show Measurable (fun x : ℕ → ℝ => (y, x)) from
          measurable_const.prodMk measurable_id)
      change (∫ ω, pathProduct θ n (shift y) (fun i => X i ω) ∂P) ≤ q ^ n
      exact integral_pathProduct_le (X := X) (m := m) (v := v) hX hind hlaw hθ hδ hv n
        (shift := shift y) hsy
    _ = q ^ n := by simp

theorem exists_uniform_path_attenuation_gap
    {θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ) :
    ∃ κ : ℝ, 0 < κ ∧ κ < 1 ∧
      ∀ (n : ℕ) (m : ℕ → ℝ) (v : ℕ → ℝ≥0), (∀ i, δ ≤ v i) →
        ∀ (shift : ℕ → (ℕ → ℝ) → ℝ), (∀ i, Measurable (shift i)) →
          (∫ x, pathProduct θ n shift x
            ∂Measure.infinitePi (fun i => gaussianReal (m i) (v i))) ≤ (1 - κ) ^ n := by
  let κ := ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
  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 _)
  refine ⟨κ, by dsimp [κ]; positivity, ?_, ?_⟩
  · have hp := mul_nonneg (sq_nonneg θ) (sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
    have hlt := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
    dsimp [κ]
    nlinarith
  · intro n m v hv shift hs
    apply integral_pathProduct_le (m := m) (v := v) (fun i => measurable_pi_apply i)
      (iIndepFun_infinitePi (fun _ => measurable_id)) _ hθ hδ hv n hs
    intro i
    exact ⟨(measurable_pi_apply i).aemeasurable, Measure.infinitePi_map_eval _ i⟩

theorem integral_pathProduct_environment_le
    {Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
    {P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
    (hX : ∀ i, Measurable (X i)) (hZ : Measurable Z) (hind : iIndepFun X P)
    (henv : IndepFun Z (fun ω i => X i ω) P)
    {m : ℕ → ℝ} {v : ℕ → ℝ≥0}
    (hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
    {θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
    (hv : ∀ i, δ ≤ v i) (n : ℕ)
    {shift : Y → ℕ → (ℕ → ℝ) → ℝ}
    (hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
    (∫ ω, pathProduct θ n (shift (Z ω)) (fun i => X i ω) ∂P) ≤
      (1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
  let μ := Measure.infinitePi (fun i => gaussianReal (m i) (v i))
  have hXm : Measurable (fun ω i => X i ω) := Measurable.of_eval hX
  have harray : HasLaw (fun ω i => X i ω) μ P :=
    hind.hasLaw_infinitePi hlaw hXm.aemeasurable
  have hpair : HasLaw (fun ω => (Z ω, fun i => X i ω)) ((P.map Z).prod μ) P := by
    refine ⟨(hZ.prodMk hXm).aemeasurable, ?_⟩
    rw [henv.map_prod_eq_prod_map_map hZ.aemeasurable hXm.aemeasurable, harray.map_eq]
  have heval (i : ℕ) : HasLaw (fun x : ℕ → ℝ => x i) (gaussianReal (m i) (v i)) μ :=
    ⟨(measurable_pi_apply i).aemeasurable, Measure.infinitePi_map_eval _ i⟩
  have hb := integral_pathProduct_parameter_le (P := μ) (P.map Z)
    (fun i => measurable_pi_apply i) (iIndepFun_infinitePi (fun _ => measurable_id))
    heval hθ hδ hv n (shift := shift) hs
  calc
    (∫ ω, pathProduct θ n (shift (Z ω)) (fun i => X i ω) ∂P) =
        ∫ p : Y × (ℕ → ℝ), pathProduct θ n (shift p.1) p.2 ∂(P.map Z).prod μ :=
      hpair.integral_comp (measurable_pathProduct_parameter θ n hs).aestronglyMeasurable
    _ ≤ _ := hb

def pathResponse (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
    (x : ℕ → ℝ) : ℝ :=
  θ ^ n * ∏ i ∈ Finset.range n, responseAttenuation θ (x i + shift i (pathTail i x))

lemma pathResponse_sq (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
    (x : ℕ → ℝ) : pathResponse θ n shift x ^ 2 = (θ ^ 2) ^ n * pathProduct θ n shift x := by
  simp only [pathResponse, pathProduct, mul_pow, Finset.prod_pow]
  rw [← pow_mul, ← pow_mul, Nat.mul_comm n 2]

theorem integral_pathResponse_sq_environment_le
    {Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
    {P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
    (hX : ∀ i, Measurable (X i)) (hZ : Measurable Z) (hind : iIndepFun X P)
    (henv : IndepFun Z (fun ω i => X i ω) P)
    {m : ℕ → ℝ} {v : ℕ → ℝ≥0}
    (hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
    {θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
    (hv : ∀ i, δ ≤ v i) (n : ℕ)
    {shift : Y → ℕ → (ℕ → ℝ) → ℝ}
    (hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
    (∫ ω, pathResponse θ n (shift (Z ω)) (fun i => X i ω) ^ 2 ∂P) ≤
      (θ ^ 2 * (1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2)) ^ n := by
  simp_rw [pathResponse_sq]
  rw [integral_const_mul, mul_pow]
  exact mul_le_mul_of_nonneg_left
    (integral_pathProduct_environment_le (X := X) (Z := Z) hX hZ hind henv hlaw hθ hδ hv n hs)
    (pow_nonneg (sq_nonneg θ) n)

lemma integrable_pathResponse_sq_environment
    {Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
    {P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
    (hX : ∀ i, Measurable (X i)) (hZ : Measurable Z)
    {θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
    {shift : Y → ℕ → (ℕ → ℝ) → ℝ}
    (hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
    Integrable (fun ω => pathResponse θ n (shift (Z ω)) (fun i => X i ω) ^ 2) P := by
  simp_rw [pathResponse_sq]
  apply Integrable.const_mul
  have hm : Measurable (fun ω => pathProduct θ n (shift (Z ω)) (fun i => X i ω)) := by
    unfold pathProduct
    apply Finset.measurable_fun_prod
    intro i hi
    have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
      simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
      fun_prop
    have hg : Measurable (fun ω => (Z ω, pathTail i (fun j => X j ω))) := by
      apply Measurable.prodMk hZ
      exact Measurable.of_eval (fun index => hX (i + 1 + index))
    exact ha.comp ((hX i).add ((hs i).comp hg))
  apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
  exact Filter.Eventually.of_forall fun ω => by
    rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
    exact pathProduct_le_one hθ _ _ _

end Problem367.GaussianAttenuation

end

end OAI

	import OAI.Probability.FreeIsing.Estimates.InfiniteBlockIndependence
	import OAI.Probability.FreeIsing.Gaussian.Attenuation

	namespace OAI

	noncomputable section
	open MeasureTheory ProbabilityTheory
	open scoped BigOperators NNReal

	namespace Problem367.GaussianAttenuation

	def pathTail (i : ℕ) (x : ℕ → ℝ) : ℕ → ℝ := fun j => x (i + 1 + j)

	def pathProduct (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
	(x : ℕ → ℝ) : ℝ :=
	∏ i ∈ Finset.range n, responseAttenuation θ (x i + shift i (pathTail i x)) ^ 2

	@[fun_prop] lemma measurable_pathTail (i : ℕ) : Measurable (pathTail i) := by
	exact Measurable.of_eval (fun index => measurable_pi_apply _)

	lemma measurable_pathProduct (θ : ℝ) (n : ℕ)
	{shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
	Measurable (pathProduct θ n shift) := by
	unfold pathProduct
	apply Finset.measurable_fun_prod
	intro i hi
	have h := (measurable_pi_apply i).add ((hs i).comp (measurable_pathTail i))
	simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
	fun_prop

	lemma pathProduct_nonneg (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
	(x : ℕ → ℝ) : 0 ≤ pathProduct θ n shift x := by
	exact Finset.prod_nonneg (fun i _ => sq_nonneg _)

	lemma pathProduct_le_one {θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
	(shift : ℕ → (ℕ → ℝ) → ℝ) (x : ℕ → ℝ) : pathProduct θ n shift x ≤ 1 := by
	exact Finset.prod_le_one₀ (fun index _ => sq_nonneg _)
	(fun i _ => attenuation_sq_le_one hθ _)

	lemma integrable_pathProduct_comp {Ω : Type*} [MeasurableSpace Ω]
	{P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ}
	(hX : ∀ i, Measurable (X i)) {θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
	{shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
	Integrable (fun ω => pathProduct θ n shift (fun i => X i ω)) P := by
	apply (integrable_const (1 : ℝ)).mono'
	((measurable_pathProduct θ n hs).comp (Measurable.of_eval hX)).aestronglyMeasurable
	exact Filter.Eventually.of_forall fun ω => by
	dsimp only [Function.comp_def]
	rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
	exact pathProduct_le_one hθ _ _ _

	lemma pathProduct_succ (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
	(x : ℕ → ℝ) :
	pathProduct θ (n + 1) shift x =
	responseAttenuation θ (x 0 + shift 0 (pathTail 0 x)) ^ 2 *
	pathProduct θ n (fun i => shift (i + 1)) (pathTail 0 x) := by
	unfold pathProduct
	rw [Finset.prod_range_succ', mul_comm]
	congr 1
	apply Finset.prod_congr rfl
	intro i hi
	have hh : pathTail 0 x i = x (i + 1) := by
	simp only [pathTail, Nat.zero_add, Nat.add_comm]
	have ht : pathTail i (pathTail 0 x) = pathTail (i + 1) x := by
	funext j
	simp only [pathTail]
	congr 1
	omega
	rw [hh, ht]

	lemma head_indep_tail {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
	{X : ℕ → Ω → ℝ} (hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P) :
	IndepFun (X 0) (fun ω j => X (j + 1) ω) P := by
	have h := indepFun_of_disjoint_blocks hind hX ({0} : Set ℕ) (Set.Ioi 0)
	(by simp : Disjoint ({0} : Set ℕ) (Set.Ioi 0))
	(fun y => y ⟨0, by simp⟩)
	(fun y : Set.Ioi (0 : ℕ) → ℝ => fun j => y ⟨j + 1, show 0 < j + 1 from Nat.succ_pos j⟩)
	(measurable_pi_apply _) (by fun_prop)
	exact h

	theorem integral_pathProduct_le {Ω : Type*} [MeasurableSpace Ω]
	{P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ}
	(hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P)
	{m : ℕ → ℝ} {v : ℕ → ℝ≥0}
	(hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
	{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
	(hv : ∀ i, δ ≤ v i) (n : ℕ)
	{shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
	(∫ ω, pathProduct θ n shift (fun i => X i ω) ∂P) ≤
	(1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
	let q := 1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
	have hq : 0 ≤ q := by
	have ht := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
	have hp := mul_nonneg (sq_nonneg θ) (sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
	dsimp [q]
	nlinarith
	change (∫ ω, pathProduct θ n shift (fun i => X i ω) ∂P) ≤ q ^ n
	induction n generalizing X m v shift with
	\| zero => simp [pathProduct]
	\| succ n ih =>
	have hi := ih (fun i => hX (i + 1)) (hind.precomp (g := fun i : ℕ => i + 1) (by intro i j hij; exact Nat.add_right_cancel hij))
	(fun i => hlaw (i + 1)) (fun i => hv (i + 1)) (fun i => hs (i + 1))
	have he := integral_shifted_attenuation_sq_mul_le_of_indepFun (hlaw 0)
	(show AEMeasurable (fun ω j => X (j + 1) ω) P from
	(Measurable.of_eval (fun index => hX (index + 1))).aemeasurable)
	(head_indep_tail hX hind) (hs 0)
	(measurable_pathProduct θ n (fun i => hs (i + 1)))
	(integrable_pathProduct_comp (fun i => hX (i + 1)) hθ.le n (fun i => hs (i + 1)))
	(pathProduct_nonneg θ n (fun i => shift (i + 1))) hθ hδ (hv 0)
	calc
	(∫ ω, pathProduct θ (n + 1) shift (fun i => X i ω) ∂P) =
	∫ ω, responseAttenuation θ (X 0 ω + shift 0 (fun j => X (j + 1) ω)) ^ 2 *
	pathProduct θ n (fun i => shift (i + 1)) (fun j => X (j + 1) ω) ∂P := by
	apply integral_congr_ae
	exact Filter.Eventually.of_forall fun ω => by
	have hz : pathTail 0 (fun i => X i ω) = (fun j => X (j + 1) ω) := by
	funext j
	simp only [pathTail, Nat.zero_add, Nat.add_comm]
	simpa only [hz] using pathProduct_succ θ n shift (fun i => X i ω)
	_ ≤ q * ∫ ω, pathProduct θ n (fun i => shift (i + 1))
	(fun j => X (j + 1) ω) ∂P := he
	_ ≤ q * q ^ n := mul_le_mul_of_nonneg_left hi hq
	_ = q ^ (n + 1) := (pow_succ' q n).symm

	lemma measurable_pathProduct_parameter {Y : Type*} [MeasurableSpace Y]
	(θ : ℝ) (n : ℕ) {shift : Y → ℕ → (ℕ → ℝ) → ℝ}
	(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
	Measurable (fun p : Y × (ℕ → ℝ) => pathProduct θ n (shift p.1) p.2) := by
	unfold pathProduct
	apply Finset.measurable_fun_prod
	intro i hi
	have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
	simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
	fun_prop
	exact ha.comp (((measurable_pi_apply i).comp measurable_snd).add
	((hs i).comp (measurable_fst.prodMk ((measurable_pathTail i).comp measurable_snd))))

	theorem integral_pathProduct_parameter_le
	{Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
	(ν : Measure Y) [IsProbabilityMeasure ν]
	{P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ}
	(hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P)
	{m : ℕ → ℝ} {v : ℕ → ℝ≥0}
	(hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
	{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
	(hv : ∀ i, δ ≤ v i) (n : ℕ)
	{shift : Y → ℕ → (ℕ → ℝ) → ℝ}
	(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
	(∫ p : Y × Ω, pathProduct θ n (shift p.1) (fun i => X i p.2) ∂ν.prod P) ≤
	(1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
	let F : Y × Ω → ℝ := fun p => pathProduct θ n (shift p.1) (fun i => X i p.2)
	have hFm : Measurable F := by
	dsimp only [F, pathProduct]
	apply Finset.measurable_fun_prod
	intro i hi
	have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
	simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
	fun_prop
	have hg : Measurable (fun p : Y × Ω => (p.1, pathTail i (fun j => X j p.2))) := by
	apply Measurable.prodMk measurable_fst
	apply Measurable.of_eval
	intro j
	exact (hX (i + 1 + j)).comp measurable_snd
	exact ha.comp (((hX i).comp measurable_snd).add ((hs i).comp hg))
	have hFi : Integrable F (ν.prod P) := by
	apply (integrable_const (1 : ℝ)).mono' hFm.aestronglyMeasurable
	exact Filter.Eventually.of_forall fun p => by
	change ‖pathProduct θ n (shift p.1) (fun i => X i p.2)‖ ≤ 1
	rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
	exact pathProduct_le_one hθ.le _ _ _
	let q := 1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
	change (∫ p, F p ∂ν.prod P) ≤ q ^ n
	rw [integral_prod F hFi]
	calc
	(∫ y, ∫ ω, F (y, ω) ∂P ∂ν) ≤ ∫ _y, q ^ n ∂ν := by
	apply integral_mono hFi.integral_prod_left (integrable_const _)
	intro y
	have hsy (i : ℕ) : Measurable (shift y i) := by
	exact (hs i).comp (show Measurable (fun x : ℕ → ℝ => (y, x)) from
	measurable_const.prodMk measurable_id)
	change (∫ ω, pathProduct θ n (shift y) (fun i => X i ω) ∂P) ≤ q ^ n
	exact integral_pathProduct_le (X := X) (m := m) (v := v) hX hind hlaw hθ hδ hv n
	(shift := shift y) hsy
	_ = q ^ n := by simp

	theorem exists_uniform_path_attenuation_gap
	{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ) :
	∃ κ : ℝ, 0 < κ ∧ κ < 1 ∧
	∀ (n : ℕ) (m : ℕ → ℝ) (v : ℕ → ℝ≥0), (∀ i, δ ≤ v i) →
	∀ (shift : ℕ → (ℕ → ℝ) → ℝ), (∀ i, Measurable (shift i)) →
	(∫ x, pathProduct θ n shift x
	∂Measure.infinitePi (fun i => gaussianReal (m i) (v i))) ≤ (1 - κ) ^ n := by
	let κ := ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
	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 _)
	refine ⟨κ, by dsimp [κ]; positivity, ?_, ?_⟩
	· have hp := mul_nonneg (sq_nonneg θ) (sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
	have hlt := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
	dsimp [κ]
	nlinarith
	· intro n m v hv shift hs
	apply integral_pathProduct_le (m := m) (v := v) (fun i => measurable_pi_apply i)
	(iIndepFun_infinitePi (fun _ => measurable_id)) _ hθ hδ hv n hs
	intro i
	exact ⟨(measurable_pi_apply i).aemeasurable, Measure.infinitePi_map_eval _ i⟩

	theorem integral_pathProduct_environment_le
	{Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
	{P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
	(hX : ∀ i, Measurable (X i)) (hZ : Measurable Z) (hind : iIndepFun X P)
	(henv : IndepFun Z (fun ω i => X i ω) P)
	{m : ℕ → ℝ} {v : ℕ → ℝ≥0}
	(hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
	{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
	(hv : ∀ i, δ ≤ v i) (n : ℕ)
	{shift : Y → ℕ → (ℕ → ℝ) → ℝ}
	(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
	(∫ ω, pathProduct θ n (shift (Z ω)) (fun i => X i ω) ∂P) ≤
	(1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
	let μ := Measure.infinitePi (fun i => gaussianReal (m i) (v i))
	have hXm : Measurable (fun ω i => X i ω) := Measurable.of_eval hX
	have harray : HasLaw (fun ω i => X i ω) μ P :=
	hind.hasLaw_infinitePi hlaw hXm.aemeasurable
	have hpair : HasLaw (fun ω => (Z ω, fun i => X i ω)) ((P.map Z).prod μ) P := by
	refine ⟨(hZ.prodMk hXm).aemeasurable, ?_⟩
	rw [henv.map_prod_eq_prod_map_map hZ.aemeasurable hXm.aemeasurable, harray.map_eq]
	have heval (i : ℕ) : HasLaw (fun x : ℕ → ℝ => x i) (gaussianReal (m i) (v i)) μ :=
	⟨(measurable_pi_apply i).aemeasurable, Measure.infinitePi_map_eval _ i⟩
	have hb := integral_pathProduct_parameter_le (P := μ) (P.map Z)
	(fun i => measurable_pi_apply i) (iIndepFun_infinitePi (fun _ => measurable_id))
	heval hθ hδ hv n (shift := shift) hs
	calc
	(∫ ω, pathProduct θ n (shift (Z ω)) (fun i => X i ω) ∂P) =
	∫ p : Y × (ℕ → ℝ), pathProduct θ n (shift p.1) p.2 ∂(P.map Z).prod μ :=
	hpair.integral_comp (measurable_pathProduct_parameter θ n hs).aestronglyMeasurable
	_ ≤ _ := hb

	def pathResponse (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
	(x : ℕ → ℝ) : ℝ :=
	θ ^ n * ∏ i ∈ Finset.range n, responseAttenuation θ (x i + shift i (pathTail i x))

	lemma pathResponse_sq (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
	(x : ℕ → ℝ) : pathResponse θ n shift x ^ 2 = (θ ^ 2) ^ n * pathProduct θ n shift x := by
	simp only [pathResponse, pathProduct, mul_pow, Finset.prod_pow]
	rw [← pow_mul, ← pow_mul, Nat.mul_comm n 2]

	theorem integral_pathResponse_sq_environment_le
	{Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
	{P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
	(hX : ∀ i, Measurable (X i)) (hZ : Measurable Z) (hind : iIndepFun X P)
	(henv : IndepFun Z (fun ω i => X i ω) P)
	{m : ℕ → ℝ} {v : ℕ → ℝ≥0}
	(hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
	{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
	(hv : ∀ i, δ ≤ v i) (n : ℕ)
	{shift : Y → ℕ → (ℕ → ℝ) → ℝ}
	(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
	(∫ ω, pathResponse θ n (shift (Z ω)) (fun i => X i ω) ^ 2 ∂P) ≤
	(θ ^ 2 * (1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2)) ^ n := by
	simp_rw [pathResponse_sq]
	rw [integral_const_mul, mul_pow]
	exact mul_le_mul_of_nonneg_left
	(integral_pathProduct_environment_le (X := X) (Z := Z) hX hZ hind henv hlaw hθ hδ hv n hs)
	(pow_nonneg (sq_nonneg θ) n)

	lemma integrable_pathResponse_sq_environment
	{Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
	{P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
	(hX : ∀ i, Measurable (X i)) (hZ : Measurable Z)
	{θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
	{shift : Y → ℕ → (ℕ → ℝ) → ℝ}
	(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
	Integrable (fun ω => pathResponse θ n (shift (Z ω)) (fun i => X i ω) ^ 2) P := by
	simp_rw [pathResponse_sq]
	apply Integrable.const_mul
	have hm : Measurable (fun ω => pathProduct θ n (shift (Z ω)) (fun i => X i ω)) := by
	unfold pathProduct
	apply Finset.measurable_fun_prod
	intro i hi
	have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
	simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
	fun_prop
	have hg : Measurable (fun ω => (Z ω, pathTail i (fun j => X j ω))) := by
	apply Measurable.prodMk hZ
	exact Measurable.of_eval (fun index => hX (i + 1 + index))
	exact ha.comp ((hX i).add ((hs i).comp hg))
	apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
	exact Filter.Eventually.of_forall fun ω => by
	rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
	exact pathProduct_le_one hθ _ _ _

	end Problem367.GaussianAttenuation

	end

	end OAI