import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring

namespace OAI

namespace FiniteFactor
open MeasureTheory
open scoped ENNReal
noncomputable section

/-- Normalized restriction density, relative to the probability base measure. -/
def intervalDensity {X : Type*} (U : Set X) (length : ℝ) : X → ℝ :=
  U.indicator (fun _ => length⁻¹)

variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ]

theorem intervalDensity_integrable {U : Set X} (hU : MeasurableSet U) (l : ℝ) :
    Integrable (intervalDensity U l) μ := (integrable_const _).indicator hU

omit [MeasurableSpace X] in
theorem intervalDensity_nonneg [MeasurableSpace X] {U : Set X} {l : ℝ} (hl : 0 ≤ l) (x : X) :
    0 ≤ intervalDensity U l x := by
  classical
  by_cases hx : x ∈ U <;> simp [intervalDensity, hx, inv_nonneg.mpr hl]

omit [MeasurableSpace X] in
theorem intervalDensity_le [MeasurableSpace X] {U : Set X} {l : ℝ} (hl : 0 ≤ l) (x : X) :
    intervalDensity U l x ≤ l⁻¹ := by
  classical
  by_cases hx : x ∈ U <;> simp [intervalDensity, hx, inv_nonneg.mpr hl]

omit [IsProbabilityMeasure μ] in
theorem intervalDensity_integral [IsProbabilityMeasure μ] {U : Set X} (hU : MeasurableSet U)
    {l : ℝ} (hl : 0 < l) (hm : μ.real U = l) :
    ∫ x, intervalDensity U l x ∂μ = 1 := by
  rw [intervalDensity, integral_indicator hU, setIntegral_const, smul_eq_mul, hm]
  exact mul_inv_cancel₀ hl.ne'

theorem intervalDensity_sq_integrable {U : Set X} (hU : MeasurableSet U) (l : ℝ) :
    Integrable (fun x => intervalDensity U l x ^ 2) μ := by
  have heq : (fun x => intervalDensity U l x ^ 2) = U.indicator (fun _ => (l⁻¹)^2) := by
    funext x
    by_cases hx : x ∈ U <;> simp [intervalDensity, hx]
  rw [heq]
  exact (integrable_const _).indicator hU

omit [IsProbabilityMeasure μ] in
theorem intervalDensity_sq_integral [IsProbabilityMeasure μ] {U : Set X} (hU : MeasurableSet U)
    {l : ℝ} (hl : 0 < l) (hm : μ.real U = l) :
    ∫ x, intervalDensity U l x ^ 2 ∂μ = l⁻¹ := by
  have heq : (fun x => intervalDensity U l x ^ 2) = U.indicator (fun _ => (l⁻¹)^2) := by
    funext x
    by_cases hx : x ∈ U <;> simp [intervalDensity, hx]
  rw [heq, integral_indicator hU, setIntegral_const, smul_eq_mul, hm]
  field_simp

/-- The mass-one identity cancels the linear term exactly. -/
theorem density_energy_identity {h : X → ℝ} (hh : Integrable h μ)
    (hh2 : Integrable (fun x => h x ^ 2) μ) (hm : ∫ x, h x ∂μ = 1) :
    (∫ x, (h x - 1)^2 ∂μ) = (∫ x, h x ^ 2 ∂μ) - 1 := by
  have heq : (fun x => (h x - 1)^2) = fun x => h x ^ 2 - 2*h x + 1 := by
    funext x; ring
  have h₁ := integral_add (hh2.sub (hh.const_mul 2)) (integrable_const (1 : ℝ))
  have h₂ := integral_sub hh2 (hh.const_mul 2)
  dsimp only [Pi.sub_apply, Pi.add_apply] at h₁ h₂
  rw [heq, h₁, h₂, integral_const_mul, hm]
  simp
  ring

/-- Each normalized interval perturbation has energy exactly 1/length - 1. -/
theorem intervalDensity_energy {U : Set X} (hU : MeasurableSet U)
    {l : ℝ} (hl : 0 < l) (hm : μ.real U = l) :
    (∫ x, (intervalDensity U l x - 1)^2 ∂μ) = l⁻¹ - 1 := by
  rw [density_energy_identity (intervalDensity_integrable hU l)
    (intervalDensity_sq_integrable hU l) (intervalDensity_integral hU hl hm),
    intervalDensity_sq_integral hU hl hm]

/-- Finite independent endpoint selections: the square integral factors exactly. -/
theorem product_density_sq_integral {ι : Type*} [Fintype ι]
    {E : ι → Type*} [∀ i, MeasurableSpace (E i)]
    (ν : ∀ i, Measure (E i)) [∀ i, SigmaFinite (ν i)]
    (h : ∀ i, E i → ℝ) :
    (∫ x, (∏ i, h i (x i))^2 ∂Measure.pi ν) =
      ∏ i, ∫ x, h i x ^ 2 ∂ν i := by
  simp_rw [← Finset.prod_pow]
  exact integral_fintype_prod_eq_prod (fun i x => h i x ^ 2)

/-- Finite product estimate used in uniform-product-density; the bound has
no dependence on the choice of endpoint orientation. -/
theorem product_energy_le_exp {ι : Type*} [Fintype ι]
    (A e : ι → ℝ) (hA0 : ∀ i, 0 ≤ A i) (hA : ∀ i, A i ≤ 1 + e i) :
    (∏ i, A i) ≤ Real.exp (∑ i, e i) := by
  calc
    (∏ i, A i) ≤ ∏ i, Real.exp (e i) := by
      apply Finset.prod_le_prod₀ (fun i _ => hA0 i)
      intro i _
      exact (hA i).trans (by linarith [Real.add_one_le_exp (e i)])
    _ = _ := (Real.exp_sum _ _).symm

end
end FiniteFactor

end OAI