import Mathlib
import OAI.Computability.VertexCover.Information.Conditionals
import OAI.Computability.VertexCover.Information.Kernels

namespace OAI

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namespace UniqueGames.Foundations.Repetition.ProfileCorrection
open scoped BigOperators
open UniqueGames.Foundations.Information
noncomputable section
variable {S X Y : Type*} [Fintype S] [Fintype X] [Fintype Y]

def seedQuestionMarginal (p : X × (S × Y) → ℝ) : X × S → ℝ :=
  firstMarginal (fun z : (X × S) × Y => p (z.1.1, (z.1.2, z.2)))

theorem seedQuestionMarginal_isProbability (p : X × (S × Y) → ℝ)
    (hp : IsProbability p) : IsProbability (seedQuestionMarginal p) := by
  apply firstMarginal_isProbability
  constructor
  · intro z
    exact hp.1 _
  · have h := hp.2
    simp only [Fintype.sum_prod_type] at h ⊢
    exact h

omit [Fintype X] in
theorem seedQuestionMarginal_first (p : X × (S × Y) → ℝ) :
    firstMarginal (seedQuestionMarginal p) = firstMarginal p := by
  funext x
  simp only [seedQuestionMarginal, firstMarginal, Fintype.sum_prod_type]

theorem totalVariation_triangle {Ω : Type*} [Fintype Ω]
    (p «c» d : Ω → ℝ) :
    totalVariation p d ≤ totalVariation p «c» + totalVariation «c» d := by
  unfold totalVariation
  have h := Finset.sum_le_sum (s := Finset.univ)
    (fun x _ => abs_sub_le (p x) («c» x) (d x))
  rw [Finset.sum_add_distrib] at h
  linarith

theorem left_profile_correction
    (p : X × (S × Y) → ℝ) (μ : X × Y → ℝ)
    (fallbackS : S → ℝ) (fallbackY : Y → ℝ)
    (hp : IsProbability p) (hμ : IsProbability μ)
    (hS : IsProbability fallbackS) (hY : IsProbability fallbackY) :
    totalVariation p (fun z => μ (z.1, z.2.2) *
      conditionalKernel (seedQuestionMarginal p) fallbackS z.1 z.2.1) ≤
    totalVariation p (fun z => seedQuestionMarginal p (z.1, z.2.1) *
      conditionalKernel μ fallbackY z.1 z.2.2) +
    totalVariation (firstMarginal p) (firstMarginal μ) := by
  let σ := seedQuestionMarginal p
  let L := conditionalKernel σ fallbackS
  let M := conditionalKernel μ fallbackY
  let k : X → S × Y → ℝ := fun x sy => L x sy.1 * M x sy.2
  let «c» : X × (S × Y) → ℝ := fun z => σ (z.1,z.2.1) * M z.1 z.2.2
  let d : X × (S × Y) → ℝ := fun z => μ (z.1,z.2.2) * L z.1 z.2.1
  have hσ : IsProbability σ := seedQuestionMarginal_isProbability p hp
  have hk : ∀ x, IsProbability (k x) := by
    intro x
    exact kernelProduct_isProbability (L x) (fun _ : S => M x)
      (conditionalKernel_isProbability σ fallbackS hσ hS x)
      (fun _ => conditionalKernel_isProbability μ fallbackY hμ hY x)
  have hc : «c» = fun z => firstMarginal σ z.1 * k z.1 z.2 := by
    funext z
    change σ (z.1,z.2.1) * M z.1 z.2.2 =
      firstMarginal σ z.1 * (L z.1 z.2.1 * M z.1 z.2.2)
    rw [← mul_assoc, marginal_mul_conditionalKernel σ fallbackS hσ]
  have hd : d = fun z => firstMarginal μ z.1 * k z.1 z.2 := by
    funext z
    change μ (z.1,z.2.2) * L z.1 z.2.1 =
      firstMarginal μ z.1 * (L z.1 z.2.1 * M z.1 z.2.2)
    calc
      _ = (firstMarginal μ z.1 * M z.1 z.2.2) * L z.1 z.2.1 := by
        rw [marginal_mul_conditionalKernel μ fallbackY hμ]
      _ = _ := by ring
  have hmiddle : totalVariation «c» d =
      totalVariation (firstMarginal σ) (firstMarginal μ) := by
    rw [hc, hd]
    exact totalVariation_joint_common_kernel _ _ k hk
  have result := totalVariation_triangle p «c» d
  rw [hmiddle] at result
  have hmarg : firstMarginal σ = firstMarginal p := seedQuestionMarginal_first p
  rw [hmarg] at result
  exact result
end
end UniqueGames.Foundations.Repetition.ProfileCorrection
end


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end OAI