Datasets:

roseking
/

openAI-math-dataset

import OAI.Computability.UniqueGames.Games.FactorizationLemmas
import OAI.Computability.UniqueGames.Games.FinishBoundsLemmas
import OAI.Computability.UniqueGames.Games.PartialRevealLemmas
import OAI.Computability.UniqueGames.Games.SelectedCommonMarginalLemmas
import OAI.Computability.UniqueGames.Games.SelectionSequence

namespace OAI

section

/-!
# Finite local completion after correlated sampling

One independent seed contains the original shared seed and two full response
tables. Its local reads give exactly the mixture of the requested completion
kernels. Applying those kernels contracts total variation. On a diagonal
target the same construction uses one common selected label.
-/

namespace UniqueGamesTheorem.Foundations.Repetition.CompletedSampling

open scoped BigOperators
open Games

noncomputable section

section DistributionIdentities

variable {A B C D : Type*} [Fintype A] [Fintype B] [Fintype C] [Fintype D]

theorem mixture_pushforward_base (μ : FiniteDistribution A) (f : A → B)
    (K : B → FiniteDistribution C) :
    (μ.pushforward f).mixture K = μ.mixture (fun a => K (f a)) := by
  apply FiniteDistribution.eq_of_weight_eq
  intro c
  exact FiniteDistribution.expectation_pushforward μ f (fun b => (K b).weight c)

theorem product_pushforward_eq_mixture_right (μ : FiniteDistribution A)
    (ν : FiniteDistribution B) (f : A × B → C) :
    (μ.product ν).pushforward f =
      ν.mixture (fun b => μ.pushforward (fun a => f (a, b))) := by
  classical
  apply FiniteDistribution.eq_of_weight_eq
  intro c
  simp only [FiniteDistribution.pushforward, FiniteDistribution.product,
    FiniteDistribution.mixture, Fintype.sum_prod_type]
  rw [Finset.sum_comm]
  apply Finset.sum_congr rfl
  intro b _
  rw [Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro a _
  by_cases h : f (a, b) = c <;> simp [h, mul_comm]

theorem product_product_pushforward_eq_mixture (μ : FiniteDistribution A)
    (ν : FiniteDistribution B) (κ : FiniteDistribution C) (f : A → B → C → D) :
    (μ.product (ν.product κ)).pushforward (fun z => f z.1 z.2.1 z.2.2) =
      (μ.product ν).mixture
        (fun z => κ.pushforward (fun c => f z.1 z.2 c)) := by
  classical
  apply FiniteDistribution.eq_of_weight_eq
  intro d
  simp only [FiniteDistribution.pushforward, FiniteDistribution.product,
    FiniteDistribution.mixture, Fintype.sum_prod_type]
  simp only [Finset.mul_sum, mul_ite, mul_zero, mul_assoc]

theorem mixture_totalVariation_le (μ ν : FiniteDistribution A)
    (K : A → FiniteDistribution B) :
    (μ.mixture K).totalVariation (ν.mixture K) ≤ μ.totalVariation ν := by
  simpa only [FiniteDistribution.totalVariation, FiniteDistribution.mixture,
    Information.totalVariation, kernelPushforward] using
      totalVariation_kernelPushforward_le μ.weight ν.weight
        (fun a => (K a).weight) (fun a => Information.gameLaw_isProbability (K a))

end DistributionIdentities

section Diagonal

variable {Z S : Type*} [Fintype Z] [Fintype S]

def diagonalLaw (σ : FiniteDistribution (Z × S)) : FiniteDistribution (Z × S × S) :=
  Information.toGameLaw (diagonalWeights σ.weight)
    (diagonalWeights_isProbability _ (Information.gameLaw_isProbability σ))

theorem diagonalLaw_eq_pushforward (σ : FiniteDistribution (Z × S)) :
    diagonalLaw σ = σ.pushforward (fun z => (z.1, z.2, z.2)) := by
  classical
  apply FiniteDistribution.eq_of_weight_eq
  rintro ⟨z, a, b⟩
  change diagonalWeights σ.weight (z, a, b) =
    (σ.pushforward (fun t => (t.1, t.2, t.2))).weight (z, a, b)
  simp only [diagonalWeights, FiniteDistribution.pushforward,
    Fintype.sum_prod_type, Prod.mk.injEq]
  by_cases hab : a = b
  · subst b
    simp [ite_and]
  · simp [hab, ite_and]

theorem diagonalLaw_mixture {T : Type*} [Fintype T]
    (σ : FiniteDistribution (Z × S)) (K : Z × S × S → FiniteDistribution T) :
    (diagonalLaw σ).mixture K = σ.mixture (fun z => K (z.1, z.2, z.2)) := by
  rw [diagonalLaw_eq_pushforward, mixture_pushforward_base]

end Diagonal

section Completion

variable {X Y S Γ U V : Type*}
  [Fintype X] [Fintype Y] [Fintype S] [Fintype Γ] [Fintype U] [Fintype V]
  [DecidableEq X] [DecidableEq Y] [DecidableEq S]

/-- The independent original seed and two full local completion tables. -/
abbrev Seed (Γ X Y S U V : Type*) :=
  Γ × KernelSampling.Seed (X × S) (Y × S) U V

def seedLaw (γ : FiniteDistribution Γ)
    (L : X × S → FiniteDistribution U) (R : Y × S → FiniteDistribution V) :
    FiniteDistribution (Seed Γ X Y S U V) :=
  γ.product (KernelSampling.seedLaw L R)

def left (sL : Γ → X → S) (seed : Seed Γ X Y S U V) (x : X) : U :=
  KernelSampling.readLeft seed.2 (x, sL seed.1 x)

def right (sR : Γ → Y → S) (seed : Seed Γ X Y S U V) (y : Y) : V :=
  KernelSampling.readRight seed.2 (y, sR seed.1 y)

def completionKernel (L : X × S → FiniteDistribution U)
    (R : Y × S → FiniteDistribution V) (z : (X × Y) × S × S) :
    FiniteDistribution (U × V) :=
  (L (z.1.1, z.2.1)).product (R (z.1.2, z.2.2))

/-- This is the shared-seed mixture used by `Game.localEmbeddingLaw`. -/
def outputLaw (μ : FiniteDistribution (X × Y)) (γ : FiniteDistribution Γ)
    (sL : Γ → X → S) (sR : Γ → Y → S)
    (L : X × S → FiniteDistribution U) (R : Y × S → FiniteDistribution V) :
    FiniteDistribution (U × V) :=
  (seedLaw γ L R).mixture (fun seed =>
    μ.pushforward (fun q => (left sL seed q.1, right sR seed q.2)))

theorem outputLaw_eq_product_pushforward
    (μ : FiniteDistribution (X × Y)) (γ : FiniteDistribution Γ)
    (sL : Γ → X → S) (sR : Γ → Y → S)
    (L : X × S → FiniteDistribution U) (R : Y × S → FiniteDistribution V) :
    outputLaw μ γ sL sR L R =
      (μ.product (seedLaw γ L R)).pushforward
        (fun z => (left sL z.2 z.1.1, right sR z.2 z.1.2)) := by
  exact (product_pushforward_eq_mixture_right μ (seedLaw γ L R)
    (fun z : (X × Y) × Seed Γ X Y S U V =>
      (left sL z.2 z.1.1, right sR z.2 z.1.2))).symm

/-- The completion mixture is derived from actual local table reads. -/
theorem outputLaw_eq_mixture
    (μ : FiniteDistribution (X × Y)) (γ : FiniteDistribution Γ)
    (sL : Γ → X → S) (sR : Γ → Y → S)
    (L : X × S → FiniteDistribution U) (R : Y × S → FiniteDistribution V) :
    outputLaw μ γ sL sR L R =
      (sharedOutputLaw μ γ sL sR).mixture (completionKernel L R) := by
  calc
    outputLaw μ γ sL sR L R =
        (μ.product γ).mixture (fun z =>
          (KernelSampling.seedLaw L R).pushforward (fun table =>
            (KernelSampling.readLeft table (z.1.1, sL z.2 z.1.1),
             KernelSampling.readRight table (z.1.2, sR z.2 z.1.2)))) := by
      rw [outputLaw_eq_product_pushforward]
      exact product_product_pushforward_eq_mixture μ γ (KernelSampling.seedLaw L R)
        (fun q seed table =>
          (KernelSampling.readLeft table (q.1, sL seed q.1),
           KernelSampling.readRight table (q.2, sR seed q.2)))
    _ = (μ.product γ).mixture (fun z =>
        completionKernel L R (z.1, sL z.2 z.1.1, sR z.2 z.1.2)) := by
      apply congrArg ((μ.product γ).mixture)
      funext z
      exact KernelSampling.read_joint_pushforward L R
        (z.1.1, sL z.2 z.1.1) (z.1.2, sR z.2 z.1.2)
    _ = (sharedOutputLaw μ γ sL sR).mixture (completionKernel L R) := by
      exact (mixture_pushforward_base (μ.product γ)
        (fun z => (z.1, sL z.2 z.1.1, sR z.2 z.1.2)) (completionKernel L R)).symm

omit [DecidableEq X] [DecidableEq Y] [DecidableEq S] in
/-- The diagonal target uses the same selected label for both completion rows. -/
theorem diagonalLaw_completion_mixture
    (σ : FiniteDistribution ((X × Y) × S))
    (L : X × S → FiniteDistribution U) (R : Y × S → FiniteDistribution V) :
    (diagonalLaw σ).mixture (completionKernel L R) =
      σ.mixture (fun z => (L (z.1.1, z.2)).product (R (z.1.2, z.2))) := by
  exact diagonalLaw_mixture σ (completionKernel L R)

/-- Completing local outputs incurs no additional total-variation loss. -/
theorem outputLaw_totalVariation_le
    (μ : FiniteDistribution (X × Y)) (γ : FiniteDistribution Γ)
    (sL : Γ → X → S) (sR : Γ → Y → S)
    (L : X × S → FiniteDistribution U) (R : Y × S → FiniteDistribution V)
    (σ : FiniteDistribution ((X × Y) × S)) :
    (outputLaw μ γ sL sR L R).totalVariation
        (σ.mixture (fun z => (L (z.1.1, z.2)).product (R (z.1.2, z.2)))) ≤
      (sharedOutputLaw μ γ sL sR).totalVariation (diagonalLaw σ) := by
  rw [outputLaw_eq_mixture, ← diagonalLaw_completion_mixture σ L R]
  exact mixture_totalVariation_le _ _ (completionKernel L R)

end Completion

section CoordinateCompletion

variable {I X Y S Γ : Type*}
  [Fintype I] [Fintype X] [Fintype Y] [Fintype S] [Fintype Γ]
  [DecidableEq I] [DecidableEq X] [DecidableEq Y] [DecidableEq S]

/-- Restore the left input coordinate even on completion tables of weight zero. -/
def coordinateLeft (j : I) (sL : Γ → X → S)
    (seed : Seed Γ X Y S (I → X) (I → Y)) (x : X) : I → X :=
  KernelSampling.completedLeft j seed.2 (x, sL seed.1 x)

/-- Restore the right input coordinate even on completion tables of weight zero. -/
def coordinateRight (j : I) (sR : Γ → Y → S)
    (seed : Seed Γ X Y S (I → X) (I → Y)) (y : Y) : I → Y :=
  KernelSampling.completedRight j seed.2 (y, sR seed.1 y)

omit [Fintype I] [Fintype X] [Fintype Y] [Fintype S] [Fintype Γ]
  [DecidableEq X] [DecidableEq Y] [DecidableEq S] in
@[simp] theorem coordinateLeft_preserves (j : I) (sL : Γ → X → S)
    (seed : Seed Γ X Y S (I → X) (I → Y)) (x : X) :
    coordinateLeft j sL seed x j = x := by
  simp [coordinateLeft]

omit [Fintype I] [Fintype X] [Fintype Y] [Fintype S] [Fintype Γ]
  [DecidableEq X] [DecidableEq Y] [DecidableEq S] in
@[simp] theorem coordinateRight_preserves (j : I) (sR : Γ → Y → S)
    (seed : Seed Γ X Y S (I → X) (I → Y)) (y : Y) :
    coordinateRight j sR seed y j = y := by
  simp [coordinateRight]

def coordinateOutputLaw (μ : FiniteDistribution (X × Y)) (γ : FiniteDistribution Γ)
    (j : I) (sL : Γ → X → S) (sR : Γ → Y → S)
    (L : X × S → FiniteDistribution (I → X))
    (R : Y × S → FiniteDistribution (I → Y)) :
    FiniteDistribution ((I → X) × (I → Y)) :=
  (seedLaw γ L R).mixture (fun seed => μ.pushforward
    (fun q => (coordinateLeft j sL seed q.1, coordinateRight j sR seed q.2)))

/-- Supported coordinate repair preserves the exact completion mixture. -/
theorem coordinateOutputLaw_eq_mixture
    (μ : FiniteDistribution (X × Y)) (γ : FiniteDistribution Γ)
    (j : I) (sL : Γ → X → S) (sR : Γ → Y → S)
    (L : X × S → FiniteDistribution (I → X))
    (R : Y × S → FiniteDistribution (I → Y))
    (hL : ∀ q xs, (L q).weight xs ≠ 0 → xs j = q.1)
    (hR : ∀ q ys, (R q).weight ys ≠ 0 → ys j = q.1) :
    coordinateOutputLaw μ γ j sL sR L R =
      (sharedOutputLaw μ γ sL sR).mixture (completionKernel L R) := by
  calc
    coordinateOutputLaw μ γ j sL sR L R =
        (μ.product (seedLaw γ L R)).pushforward
          (fun z => (coordinateLeft j sL z.2 z.1.1,
            coordinateRight j sR z.2 z.1.2)) := by
      exact (product_pushforward_eq_mixture_right μ (seedLaw γ L R)
        (fun z : (X × Y) × Seed Γ X Y S (I → X) (I → Y) =>
          (coordinateLeft j sL z.2 z.1.1, coordinateRight j sR z.2 z.1.2))).symm
    _ = (μ.product γ).mixture (fun z =>
        (KernelSampling.seedLaw L R).pushforward (fun table =>
          (KernelSampling.completedLeft j table (z.1.1, sL z.2 z.1.1),
           KernelSampling.completedRight j table (z.1.2, sR z.2 z.1.2)))) := by
      exact product_product_pushforward_eq_mixture μ γ (KernelSampling.seedLaw L R)
        (fun q seed table =>
          (KernelSampling.completedLeft j table (q.1, sL seed q.1),
           KernelSampling.completedRight j table (q.2, sR seed q.2)))
    _ = (μ.product γ).mixture (fun z =>
        completionKernel L R (z.1, sL z.2 z.1.1, sR z.2 z.1.2)) := by
      apply congrArg ((μ.product γ).mixture)
      funext z
      exact KernelSampling.completed_joint_pushforward L R j hL hR
        (z.1.1, sL z.2 z.1.1) (z.1.2, sR z.2 z.1.2)
    _ = (sharedOutputLaw μ γ sL sR).mixture (completionKernel L R) := by
      exact (mixture_pushforward_base (μ.product γ)
        (fun z => (z.1, sL z.2 z.1.1, sR z.2 z.1.2)) (completionKernel L R)).symm

theorem coordinateOutputLaw_eq_outputLaw
    (μ : FiniteDistribution (X × Y)) (γ : FiniteDistribution Γ)
    (j : I) (sL : Γ → X → S) (sR : Γ → Y → S)
    (L : X × S → FiniteDistribution (I → X))
    (R : Y × S → FiniteDistribution (I → Y))
    (hL : ∀ q xs, (L q).weight xs ≠ 0 → xs j = q.1)
    (hR : ∀ q ys, (R q).weight ys ≠ 0 → ys j = q.1) :
    coordinateOutputLaw μ γ j sL sR L R = outputLaw μ γ sL sR L R := by
  rw [coordinateOutputLaw_eq_mixture μ γ j sL sR L R hL hR, outputLaw_eq_mixture]

theorem coordinateOutputLaw_totalVariation_le
    (μ : FiniteDistribution (X × Y)) (γ : FiniteDistribution Γ)
    (j : I) (sL : Γ → X → S) (sR : Γ → Y → S)
    (L : X × S → FiniteDistribution (I → X))
    (R : Y × S → FiniteDistribution (I → Y))
    (hL : ∀ q xs, (L q).weight xs ≠ 0 → xs j = q.1)
    (hR : ∀ q ys, (R q).weight ys ≠ 0 → ys j = q.1)
    (σ : FiniteDistribution ((X × Y) × S)) :
    (coordinateOutputLaw μ γ j sL sR L R).totalVariation
        (σ.mixture (fun z => (L (z.1.1, z.2)).product (R (z.1.2, z.2)))) ≤
      (sharedOutputLaw μ γ sL sR).totalVariation (diagonalLaw σ) := by
  rw [coordinateOutputLaw_eq_outputLaw μ γ j sL sR L R hL hR]
  exact outputLaw_totalVariation_le μ γ sL sR L R σ

end CoordinateCompletion

end

end UniqueGamesTheorem.Foundations.Repetition.CompletedSampling

end

section

/-! An explicit finite shared sampler followed by supported local completion
gives the coordinate success bound. No embedding-existence premise remains. -/
namespace UniqueGamesTheorem.Foundations.Repetition
open scoped BigOperators
open Games CorrelatedSampling
noncomputable section
variable {Q₁ Q₂ A₁ A₂ S : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] [Fintype S]
  [Nonempty A₁] [Nonempty A₂] [Nonempty S]
  [DecidableEq Q₁] [DecidableEq Q₂] [DecidableEq S] {n : Nat}

theorem coordinate_probability_le_value_of_local_completion
    (G : Game Q₁ Q₂ A₁ A₂) (j : Fin n)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (σ : FiniteDistribution ((Q₁ × Q₂) × S))
    (profileL : Q₁ → FiniteDistribution S) (profileR : Q₂ → FiniteDistribution S)
    (completionL : Q₁ × S → FiniteDistribution (Fin n → Q₁))
    (completionR : Q₂ × S → FiniteDistribution (Fin n → Q₂))
    (supportL : ∀ q xs, (completionL q).weight xs ≠ 0 → xs j = q.1)
    (supportR : ∀ q ys, (completionR q).weight ys ≠ 0 → ys j = q.1)
    (fallback : S) :
    (σ.mixture (fun z => (completionL (z.1.1,z.2)).product
      (completionR (z.1.2,z.2)))).probability (G.coordinateWin strategy j) ≤
      G.value +
        (2 * Information.totalVariation σ.weight
          (fun z => G.questions.weight z.1 * (profileL z.1.1).weight z.2) +
         2 * Information.totalVariation σ.weight
          (fun z => G.questions.weight z.1 * (profileR z.1.2).weight z.2)) := by
  let targetLaw := σ.mixture (fun z => (completionL (z.1.1,z.2)).product (completionR (z.1.2,z.2)))
  let δ := 2 * Information.totalVariation σ.weight
      (fun z => G.questions.weight z.1 * (profileL z.1.1).weight z.2) +
    2 * Information.totalVariation σ.weight
      (fun z => G.questions.weight z.1 * (profileR z.1.2).weight z.2)
  change targetLaw.probability (G.coordinateWin strategy j) ≤ G.value + δ
  by_contra h
  have gap : 0 < targetLaw.probability (G.coordinateWin strategy j) - (G.value + δ) :=
    sub_pos.mpr (lt_of_not_ge h)
  let η := (targetLaw.probability (G.coordinateWin strategy j) - (G.value + δ)) / 2
  have hη : 0 < η := by dsimp [η]; linarith
  obtain ⟨N,hN⟩ := samplerOutputLaw_arbitrarily_close σ G.questions profileL profileR fallback η hη
  let thresholds := sharedProfileThresholds profileL profileR
  let γ := traceSeedLaw (rectangleDistribution (α := S) thresholds) N
  let sL := samplerLocal thresholds profileL fallback N
  let sR := samplerLocal thresholds profileR fallback N
  have hvariation := CompletedSampling.coordinateOutputLaw_totalVariation_le
    G.questions γ j sL sR completionL completionR supportL supportR σ
  have hclose : (CompletedSampling.coordinateOutputLaw G.questions γ j sL sR completionL completionR).totalVariation targetLaw ≤ δ + η :=
    hvariation.trans hN
  have hsuccess := G.coordinate_probability_le_value_add_embedding_distance j targetLaw
    (CompletedSampling.seedLaw γ completionL completionR)
    (CompletedSampling.coordinateLeft j sL) (CompletedSampling.coordinateRight j sR)
    (CompletedSampling.coordinateLeft_preserves j sL)
    (CompletedSampling.coordinateRight_preserves j sR) strategy
  have hsym : targetLaw.totalVariation
      (G.localEmbeddingLaw (CompletedSampling.seedLaw γ completionL completionR)
        (CompletedSampling.coordinateLeft j sL) (CompletedSampling.coordinateRight j sR)) =
      (CompletedSampling.coordinateOutputLaw G.questions γ j sL sR completionL completionR).totalVariation targetLaw := by
    unfold FiniteDistribution.totalVariation
    congr 1
    apply Finset.sum_congr rfl
    intro q _
    exact abs_sub_comm _ _
  rw [hsym] at hsuccess
  dsimp [η] at hclose
  linarith

end
end UniqueGamesTheorem.Foundations.Repetition

end

section

/-! The two actual conditional common-data profiles used by the shared
sampler, with explicit normalized default laws on all null inputs. -/
namespace UniqueGamesTheorem.Foundations.Repetition
open scoped BigOperators
open Games Information
noncomputable section

theorem finiteDistribution_nonempty {A : Type*} [Fintype A] (μ : FiniteDistribution A) :
    Nonempty A := by
  have hsum : (∑ a, μ.weight a) ≠ 0 := by rw [μ.normalized]; norm_num
  obtain ⟨a, _, _⟩ := Finset.exists_ne_zero_of_sum_ne_zero hsum
  exact ⟨a⟩

variable {Q₁ Q₂ A₁ A₂ : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
  [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat}

def selectedProfileFallback (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    FiniteDistribution (SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂)
      (A₁ := A₁) (A₂ := A₂) selected j) :=
  (selectedCommonLaw G strategy selected positive j).pushforward Prod.fst

def selectedLeftProfile (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) (x : Q₁) :
    FiniteDistribution (SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂)
      (A₁ := A₁) (A₂ := A₂) selected j) :=
  toGameLaw
    (leftCommonProfile (selectedCommonLaw G strategy selected positive j).weight
      (selectedProfileFallback G strategy selected positive j).weight x)
    (leftCommonProfile_isProbability _ _
      (gameLaw_isProbability (selectedCommonLaw G strategy selected positive j))
      (gameLaw_isProbability (selectedProfileFallback G strategy selected positive j)) x)

def selectedRightProfile (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) (y : Q₂) :
    FiniteDistribution (SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂)
      (A₁ := A₁) (A₂ := A₂) selected j) :=
  toGameLaw
    (rightCommonProfile (selectedCommonLaw G strategy selected positive j).weight
      (selectedProfileFallback G strategy selected positive j).weight y)
    (rightCommonProfile_isProbability _ _
      (gameLaw_isProbability (selectedCommonLaw G strategy selected positive j))
      (gameLaw_isProbability (selectedProfileFallback G strategy selected positive j)) y)

def selectedLeftProfileError (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) : ℝ :=
  totalVariation (selectedCommonLaw G strategy selected positive j).weight
    (fun z => G.questions.weight z.2 * (selectedLeftProfile G strategy selected positive j z.2.1).weight z.1)

def selectedRightProfileError (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) : ℝ :=
  totalVariation (selectedCommonLaw G strategy selected positive j).weight
    (fun z => G.questions.weight z.2 * (selectedRightProfile G strategy selected positive j z.2.2).weight z.1)

theorem selectedLeftProfileError_le (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    selectedLeftProfileError G strategy selected positive j ≤
      totalVariation (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected))
        (leftRevealModel G.questions
          (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected))) +
      totalVariation (secondMarginal (selectedCommonLaw G strategy selected positive j).weight)
        G.questions.weight :=
  leftCommonProfile_error _ G.questions _
    (gameLaw_isProbability (selectedCommonLaw G strategy selected positive j))
    (gameLaw_isProbability (selectedProfileFallback G strategy selected positive j))

theorem selectedRightProfileError_le (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    selectedRightProfileError G strategy selected positive j ≤
      totalVariation (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected))
        (rightRevealModel G.questions
          (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected))) +
      totalVariation (secondMarginal (selectedCommonLaw G strategy selected positive j).weight)
        G.questions.weight :=
  rightCommonProfile_error _ G.questions _
    (gameLaw_isProbability (selectedCommonLaw G strategy selected positive j))
    (gameLaw_isProbability (selectedProfileFallback G strategy selected positive j))

def selectedSamplingTarget (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    FiniteDistribution ((Q₁ × Q₂) × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂)
      (A₁ := A₁) (A₂ := A₂) selected j) :=
  (selectedCommonLaw G strategy selected positive j).transport (Equiv.prodComm _ _)

theorem selectedSamplingTarget_left_error (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    totalVariation (selectedSamplingTarget G strategy selected positive j).weight
      (fun z => G.questions.weight z.1 * (selectedLeftProfile G strategy selected positive j z.1.1).weight z.2) =
      selectedLeftProfileError G strategy selected positive j := by
  exact totalVariation_comp_equiv
    (Equiv.prodComm (Q₁ × Q₂) (SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂)
      (A₁ := A₁) (A₂ := A₂) selected j))
    (selectedCommonLaw G strategy selected positive j).weight
    (fun z => G.questions.weight z.2 * (selectedLeftProfile G strategy selected positive j z.2.1).weight z.1)

theorem selectedSamplingTarget_right_error (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    totalVariation (selectedSamplingTarget G strategy selected positive j).weight
      (fun z => G.questions.weight z.1 * (selectedRightProfile G strategy selected positive j z.1.2).weight z.2) =
      selectedRightProfileError G strategy selected positive j := by
  exact totalVariation_comp_equiv
    (Equiv.prodComm (Q₁ × Q₂) (SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂)
      (A₁ := A₁) (A₂ := A₂) selected j))
    (selectedCommonLaw G strategy selected positive j).weight
    (fun z => G.questions.weight z.2 * (selectedRightProfile G strategy selected positive j z.2.2).weight z.1)

end
end UniqueGamesTheorem.Foundations.Repetition

end

section

namespace UniqueGamesTheorem.Foundations.Repetition
open scoped BigOperators
open Games Information
noncomputable section
variable {Q₁ Q₂ A₁ A₂ : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
  [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat}

omit [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedQuestionRadius_le_informationRadius [Nonempty A₁] [Nonempty A₂]
    (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) :
    Real.sqrt ((Fintype.card {i : Fin n // i ∉ selected} : ℝ) *
      Real.log (1 / G.selectedSuccess strategy selected)) ≤
      selectedInformationRadius G strategy selected := by
  have hcard : (1 : ℝ) ≤ Fintype.card (SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) := by
    exact_mod_cast (Nat.succ_le_of_lt
      (Fintype.card_pos : 0 < Fintype.card (SelectedLabels (A₁ := A₁) (A₂ := A₂) selected)))
  have hlog := Real.log_nonneg hcard
  have hnonneg : 0 ≤ (Fintype.card {i : Fin n // i ∉ selected} : ℝ) := Nat.cast_nonneg _
  unfold selectedInformationRadius
  apply Real.sqrt_le_sqrt
  nlinarith

theorem selectedQuestionMarginal_error_sum [Nonempty A₁] [Nonempty A₂]
    (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) :
    (∑ j : {i : Fin n // i ∉ selected},
      (selectedQuestionMarginal G strategy selected positive j.1).totalVariation G.questions) ≤
      selectedInformationRadius G strategy selected :=
  (unselectedQuestionMarginal_totalVariation_sum_le_sqrt G strategy selected positive).trans
    (selectedQuestionRadius_le_informationRadius G strategy selected)

def selectedEmbeddingError (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) : ℝ :=
  2 * selectedLeftProfileError G strategy selected positive j +
    2 * selectedRightProfileError G strategy selected positive j

theorem selectedEmbeddingError_nonnegative (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    0 ≤ selectedEmbeddingError G strategy selected positive j := by
  exact add_nonneg
    (mul_nonneg (by norm_num) (Information.totalVariation_nonneg _ _))
    (mul_nonneg (by norm_num) (Information.totalVariation_nonneg _ _))

theorem selectedLeftProfileError_sum [Nonempty A₁] [Nonempty A₂]
    (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) :
    (∑ j : {i : Fin n // i ∉ selected}, selectedLeftProfileError G strategy selected positive j) ≤
      3 * selectedInformationRadius G strategy selected := by
  have hsum := Finset.sum_le_sum
    (fun j (_ : j ∈ (Finset.univ : Finset {i : Fin n // i ∉ selected})) =>
      selectedLeftProfileError_le G strategy selected positive j)
  simp_rw [selectedCommonLaw_secondMarginal] at hsum
  rw [Finset.sum_add_distrib] at hsum
  have hleft := selected_leftReveal_error_sum G strategy selected positive
  have hquestion := selectedQuestionMarginal_error_sum G strategy selected positive
  change (∑ j : {i : Fin n // i ∉ selected},
    Information.totalVariation (selectedQuestionMarginal G strategy selected positive j.1).weight
      G.questions.weight) ≤ _ at hquestion
  linarith

theorem selectedRightProfileError_sum [Nonempty A₁] [Nonempty A₂]
    (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) :
    (∑ j : {i : Fin n // i ∉ selected}, selectedRightProfileError G strategy selected positive j) ≤
      3 * selectedInformationRadius G strategy selected := by
  have hsum := Finset.sum_le_sum
    (fun j (_ : j ∈ (Finset.univ : Finset {i : Fin n // i ∉ selected})) =>
      selectedRightProfileError_le G strategy selected positive j)
  simp_rw [selectedCommonLaw_secondMarginal] at hsum
  rw [Finset.sum_add_distrib] at hsum
  have hright := selected_rightReveal_error_sum G strategy selected positive
  have hquestion := selectedQuestionMarginal_error_sum G strategy selected positive
  change (∑ j : {i : Fin n // i ∉ selected},
    Information.totalVariation (selectedQuestionMarginal G strategy selected positive j.1).weight
      G.questions.weight) ≤ _ at hquestion
  linarith

theorem selectedEmbeddingError_sum_le_fifteen [Nonempty A₁] [Nonempty A₂]
    (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) :
    (∑ j : {i : Fin n // i ∉ selected}, selectedEmbeddingError G strategy selected positive j) ≤
      15 * selectedInformationRadius G strategy selected := by
  have hl := selectedLeftProfileError_sum G strategy selected positive
  have hr := selectedRightProfileError_sum G strategy selected positive
  have hn : 0 ≤ selectedInformationRadius G strategy selected := Real.sqrt_nonneg _
  simp only [selectedEmbeddingError, Finset.sum_add_distrib, ← Finset.mul_sum]
  linarith

end
end UniqueGamesTheorem.Foundations.Repetition

end

section

/-! Quantitative parallel repetition for actual finite two-player games.
The coordinate bound is derived from the explicitly conditioned question law,
finite reveal variables, information estimates, shared rejection sampler,
and normalized local completions. The scalar recurrence is then applied to
the actual repeated strategy's success probability. -/
namespace UniqueGamesTheorem.Foundations.Repetition
open scoped BigOperators
open Games
noncomputable section
variable {Q₁ Q₂ A₁ A₂ : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
  [Nonempty A₁] [Nonempty A₂]
  [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat}
theorem selected_coordinate_probability_le_value_add_error
    (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    ((G.repetition n).questions.condition (G.selectedWins strategy selected) positive).probability
      (G.coordinateWin strategy j.1) ≤
        G.value + selectedEmbeddingError G strategy selected positive j := by
  classical
  let : Nonempty (SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂)
      (A₁ := A₁) (A₂ := A₂) selected j) :=
    finiteDistribution_nonempty (selectedProfileFallback G strategy selected positive j)
  have h := coordinate_probability_le_value_of_local_completion G j.1 strategy
    (selectedSamplingTarget G strategy selected positive j)
    (selectedLeftProfile G strategy selected positive j)
    (selectedRightProfile G strategy selected positive j)
    (selectedFullLeftCompletion G strategy selected j)
    (selectedFullRightCompletion G strategy selected j)
    (selectedFullLeftCompletion_support G strategy selected j)
    (selectedFullRightCompletion_support G strategy selected j)
    (Classical.choice inferInstance)
  have htarget : (selectedSamplingTarget G strategy selected positive j).mixture
      (fun z => (selectedFullLeftCompletion G strategy selected j (z.1.1,z.2)).product
        (selectedFullRightCompletion G strategy selected j (z.1.2,z.2))) =
      (G.repetition n).questions.condition (G.selectedWins strategy selected) positive :=
    selectedFullCompletionMixture_eq_conditionedQuestions G strategy selected positive j
  rw [htarget, selectedSamplingTarget_left_error, selectedSamplingTarget_right_error] at h
  exact h
theorem holenstein_conditionalCoordinateBound
    (G : Game Q₁ Q₂ A₁ A₂) (n : Nat)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (ell : ℝ)
    (halphabet : logTwo ((Fintype.card A₁ : ℝ) * (Fintype.card A₂ : ℝ)) ≤ ell) :
    SelectionSequence.ConditionalCoordinateBound G n strategy ell := by
  intro selected hcard positive
  obtain ⟨j,hj⟩ := exists_coordinate_le_information_budget G strategy selected positive ell
    halphabet hcard (selectedEmbeddingError G strategy selected positive)
    (selectedEmbeddingError_sum_le_fifteen G strategy selected positive)
  exact ⟨j.1,j.property,
    (selected_coordinate_probability_le_value_add_error G strategy selected positive j).trans
      (add_le_add le_rfl hj)⟩
/-- Holenstein's explicit constant-6000 repetition estimate, for the actual
success probability of every finite deterministic repeated strategy. -/
theorem holenstein_repetition_success
    (G : Game Q₁ Q₂ A₁ A₂) (n : Nat)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    {v ell : ℝ} (hvalue : G.value ≤ v) (hv : v < 1) (hell : 1 ≤ ell)
    (halphabet : logTwo ((Fintype.card A₁ : ℝ) * (Fintype.card A₂ : ℝ)) ≤ ell) :
    (G.repetition n).success strategy ≤
      (1 - (1 - v)^3 / 6000)^((n : ℝ) / ell) :=
  SelectionSequence.repetition_success_le_of_conditionalCoordinateBound G n strategy
    hvalue hv hell (holenstein_conditionalCoordinateBound G n strategy ell halphabet)

/-- Quantitative parallel repetition of the actual maximum game value.
There is no repetition, embedding, information-budget, or independence premise. -/
theorem holenstein_repetition_value
    (G : Game Q₁ Q₂ A₁ A₂) (n : Nat)
    {v ell : ℝ} (hvalue : G.value ≤ v) (hv : v < 1) (hell : 1 ≤ ell)
    (halphabet : logTwo ((Fintype.card A₁ : ℝ) * (Fintype.card A₂ : ℝ)) ≤ ell) :
    (G.repetition n).value ≤
      (1 - (1 - v)^3 / 6000)^((n : ℝ) / ell) := by
  apply ((G.repetition n).value_le_iff _).2
  intro strategy
  exact holenstein_repetition_success G n strategy hvalue hv hell halphabet

end
end UniqueGamesTheorem.Foundations.Repetition

end

end OAI