lean/OAI/Computability/MinUncut/PCP/CompletedSampling.lean

import Mathlib
import OAI.Computability.MinUncut.PCP.Overlap

namespace OAI

namespace MinUncutGames.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]

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))

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

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

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)

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

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]

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)

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)))

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

end MinUncutGames.Foundations.Repetition.CompletedSampling

namespace MinUncutGames.Foundations.Games.Game

noncomputable section

variable {Q₁ Q₂ A₁ A₂ R₁ R₂ B₁ B₂ Seed : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
  [Fintype R₁] [Fintype R₂] [Fintype B₁] [Fintype B₂] [Fintype Seed]
  [Nonempty A₁] [Nonempty A₂]

def localEmbeddingLaw (G : Game Q₁ Q₂ A₁ A₂)
    (seedLaw : FiniteDistribution Seed)
    (left : Seed → Q₁ → R₁) (right : Seed → Q₂ → R₂) :
    FiniteDistribution (R₁ × R₂) :=
  seedLaw.mixture fun seed =>
    G.questions.pushforward fun q => (left seed q.1, right seed q.2)

theorem success_le_value_add_embedding_distance
    (G : Game Q₁ Q₂ A₁ A₂) (H : Game R₁ R₂ B₁ B₂)
    (seedLaw : FiniteDistribution Seed)
    (left : Seed → Q₁ → R₁) (right : Seed → Q₂ → R₂)
    (answerLeft : Seed → Q₁ → B₁ → A₁)
    (answerRight : Seed → Q₂ → B₂ → A₂)
    (acceptance : ∀ seed x y a b,
      H.accepts (left seed x) (right seed y) a b = true →
      G.accepts x y (answerLeft seed x a) (answerRight seed y b) = true)
    (strategy : Strategy R₁ R₂ B₁ B₂) :
    H.success strategy ≤ G.value +
      H.questions.totalVariation (G.localEmbeddingLaw seedLaw left right) := by
  have localBound (seed : Seed) :
      (G.questions.pushforward (fun q => (left seed q.1, right seed q.2))).probability
        (H.wins strategy) ≤ G.value := by
    let embedded : Game R₁ R₂ B₁ B₂ :=
      { questions := G.questions.pushforward fun q => (left seed q.1, right seed q.2)
        accepts := H.accepts }
    change embedded.success strategy ≤ G.value
    exact (G.success_le_of_localSimulation embedded (left seed) (right seed)
      (answerLeft seed) (answerRight seed) rfl (acceptance seed) strategy).trans
      (G.success_le_value _)
  have mixtureBound :
      (G.localEmbeddingLaw seedLaw left right).probability (H.wins strategy) ≤ G.value :=
    seedLaw.probability_mixture_le _ _ _ localBound
  exact (H.questions.probability_le_add_totalVariation
    (G.localEmbeddingLaw seedLaw left right) (H.wins strategy)).trans
    (add_le_add mixtureBound (le_refl _))

def coordinateGame (G : Game Q₁ Q₂ A₁ A₂) {n : Nat} (coordinate : Fin n)
    (law : FiniteDistribution ((Fin n → Q₁) × (Fin n → Q₂))) :
    Game (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂) where
  questions := law
  accepts x y a b := G.accepts (x coordinate) (y coordinate) (a coordinate) (b coordinate)

theorem coordinate_probability_le_value_add_embedding_distance
    (G : Game Q₁ Q₂ A₁ A₂) {n : Nat} (coordinate : Fin n)
    (law : FiniteDistribution ((Fin n → Q₁) × (Fin n → Q₂)))
    (seedLaw : FiniteDistribution Seed)
    (left : Seed → Q₁ → Fin n → Q₁) (right : Seed → Q₂ → Fin n → Q₂)
    (left_preserves : ∀ seed x, left seed x coordinate = x)
    (right_preserves : ∀ seed y, right seed y coordinate = y)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) :
    law.probability (G.coordinateWin strategy coordinate) ≤ G.value +
      law.totalVariation (G.localEmbeddingLaw seedLaw left right) := by
  apply G.success_le_value_add_embedding_distance (G.coordinateGame coordinate law)
    seedLaw left right (fun _ _ a => a coordinate) (fun _ _ b => b coordinate)
  intro seed x y a b h
  simpa only [coordinateGame, left_preserves, right_preserves] using h

theorem coordinate_probability_le_value_of_approximate_embeddings
    {SeedFamily : Nat → Type*} [∀ t, Fintype (SeedFamily t)]
    (G : Game Q₁ Q₂ A₁ A₂) {n : Nat} (coordinate : Fin n)
    (law : FiniteDistribution ((Fin n → Q₁) × (Fin n → Q₂)))
    (seedLaw : (t : Nat) → FiniteDistribution (SeedFamily t))
    (left : (t : Nat) → SeedFamily t → Q₁ → Fin n → Q₁)
    (right : (t : Nat) → SeedFamily t → Q₂ → Fin n → Q₂)
    (left_preserves : ∀ t seed x, left t seed x coordinate = x)
    (right_preserves : ∀ t seed y, right t seed y coordinate = y)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (distance : ℝ)
    (close : ∀ η : ℝ, 0 < η → ∃ t,
      law.totalVariation (G.localEmbeddingLaw (seedLaw t) (left t) (right t)) ≤
        distance + η) :
    law.probability (G.coordinateWin strategy coordinate) ≤ G.value + distance := by
  by_contra h
  have gap : 0 < law.probability (G.coordinateWin strategy coordinate) -
      (G.value + distance) := sub_pos.mpr (lt_of_not_ge h)
  obtain ⟨t, ht⟩ := close
    ((law.probability (G.coordinateWin strategy coordinate) - (G.value + distance)) / 2)
    (by linarith)
  have bound := G.coordinate_probability_le_value_add_embedding_distance coordinate law
    (seedLaw t) (left t) (right t) (left_preserves t) (right_preserves t) strategy
  linarith

end
end MinUncutGames.Foundations.Games.Game

namespace MinUncutGames.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

noncomputable def logTwo (x : ℝ) : ℝ := Real.log x / Real.log 2

@[simp] theorem logTwo_one : logTwo 1 = 0 := by simp [logTwo]

theorem logTwo_inv_le_of_half_pow_le {p : ℝ} (m : ℕ)
    (h : (1 / 2 : ℝ) ^ m ≤ p) : logTwo (1 / p) ≤ m := by
  have hp : 0 < p := (pow_pos (by norm_num : (0 : ℝ) < 1 / 2) m).trans_le h
  have hlog := Real.log_le_log (pow_pos (by norm_num : (0 : ℝ) < 1 / 2) m) h
  have hhalf : Real.log (1 / 2 : ℝ) = -Real.log 2 := by
    rw [one_div, Real.log_inv]
  rw [Real.log_pow, hhalf] at hlog
  unfold logTwo
  rw [one_div, Real.log_inv]
  apply (div_le_iff₀ (Real.log_pos (by norm_num : (1 : ℝ) < 2))).2
  nlinarith

theorem contraction_step {v ell : ℝ} {n m : ℕ} {p next : ℝ}
    (hv0 : 0 ≤ v) (hv1 : v < 1) (hell : 1 ≤ ell)
    (hm : 1 ≤ m) (hmn : m < n) (hp : 0 ≤ p)
    (hmono : next ≤ p)
    (hprev : p ≤ ((1 + v) / 2) ^ m)
    (hcut : 2700 * (m : ℝ) * ell ≤ (1 - v) ^ 2 * ((n : ℝ) - m))
    (hstep : next ≤ p * (v + 15 * Real.sqrt
      (((m : ℝ) * ell + logTwo (1 / p)) / ((n : ℝ) - m)))) :
    next ≤ ((1 + v) / 2) ^ (m + 1) := by
  by_cases hsmall : p ≤ ((1 + v) / 2) ^ (m + 1)
  · exact hmono.trans hsmall
  have hq : (1 / 2 : ℝ) ≤ (1 + v) / 2 := by linarith
  have hhalf : (1 / 2 : ℝ) ^ (m + 1) ≤ p :=
    (pow_le_pow_left₀ (by norm_num) hq (m + 1)).trans (le_of_not_ge hsmall)
  have hlog := logTwo_inv_le_of_half_pow_le (m + 1) hhalf
  have hmreal : (1 : ℝ) ≤ m := by exact_mod_cast hm
  have hden : 0 < (n : ℝ) - m := by
    have hmnreal : (m : ℝ) < n := by exact_mod_cast hmn
    linarith
  have hmell : (m : ℝ) ≤ (m : ℝ) * ell := by nlinarith
  have hbudget : (m : ℝ) * ell + logTwo (1 / p) ≤ 3 * (m : ℝ) * ell := by
    simp only [Nat.cast_add, Nat.cast_one] at hlog
    nlinarith
  have hsqrt : Real.sqrt
      (((m : ℝ) * ell + logTwo (1 / p)) / ((n : ℝ) - m)) ≤ (1 - v) / 30 := by
    apply (Real.sqrt_le_left (by linarith : 0 ≤ (1 - v) / 30)).2
    apply (div_le_iff₀ hden).2
    nlinarith
  have hfactor : v + 15 * Real.sqrt
      (((m : ℝ) * ell + logTwo (1 / p)) / ((n : ℝ) - m)) ≤ (1 + v) / 2 := by
    linarith
  calc
    next ≤ p * (v + 15 * Real.sqrt
        (((m : ℝ) * ell + logTwo (1 / p)) / ((n : ℝ) - m))) := hstep
    _ ≤ p * ((1 + v) / 2) := mul_le_mul_of_nonneg_left hfactor hp
    _ ≤ ((1 + v) / 2) ^ m * ((1 + v) / 2) :=
      mul_le_mul_of_nonneg_right hprev (by linarith)
    _ = ((1 + v) / 2) ^ (m + 1) := (pow_succ _ _).symm

theorem initial_step {v ell next : ℝ} {n : ℕ}
    (hstep : next ≤ 1 * (v + 15 * Real.sqrt
      (((0 : ℝ) * ell + logTwo (1 / 1)) / ((n : ℝ) - 0)))) :
    next ≤ v := by simpa using hstep

theorem cutoff_step {v ell : ℝ} {n m : ℕ}
    (hv0 : 0 ≤ v) (hv1 : v < 1) (hell : 1 ≤ ell)
    (hm : (m : ℝ) < (n : ℝ) * (1 - v) ^ 2 / (2925 * ell)) :
    2700 * (m : ℝ) * ell ≤ (1 - v) ^ 2 * ((n : ℝ) - m) := by
  have hell0 : 0 < ell := by linarith
  have hd : (1 - v) ^ 2 ≤ 1 := by nlinarith
  have hmul : (m : ℝ) * (2925 * ell) < (n : ℝ) * (1 - v) ^ 2 :=
    (lt_div_iff₀ (by positivity : 0 < 2925 * ell)).1 hm
  have hdell : (1 - v) ^ 2 ≤ 225 * ell := by linarith
  have hprod := mul_le_mul_of_nonneg_left hdell (Nat.cast_nonneg m : (0 : ℝ) ≤ m)
  nlinarith

theorem cutoff_le_length {v ell : ℝ} (n : ℕ)
    (hv0 : 0 ≤ v) (hv1 : v < 1) (hell : 1 ≤ ell) :
    (n : ℝ) * (1 - v) ^ 2 / (2925 * ell) ≤ n := by
  have hell0 : 0 < ell := by linarith
  apply (div_le_iff₀ (by positivity : 0 < 2925 * ell)).2
  have hd : (1 - v) ^ 2 ≤ 2925 * ell := by nlinarith
  nlinarith [mul_le_mul_of_nonneg_left hd (Nat.cast_nonneg n : (0 : ℝ) ≤ n)]

def ScalarRecurrence (p : ℕ → ℝ) (n : ℕ) (v ell : ℝ) : Prop :=
  ∀ m, m < n → p (m + 1) ≤ p m * (v + 15 * Real.sqrt
    (((m : ℝ) * ell + logTwo (1 / p m)) / ((n : ℝ) - m)))

theorem decay_to_cutoff (p : ℕ → ℝ) (n : ℕ) {v ell : ℝ}
    (hv0 : 0 ≤ v) (hv1 : v < 1) (hell : 1 ≤ ell)
    (hp0 : p 0 = 1) (hpnonneg : ∀ m, 0 ≤ p m)
    (hpmono : Antitone p) (hrec : ScalarRecurrence p n v ell) :
    p n ≤ ((1 + v) / 2) ^
      ((n : ℝ) * (1 - v) ^ 2 / (2925 * ell)) := by
  let T : ℝ := (n : ℝ) * (1 - v) ^ 2 / (2925 * ell)
  let M : ℕ := Nat.ceil T
  have hMn : M ≤ n := Nat.ceil_le.mpr (cutoff_le_length n hv0 hv1 hell)
  have hind : ∀ m, m ≤ M → p m ≤ ((1 + v) / 2) ^ m := by
    intro m
    induction m with
    | zero => intro _; simp [hp0]
    | succ m ih =>
        intro hmM
        have hmM' : m < M := Nat.lt_of_succ_le hmM
        have hmn : m < n := hmM'.trans_le hMn
        have hprev := ih (Nat.le_of_lt hmM')
        by_cases hm0 : m = 0
        · subst m
          have hinit : p 1 ≤ v := by
            have hs := hrec 0 hmn
            simpa [hp0] using hs
          simpa using hinit.trans (by linarith : v ≤ (1 + v) / 2)
        · have hmT : (m : ℝ) < T := Nat.lt_ceil.mp hmM'
          exact contraction_step hv0 hv1 hell (Nat.one_le_iff_ne_zero.mpr hm0)
            hmn (hpnonneg m) (hpmono (Nat.le_succ m)) hprev
            (cutoff_step hv0 hv1 hell hmT) (hrec m hmn)
  have hq0 : 0 < (1 + v) / 2 := by linarith
  have hq1 : (1 + v) / 2 ≤ 1 := by linarith
  calc
    p n ≤ p M := hpmono hMn
    _ ≤ ((1 + v) / 2) ^ M := hind M le_rfl
    _ = ((1 + v) / 2) ^ (M : ℝ) := (Real.rpow_natCast _ _).symm
    _ ≤ ((1 + v) / 2) ^ T :=
      Real.rpow_le_rpow_of_exponent_ge hq0 hq1 (Nat.le_ceil T)

theorem block_contraction {v : ℝ} (hv0 : 0 ≤ v) (hv1 : v < 1) :
    ((1 + v) / 2) ^ ((1 - v) ^ 2 / 2925 : ℝ) ≤
      1 - (1 - v) ^ 3 / 5850 := by
  have hd : (1 - v) ^ 2 ≤ 1 := by nlinarith
  have hb := rpow_one_add_le_one_add_mul_self
    (s := -(1 - v) / 2) (p := (1 - v) ^ 2 / 2925)
    (by linarith) (by positivity) (by nlinarith)
  have hbase : 1 + -(1 - v) / 2 = (1 + v) / 2 := by ring
  have hrhs : 1 + (1 - v) ^ 2 / 2925 * (-(1 - v) / 2) =
      1 - (1 - v) ^ 3 / 5850 := by ring
  rwa [hbase, hrhs] at hb

theorem scalar_bound_6000 (p : ℕ → ℝ) (n : ℕ) {v ell : ℝ}
    (hv0 : 0 ≤ v) (hv1 : v < 1) (hell : 1 ≤ ell)
    (hp0 : p 0 = 1) (hpnonneg : ∀ m, 0 ≤ p m)
    (hpmono : Antitone p) (hrec : ScalarRecurrence p n v ell) :
    p n ≤ (1 - (1 - v) ^ 3 / 6000) ^ ((n : ℝ) / ell) := by
  have hell0 : 0 < ell := by linarith
  have hq0 : 0 ≤ (1 + v) / 2 := by linarith
  have hexp : 0 ≤ (n : ℝ) / ell := by positivity
  have hblock := block_contraction hv0 hv1
  have hweak : 1 - (1 - v) ^ 3 / 5850 ≤ 1 - (1 - v) ^ 3 / 6000 := by
    have : 0 ≤ (1 - v) ^ 3 := by positivity
    nlinarith
  calc
    p n ≤ ((1 + v) / 2) ^
        ((n : ℝ) * (1 - v) ^ 2 / (2925 * ell)) :=
      decay_to_cutoff p n hv0 hv1 hell hp0 hpnonneg hpmono hrec
    _ = (((1 + v) / 2) ^ ((1 - v) ^ 2 / 2925 : ℝ)) ^ ((n : ℝ) / ell) := by
      rw [← Real.rpow_mul hq0]
      congr 1
      ring
    _ ≤ (1 - (1 - v) ^ 3 / 6000) ^ ((n : ℝ) / ell) :=
      Real.rpow_le_rpow (Real.rpow_nonneg hq0 _) (hblock.trans hweak) hexp

noncomputable def incidenceRate : ℝ := 1 - 1 / (100000 * 192 ^ 3)

theorem incidenceRate_pos : 0 < incidenceRate := by norm_num [incidenceRate]

theorem incidenceRate_lt_one : incidenceRate < 1 := by norm_num [incidenceRate]

theorem incidenceRate_fourth :
    1 - (1 / 192 : ℝ) ^ 3 / 6000 ≤ incidenceRate ^ 4 := by
  norm_num [incidenceRate]

theorem incidence_rate_comparison (n : ℕ) :
    (1 - (1 / 192 : ℝ) ^ 3 / 6000) ^ ((n : ℝ) / 4) ≤ incidenceRate ^ n := by
  calc
    (1 - (1 / 192 : ℝ) ^ 3 / 6000) ^ ((n : ℝ) / 4)
        ≤ (incidenceRate ^ 4) ^ ((n : ℝ) / 4) :=
      Real.rpow_le_rpow (by norm_num) incidenceRate_fourth (by positivity)
    _ = incidenceRate ^ ((4 : ℝ) * ((n : ℝ) / 4)) :=
      (Real.rpow_natCast_mul incidenceRate_pos.le 4 ((n : ℝ) / 4)).symm
    _ = incidenceRate ^ (n : ℝ) := by congr 1; ring
    _ = incidenceRate ^ n := Real.rpow_natCast _ _

theorem scalar_incidence_bound (p : ℕ → ℝ) (n : ℕ) {v : ℝ}
    (hv : v ≤ 1 - (1 / 192 : ℝ))
    (hp0 : p 0 = 1) (hpnonneg : ∀ m, 0 ≤ p m)
    (hpmono : Antitone p) (hrec : ScalarRecurrence p n v 4) :
    p n ≤ incidenceRate ^ n := by
  have hrec' : ScalarRecurrence p n (1 - (1 / 192 : ℝ)) 4 := by
    intro m hm
    exact (hrec m hm).trans (mul_le_mul_of_nonneg_left (by linarith) (hpnonneg m))
  have hbound := scalar_bound_6000 p n
    (v := 1 - (1 / 192 : ℝ)) (ell := 4)
    (by norm_num) (by norm_num) (by norm_num) hp0 hpnonneg hpmono hrec'
  norm_num only [sub_sub_cancel] at hbound
  have hcomparison := incidence_rate_comparison n
  norm_num at hcomparison
  exact hbound.trans hcomparison

end MinUncutGames.Foundations.Repetition

end OAI