lean/OAI/Computability/MaxCut/PCP/KernelIteration.lean

import Mathlib
import OAI.Computability.MaxCut.PCP.Parameters
import OAI.Computability.MaxCut.PCP.ProductCollisionBounds

namespace OAI

/-!
Independent repetitions of the actual conditional projection kernel. The
resulting game has precisely the original game's product question law. A
successor tuple is a reindexed independent product, so collision bounds on the
product apply to unrestricted strategies on whole tuples.
-/

namespace MaxCutGames.Repetition.ProjectionKernel

open MaxCutGames.Foundations.Games
open scoped BigOperators

noncomputable section

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

def repetition (K : ProjectionKernel Q₁ Q₂ A₁ A₂) (n : Nat) :
    ProjectionKernel (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂) := by
  classical
  exact
    { outer := K.outer.iid n
      inner := fun y => FiniteDistribution.table (fun i : Fin n => K.inner (y i))
      accepts := fun x y a b => decide (∀ i, K.accepts (x i) (y i) (a i) (b i) = true)
      projection := by
        intro x y a b b' hb hb'
        simp only [decide_eq_true_eq] at hb hb'
        funext i
        exact K.projection _ _ _ _ _ (hb i) (hb' i) }

@[simp] theorem repetition_accepts_iff (K : ProjectionKernel Q₁ Q₂ A₁ A₂) (n : Nat)
    (x : Fin n → Q₁) (y : Fin n → Q₂) (a : Fin n → A₁) (b : Fin n → A₂) :
    (K.repetition n).accepts x y a b = true ↔
      ∀ i, K.accepts (x i) (y i) (a i) (b i) = true := by
  classical
  simp [repetition]

@[simp] theorem toGame_repetition (K : ProjectionKernel Q₁ Q₂ A₁ A₂) (n : Nat) :
    (K.repetition n).toGame = K.toGame.repetition n := by
  classical
  have hq : (K.repetition n).toGame.questions = (K.toGame.repetition n).questions := by
    apply FiniteDistribution.eq_of_weight_eq
    intro q
    change ((∏ i, K.outer.weight (q.2 i)) *
        (∏ i, (K.inner (q.2 i)).weight (q.1 i))) =
      ∏ i, K.outer.weight (q.2 i) * (K.inner (q.2 i)).weight (q.1 i)
    exact Finset.prod_mul_distrib.symm
  change Game.mk (K.repetition n).toGame.questions
      (fun (x : Fin n → Q₁) (y : Fin n → Q₂) (a : Fin n → A₁) (b : Fin n → A₂) =>
        decide (∀ i, K.accepts (x i) (y i) (a i) (b i) = true)) =
    Game.mk (K.toGame.repetition n).questions
      (fun (x : Fin n → Q₁) (y : Fin n → Q₂) (a : Fin n → A₁) (b : Fin n → A₂) =>
        decide (∀ i, K.accepts (x i) (y i) (a i) (b i) = true))
  rw [hq]

theorem eq_of_data_eq {K H : ProjectionKernel Q₁ Q₂ A₁ A₂}
    (houter : K.outer = H.outer) (hinner : K.inner = H.inner)
    (haccepts : K.accepts = H.accepts) : K = H := by
  cases K
  cases H
  cases houter
  cases hinner
  cases haccepts
  rfl

/-- Separate the first coordinate without restricting answer dependence on
any coordinate of the complete local question tuple. -/
theorem repetition_succ (K : ProjectionKernel Q₁ Q₂ A₁ A₂) (n : Nat) :
    K.repetition (n + 1) =
      (K.product (K.repetition n)).reindex
        (Fin.consEquiv (fun _ : Fin (n + 1) => Q₁))
        (Fin.consEquiv (fun _ : Fin (n + 1) => Q₂))
        (Fin.consEquiv (fun _ : Fin (n + 1) => A₁))
        (Fin.consEquiv (fun _ : Fin (n + 1) => A₂)) := by
  classical
  apply eq_of_data_eq
  · apply FiniteDistribution.eq_of_weight_eq
    intro y
    change (∏ i : Fin (n + 1), K.outer.weight (y i)) =
      K.outer.weight (y 0) * ∏ i : Fin n, K.outer.weight (y i.succ)
    exact Fin.prod_univ_succ _
  · funext y
    apply FiniteDistribution.eq_of_weight_eq
    intro x
    change (∏ i : Fin (n + 1), (K.inner (y i)).weight (x i)) =
      (K.inner (y 0)).weight (x 0) *
        ∏ i : Fin n, (K.inner (y i.succ)).weight (x i.succ)
    exact Fin.prod_univ_succ _
  · funext x y a b
    apply Bool.eq_iff_iff.mpr
    change (decide (∀ i : Fin (n + 1), K.accepts (x i) (y i) (a i) (b i) = true) = true) ↔
      (K.accepts (x 0) (y 0) (a 0) (b 0) &&
        decide (∀ i : Fin n, K.accepts (x i.succ) (y i.succ) (a i.succ) (b i.succ) = true)) = true
    simp only [decide_eq_true_eq, Bool.and_eq_true]
    constructor
    · intro h
      exact ⟨h 0, fun i => h i.succ⟩
    · rintro ⟨hzero, hsucc⟩ i
      exact Fin.cases hzero hsucc i

theorem collisionValue_repetition_succ [Nonempty A₁]
    (K : ProjectionKernel Q₁ Q₂ A₁ A₂) (n : Nat) :
    (K.repetition (n + 1)).collisionValue =
      (K.product (K.repetition n)).collisionValue := by
  rw [repetition_succ, collisionValue_reindex]

theorem collisionValue_repetition_zero_le_one [Nonempty A₁] [Nonempty A₂]
    (K : ProjectionKernel Q₁ Q₂ A₁ A₂) :
    (K.repetition 0).collisionValue ≤ 1 :=
  (K.repetition 0).collisionValue_le_one

end
end MaxCutGames.Repetition.ProjectionKernel

/-!
The finite product algebra of dependency breaking.

Revealing one endpoint of each independently sampled question pair turns the
remaining question weight into a product of two separate functions, one for
each player. Further separate restrictions on the players' answers preserve
this product. The final theorem normalizes the displayed weights, rather than
assuming conditional independence as an abstract hypothesis.
-/

namespace MaxCutGames.Foundations.Repetition

open scoped BigOperators

noncomputable section

variable {ι α β : Type*} [Fintype ι] [DecidableEq α] [DecidableEq β]

/-- `mask i = false` reveals the first question, and `true` the second. -/
def revealedProductWeight (weight : ι → α → β → ℝ) (mask : ι → Bool)
    (fixedLeft : ι → α) (fixedRight : ι → β) (x : ι → α) (y : ι → β) : ℝ :=
  ∏ i, weight i (x i) (y i) *
    (if mask i then (if y i = fixedRight i then 1 else 0)
     else (if x i = fixedLeft i then 1 else 0))

def revealedLeftWeight (weight : ι → α → β → ℝ) (mask : ι → Bool)
    (fixedLeft : ι → α) (fixedRight : ι → β) (x : ι → α) : ℝ :=
  ∏ i, if mask i then weight i (x i) (fixedRight i)
    else (if x i = fixedLeft i then 1 else 0)

def revealedRightWeight (weight : ι → α → β → ℝ) (mask : ι → Bool)
    (fixedLeft : ι → α) (fixedRight : ι → β) (y : ι → β) : ℝ :=
  ∏ i, if mask i then (if y i = fixedRight i then 1 else 0)
    else weight i (fixedLeft i) (y i)

/-- Exact factorization after fixing the selected endpoint at every position.
No assumption on coordinate alphabets or probabilities is needed for this
identity, so it applies separately at every fixed outside question tuple. -/
theorem revealedProductWeight_factorizes (weight : ι → α → β → ℝ) (mask : ι → Bool)
    (fixedLeft : ι → α) (fixedRight : ι → β) (x : ι → α) (y : ι → β) :
    revealedProductWeight weight mask fixedLeft fixedRight x y =
      revealedLeftWeight weight mask fixedLeft fixedRight x *
        revealedRightWeight weight mask fixedLeft fixedRight y := by
  unfold revealedProductWeight revealedLeftWeight revealedRightWeight
  rw [← Finset.prod_mul_distrib]
  apply Finset.prod_congr rfl
  intro i _
  by_cases hm : mask i = true
  · by_cases hy : y i = fixedRight i <;> simp [hm, hy]
  · by_cases hx : x i = fixedLeft i <;> simp [hm, hx]

/-- Local answer tests are separate functions of the complete local question
vectors. Multiplying by both tests leaves a product of two local weights. -/
theorem revealedProductWeight_local_restrictions
    (weight : ι → α → β → ℝ) (mask : ι → Bool)
    (fixedLeft : ι → α) (fixedRight : ι → β)
    (leftTest : (ι → α) → ℝ) (rightTest : (ι → β) → ℝ)
    (x : ι → α) (y : ι → β) :
    revealedProductWeight weight mask fixedLeft fixedRight x y * leftTest x * rightTest y =
      (revealedLeftWeight weight mask fixedLeft fixedRight x * leftTest x) *
      (revealedRightWeight weight mask fixedLeft fixedRight y * rightTest y) := by
  rw [revealedProductWeight_factorizes]
  ring

variable {X Y : Type*} [Fintype X] [Fintype Y]

theorem productWeight_mass (left : X → ℝ) (right : Y → ℝ) :
    (∑ xy : X × Y, left xy.1 * right xy.2) =
      (∑ x, left x) * (∑ y, right y) := by
  rw [Fintype.sum_prod_type]
  simp only [← Finset.mul_sum, ← Finset.sum_mul]

def normalizedWeight (weight : X → ℝ) : X → ℝ :=
  fun x => weight x / ∑ x', weight x'

theorem normalizedWeight_isProbability (weight : X → ℝ)
    (hweight : ∀ x, 0 ≤ weight x) (hmass : 0 < ∑ x, weight x) :
    (∀ x, 0 ≤ normalizedWeight weight x) ∧ (∑ x, normalizedWeight weight x) = 1 := by
  constructor
  · intro x
    exact div_nonneg (hweight x) hmass.le
  · simp only [normalizedWeight, div_eq_mul_inv, ← Finset.sum_mul, mul_inv_cancel₀ hmass.ne']

/-- Normalizing an actual product weight gives the product of its separately
normalized marginals. This is the finite conditional-independence identity. -/
theorem normalized_productWeight (left : X → ℝ) (right : Y → ℝ)
    (hleft : 0 < ∑ x, left x) (hright : 0 < ∑ y, right y) (xy : X × Y) :
    normalizedWeight (fun z : X × Y => left z.1 * right z.2) xy =
      normalizedWeight left xy.1 * normalizedWeight right xy.2 := by
  unfold normalizedWeight
  rw [productWeight_mass]
  field_simp

/-- Conditioning a product law on separate local events keeps the two factors
independent, with each conditioned on its own event. -/
theorem normalized_rectangle_restriction
    (left : X → ℝ) (right : Y → ℝ) (leftTest : X → ℝ) (rightTest : Y → ℝ)
    (hleft : 0 < ∑ x, left x * leftTest x)
    (hright : 0 < ∑ y, right y * rightTest y) (xy : X × Y) :
    normalizedWeight
      (fun z : X × Y => left z.1 * right z.2 * leftTest z.1 * rightTest z.2) xy =
      normalizedWeight (fun x => left x * leftTest x) xy.1 *
      normalizedWeight (fun y => right y * rightTest y) xy.2 := by
  have hfun : (fun z : X × Y => left z.1 * right z.2 * leftTest z.1 * rightTest z.2) =
      (fun z : X × Y => (left z.1 * leftTest z.1) * (right z.2 * rightTest z.2)) := by
    funext z
    ring
  rw [hfun]
  exact normalized_productWeight _ _ hleft hright xy

end

end MaxCutGames.Foundations.Repetition

/-! Normalized local completion laws derived from their actual product weights.
Zero rows use explicit default laws and satisfy the same recombination identity.
-/

namespace MaxCutGames.Foundations.Repetition

open scoped BigOperators
open Games
noncomputable section

variable {X Y : Type*} [Fintype X] [Fintype Y]

def normalizeOr (w : X → ℝ) (hw : ∀ x, 0 ≤ w x)
    (fallback : FiniteDistribution X) : FiniteDistribution X :=
  if h : 0 < ∑ x, w x then
    { weight := normalizedWeight w
      nonnegative := (normalizedWeight_isProbability w hw h).1
      normalized := (normalizedWeight_isProbability w hw h).2 }
  else fallback

theorem sum_mul_normalizeOr (w : X → ℝ) (hw : ∀ x, 0 ≤ w x)
    (fallback : FiniteDistribution X) (x : X) :
    (∑ z, w z) * (normalizeOr w hw fallback).weight x = w x := by
  classical
  have hs : 0 ≤ ∑ z, w z := Finset.sum_nonneg (fun z _ => hw z)
  by_cases h : 0 < ∑ z, w z
  · simp only [normalizeOr, dite_eq_left h, normalizedWeight]
    field_simp
  · have hz : ∑ z, w z = 0 := le_antisymm (le_of_not_gt h) hs
    have hx : w x = 0 := by
      apply le_antisymm _ (hw x)
      have hl := Finset.single_le_sum (fun z _ => hw z) (Finset.mem_univ x)
      simpa only [hz] using hl
    simp [normalizeOr, hz, hx]

/-- Exact reconstruction from two separately normalized local completion
laws, valid even when either unnormalized row has mass zero. -/
theorem local_completion_recombination
    (left : X → ℝ) (right : Y → ℝ)
    (hl : ∀ x, 0 ≤ left x) (hr : ∀ y, 0 ≤ right y)
    (defaultLeft : FiniteDistribution X) (defaultRight : FiniteDistribution Y)
    (c : ℝ) (x : X) (y : Y) :
    (c * (∑ z, left z) * (∑ z, right z)) *
      ((normalizeOr left hl defaultLeft).weight x *
        (normalizeOr right hr defaultRight).weight y) = c * left x * right y := by
  calc
    _ = c * ((∑ z, left z) * (normalizeOr left hl defaultLeft).weight x) *
        ((∑ z, right z) * (normalizeOr right hr defaultRight).weight y) := by ring
    _ = _ := by rw [sum_mul_normalizeOr, sum_mul_normalizeOr]

/-- The row mass appearing in reconstruction is the actual sum of product
weights, not an assumed marginal identity. -/
theorem factorized_completion_row
    (left : X → ℝ) (right : Y → ℝ)
    (hl : ∀ x, 0 ≤ left x) (hr : ∀ y, 0 ≤ right y)
    (defaultLeft : FiniteDistribution X) (defaultRight : FiniteDistribution Y)
    (c : ℝ) (xy : X × Y) :
    (∑ z : X × Y, c * left z.1 * right z.2) *
      ((normalizeOr left hl defaultLeft).product
        (normalizeOr right hr defaultRight)).weight xy = c * left xy.1 * right xy.2 := by
  have hm : (∑ z : X × Y, c * left z.1 * right z.2) =
      c * (∑ z, left z) * (∑ z, right z) := by
    simp_rw [mul_assoc, ← Finset.mul_sum]
    rw [productWeight_mass]
  rw [hm]
  exact local_completion_recombination left right hl hr defaultLeft defaultRight c xy.1 xy.2

end
end MaxCutGames.Foundations.Repetition

/-!
Every finite projection game has an exact conditional-kernel representation.
The right question uses its original marginal. Conditional left-question laws
are normalized original rows; zero rows use the original left marginal as a
fallback. Their zero outer weight makes the reconstruction identity valid on
all rows without positivity or support hypotheses.
-/

namespace MaxCutGames.Repetition

open MaxCutGames.Foundations.Games
open scoped BigOperators

noncomputable section

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

def leftQuestionMarginal (G : Game Q₁ Q₂ A₁ A₂) : FiniteDistribution Q₁ where
  weight x := ∑ y, G.questions.weight (x, y)
  nonnegative x := Finset.sum_nonneg fun y _ => G.questions.nonnegative (x, y)
  normalized := by
    rw [← Fintype.sum_prod_type]
    exact G.questions.normalized

def rightQuestionMarginal (G : Game Q₁ Q₂ A₁ A₂) : FiniteDistribution Q₂ where
  weight y := ∑ x, G.questions.weight (x, y)
  nonnegative y := Finset.sum_nonneg fun x _ => G.questions.nonnegative (x, y)
  normalized := by
    rw [Finset.sum_comm, ← Fintype.sum_prod_type]
    exact G.questions.normalized

/-- The fallback exists even when the question types were not supplied with
`Nonempty` instances: it is a marginal of the game's actual probability law. -/
def leftGivenRight (G : Game Q₁ Q₂ A₁ A₂) (y : Q₂) : FiniteDistribution Q₁ :=
  MaxCutGames.Foundations.Repetition.normalizeOr
    (fun x => G.questions.weight (x, y))
    (fun x => G.questions.nonnegative (x, y)) (leftQuestionMarginal G)

/-- Exact disintegration, including right questions of marginal probability
zero. The conditional law is never assumed to reconstruct the question law. -/
theorem rightQuestionMarginal_mul_leftGivenRight
    (G : Game Q₁ Q₂ A₁ A₂) (x : Q₁) (y : Q₂) :
    (rightQuestionMarginal G).weight y * (leftGivenRight G y).weight x =
      G.questions.weight (x, y) :=
  MaxCutGames.Foundations.Repetition.sum_mul_normalizeOr
    (fun x => G.questions.weight (x, y))
    (fun x => G.questions.nonnegative (x, y)) (leftQuestionMarginal G) x

/-- A projection game's predicate and actual question distribution determine
the conditional kernel used by the collision argument. -/
def kernelOfGame (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G) :
    ProjectionKernel Q₁ Q₂ A₁ A₂ where
  outer := rightQuestionMarginal G
  inner := leftGivenRight G
  accepts := G.accepts
  projection := hG

@[simp] theorem kernelOfGame_question_weight
    (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G) (q : Q₁ × Q₂) :
    (kernelOfGame G hG).toGame.questions.weight q = G.questions.weight q :=
  rightQuestionMarginal_mul_leftGivenRight G q.1 q.2

@[simp] theorem toGame_kernelOfGame
    (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G) :
    (kernelOfGame G hG).toGame = G := by
  have hq : (kernelOfGame G hG).toGame.questions = G.questions :=
    FiniteDistribution.eq_of_weight_eq (kernelOfGame_question_weight G hG)
  calc
    _ = { questions := G.questions, accepts := G.accepts } := by
      change Game.mk (kernelOfGame G hG).toGame.questions G.accepts =
        Game.mk G.questions G.accepts
      rw [hq]
    _ = G := by cases G; rfl

@[simp] theorem kernelOfGame_success
    (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G)
    (strategy : Strategy Q₁ Q₂ A₁ A₂) :
    (kernelOfGame G hG).toGame.success strategy = G.success strategy := by
  rw [toGame_kernelOfGame]

@[simp] theorem kernelOfGame_value [Nonempty A₁] [Nonempty A₂]
    (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G) :
    (kernelOfGame G hG).toGame.value = G.value := by
  rw [toGame_kernelOfGame]

end
end MaxCutGames.Repetition

namespace MaxCutGames.Repetition

theorem sequence_le_power {r : ℝ} (hr : 0 ≤ r) (c : ℕ → ℝ)
    (hzero : c 0 ≤ 1) (hstep : ∀ n, c (n + 1) ≤ r * c n) :
    ∀ n, c n ≤ r ^ n := by
  intro n
  induction n with
  | zero => simpa using hzero
  | succ n ih =>
      calc
        c (n + 1) ≤ r * c n := hstep n
        _ ≤ r * r ^ n := mul_le_mul_of_nonneg_left ih hr
        _ = r ^ (n + 1) := by rw [pow_succ, mul_comm]

theorem squared_rate_le_dsRate_sq {gap : ℝ}
    (_hgap₀ : 0 ≤ gap) (_hgap₁ : gap ≤ 1) :
    1 - gap ^ 2 / 8 ≤ dsRate gap ^ 2 := by
  dsimp [dsRate]
  nlinarith only [sq_nonneg (gap ^ 2)]

theorem rate_of_squared_collision_recurrence {gap : ℝ}
    (hgap₀ : 0 ≤ gap) (hgap₁ : gap ≤ 1)
    (c v : ℕ → ℝ) (hzero : c 0 ≤ 1)
    (hstep : ∀ n, c (n + 1) ≤ (1 - gap ^ 2 / 8) * c n)
    (hvalue : ∀ n, v n ^ 2 ≤ c n) :
    ∀ n, v n ≤ dsRate gap ^ n := by
  have hr : 0 ≤ 1 - gap ^ 2 / 8 := by nlinarith
  have hpow := sequence_le_power hr c hzero hstep
  intro n
  apply le_of_sq_le_sq _ (pow_nonneg (dsRate_nonneg hgap₀ hgap₁) n)
  calc
    v n ^ 2 ≤ c n := hvalue n
    _ ≤ (1 - gap ^ 2 / 8) ^ n := hpow n
    _ ≤ (dsRate gap ^ 2) ^ n :=
      pow_le_pow_left₀ hr (squared_rate_le_dsRate_sq hgap₀ hgap₁) n
    _ = (dsRate gap ^ n) ^ 2 := by rw [← pow_mul, ← pow_mul, Nat.mul_comm]

end MaxCutGames.Repetition

/-!
Alphabet-independent repetition of ordinary finite projection games.

The game is the actual finite question distribution and its local acceptance
predicate. The proof bounds nonnegative vector assignments by finite
thresholding and correlated sampling, factors each unrestricted product
strategy through that bound, and iterates the resulting collision contraction.
No repetition bound, decoder certificate, or alphabet-size budget is a premise.

The deterministic-predicate game interface permits arbitrary real question
weights. Distinct hidden constraints on the same question pair are not encoded
by this interface. Both v2 uses have a unique predicate for each question pair:
the distinct-position equation--variable game and the final simple graph.
-/

namespace MaxCutGames.Repetition
open MaxCutGames.Foundations.Games
noncomputable section

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

omit [Nonempty B₂] in
theorem collisionValue_product_le_gap_rate
    (K : ProjectionKernel Q₁ Q₂ A₁ A₂)
    (H : ProjectionKernel R₁ R₂ B₁ B₂) {gap : ℝ}
    (hgap₀ : 0 ≤ gap) (hgap₁ : gap ≤ 1)
    (hvalue : K.toGame.value ≤ 1 - gap) :
    (K.product H).collisionValue ≤ (1 - gap ^ 2 / 8) * H.collisionValue := by
  apply ((K.product H).collisionValue_le_iff _).2
  intro labels
  rw [← vectorEnergy_tensorSlice]
  apply K.vectorEnergy_le_of_unit_bound (1 - gap ^ 2 / 8) H.collisionValue
    (fun f hf hmass => K.vectorEnergy_le_gap_rate f hf hmass hgap₀ hgap₁ hvalue)
    H.collisionValue_nonneg (H.tensorSlice labels)
    (H.tensorSlice_nonnegative labels) (H.vectorMass_tensorSlice_le labels)

theorem repetition_value_le_dsRate
    (K : ProjectionKernel Q₁ Q₂ A₁ A₂) {gap : ℝ}
    (hgap₀ : 0 ≤ gap) (hgap₁ : gap ≤ 1)
    (hvalue : K.toGame.value ≤ 1 - gap) (n : ℕ) :
    (K.toGame.repetition n).value ≤ dsRate gap ^ n := by
  have hstep : ∀ m, (K.repetition (m + 1)).collisionValue ≤
      (1 - gap ^ 2 / 8) * (K.repetition m).collisionValue := by
    intro m
    rw [K.collisionValue_repetition_succ]
    exact K.collisionValue_product_le_gap_rate (K.repetition m) hgap₀ hgap₁ hvalue
  have h := rate_of_squared_collision_recurrence hgap₀ hgap₁
    (fun m => (K.repetition m).collisionValue)
    (fun m => (K.repetition m).toGame.value)
    K.collisionValue_repetition_zero_le_one hstep
    (fun m => (K.repetition m).value_sq_le_collisionValue) n
  simpa only [toGame_repetition] using h

end ProjectionKernel

variable {Q₁ Q₂ A₁ A₂ : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
  [Nonempty A₁] [Nonempty A₂]

/-- The v2 projection-repetition estimate, including the empty product.
The repetition exponent is independent of all question and answer cardinalities. -/
theorem projection_repetition_value
    (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G) (n : ℕ)
    {gap : ℝ} (hgap₀ : 0 < gap) (hgap₁ : gap < 1)
    (hvalue : G.value ≤ 1 - gap) :
    (G.repetition n).value ≤ (1 - gap ^ 2 / 16) ^ n := by
  have h := (kernelOfGame G hG).repetition_value_le_dsRate hgap₀.le hgap₁.le
    (by simpa only [kernelOfGame_value] using hvalue) n
  simpa only [toGame_kernelOfGame, dsRate] using h

theorem projection_repetition_success
    (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G) (n : ℕ)
    {gap : ℝ} (hgap₀ : 0 < gap) (hgap₁ : gap < 1)
    (hvalue : G.value ≤ 1 - gap)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) :
    (G.repetition n).success strategy ≤ (1 - gap ^ 2 / 16) ^ n :=
  ((G.repetition n).success_le_value strategy).trans
    (projection_repetition_value G hG n hgap₀ hgap₁ hvalue)

theorem clean_repetition_value
    (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G) (n : ℕ)
    (hvalue : G.value ≤ (14 : ℝ) / 15) :
    (G.repetition n).value ≤ (1 - (1 : ℝ) / 3600) ^ n := by
  have hv : G.value ≤ 1 - (1 : ℝ) / 15 := by convert hvalue using 1 ; norm_num
  have h := projection_repetition_value G hG n
    (gap := (1 : ℝ) / 15) (by norm_num) (by norm_num) hv
  norm_num at h ⊢
  exact h

theorem final_repetition_value
    (G : Game Q₁ Q₂ A₁ A₂) (hG : IsProjection G) (n : ℕ)
    (hvalue : G.value ≤ 1 - (1 : ℝ) / 800) :
    (G.repetition n).value ≤ (1 - (1 : ℝ) / 10240000) ^ n := by
  have h := projection_repetition_value G hG n
    (gap := (1 : ℝ) / 800) (by norm_num) (by norm_num) hvalue
  norm_num at h ⊢
  exact h

end
end MaxCutGames.Repetition

end OAI