lean/OAI/Combinatorics/IndependentSets/Repetition/SharedSampling.lean

import Mathlib
import OAI.Combinatorics.IndependentSets.Repetition.Rectangle2
import OAI.Combinatorics.IndependentSets.Repetition.Overlap

namespace OAI

namespace IndependentSetsGames.Foundations.Repetition

open scoped BigOperators
open Games
noncomputable section

universe u

def TraceSeed (σ : Type u) : Nat → Type u
  | 0 => PUnit
  | n + 1 => σ × TraceSeed σ n

instance traceSeedFintype {σ : Type*} [Fintype σ] (n : Nat) : Fintype (TraceSeed σ n) := by
  induction n with
  | zero => exact inferInstanceAs (Fintype PUnit)
  | succ n ih => exact inferInstanceAs (Fintype (σ × TraceSeed σ n))

instance traceSeedUniqueZero {σ : Type*} : Unique (TraceSeed σ 0) :=
  inferInstanceAs (Unique PUnit)

def traceList {σ : Type*} : (n : Nat) → TraceSeed σ n → List σ
  | 0, _ => []
  | n + 1, seed => seed.1 :: traceList n seed.2

def traceSeedLaw {σ : Type*} [Fintype σ] (μ : FiniteDistribution σ) :
    (n : Nat) → FiniteDistribution (TraceSeed σ n)
  | 0 => { weight := fun _ => 1, nonnegative := by intro; norm_num,
           normalized := by simp }
  | n + 1 => μ.product (traceSeedLaw μ n)

theorem traceSeedLaw_expectation {σ : Type*} [Fintype σ]
    (μ : FiniteDistribution σ) (n : Nat) (f : List σ → ℝ) :
    (traceSeedLaw μ n).expectation (fun seed => f (traceList n seed)) =
      CorrelatedSampling.traceAverage μ.weight n f := by
  induction n generalizing f with
  | zero => simp [traceSeedLaw, traceList, FiniteDistribution.expectation,
      CorrelatedSampling.traceAverage]
  | succ n ih =>
      change (μ.product (traceSeedLaw μ n)).expectation
        (fun seed => f (seed.1 :: traceList n seed.2)) = _
      rw [FiniteDistribution.expectation_product]
      simp only [CorrelatedSampling.traceAverage, FiniteDistribution.expectation]
      apply Finset.sum_congr rfl
      intro s _
      congr 1
      exact ih (fun xs => f (s :: xs))

def sharedOutputLaw {X Y S Seed : Type*}
    [Fintype X] [Fintype Y] [Fintype S] [Fintype Seed] [DecidableEq S]
    (μ : FiniteDistribution (X × Y)) (seedLaw : FiniteDistribution Seed)
    (left : Seed → X → S) (right : Seed → Y → S) :
    FiniteDistribution ((X × Y) × S × S) :=
  (μ.product seedLaw).pushforward
    (fun z => (z.1, left z.2 z.1.1, right z.2 z.1.2))

theorem sharedOutputLaw_weight {X Y S Seed : Type*}
    [Fintype X] [Fintype Y] [Fintype S] [Fintype Seed] [DecidableEq S]
    (μ : FiniteDistribution (X × Y)) (seedLaw : FiniteDistribution Seed)
    (left : Seed → X → S) (right : Seed → Y → S) (x : X) (y : Y) (a b : S) :
    (sharedOutputLaw μ seedLaw left right).weight ((x,y),a,b) =
      μ.weight (x,y) * seedLaw.expectation
        (fun seed => if left seed x = a ∧ right seed y = b then 1 else 0) := by
  classical
  simp only [sharedOutputLaw, FiniteDistribution.pushforward,
    FiniteDistribution.product, Fintype.sum_prod_type, Prod.mk.injEq]
  simp only [and_assoc, ite_and]
  simp [FiniteDistribution.expectation, Finset.mul_sum, mul_ite]

open CorrelatedSampling

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

def samplerLocal (thresholds : List ℝ) (profile : X → FiniteDistribution S)
    (fallback : S) (n : Nat) (seed : TraceSeed (RectangleSeed thresholds S) n)
    (x : X) : S :=
  localSample (rectangleAccept thresholds (profile x).weight) Prod.fst fallback
    (traceList n seed)

def samplerOutputLaw (thresholds : List ℝ) (μ : FiniteDistribution (X × Y))
    (L : X → FiniteDistribution S) (R : Y → FiniteDistribution S)
    (fallback : S) (n : Nat) : FiniteDistribution ((X × Y) × S × S) :=
  sharedOutputLaw μ (traceSeedLaw (rectangleDistribution thresholds) n)
    (samplerLocal thresholds L fallback n) (samplerLocal thresholds R fallback n)

omit [Nonempty S] [DecidableEq S] in
theorem probability_weight_le_one (μ : FiniteDistribution S) (s : S) : μ.weight s ≤ 1 := by
  calc
    μ.weight s ≤ ∑ t, μ.weight t :=
      Finset.single_le_sum (fun t _ => μ.nonnegative t) (Finset.mem_univ s)
    _ = 1 := μ.normalized

theorem samplerOutputLaw_diagonal (thresholds : List ℝ)
    (μ : FiniteDistribution (X × Y))
    (L : X → FiniteDistribution S) (R : Y → FiniteDistribution S)
    (hL : ∀ x s, (L x).weight s ∈ thresholds)
    (hR : ∀ y s, (R y).weight s ∈ thresholds)
    (fallback : S) (n : Nat) (x : X) (y : Y) (a : S) :
    μ.weight (x,y) * min ((L x).weight a) ((R y).weight a) /
        (1 + Information.totalVariation (L x).weight (R y).weight) *
        (1 - eventMass (rectangleWeight thresholds)
          (fun s => !(rectangleAccept thresholds (L x).weight s ||
            rectangleAccept thresholds (R y).weight s)) ^ n) ≤
      (samplerOutputLaw thresholds μ L R fallback n).weight ((x,y),a,a) := by
  have h := rectangle_label_diagonal_bound thresholds (L x).weight (R y).weight
    fallback a (hL x) (hR y) (L x).nonnegative (R y).nonnegative
    (probability_weight_le_one (L x)) (probability_weight_le_one (R y))
    (L x).normalized (R y).normalized n
  rw [samplerOutputLaw, sharedOutputLaw_weight]
  have he : (traceSeedLaw (rectangleDistribution (α := S) thresholds) n).expectation
      (fun seed => if samplerLocal thresholds L fallback n seed x = a ∧
        samplerLocal thresholds R fallback n seed y = a then 1 else 0) =
      traceAverage (rectangleWeight thresholds) n
        (localDiagonal (rectangleAccept thresholds (L x).weight)
          (rectangleAccept thresholds (R y).weight) Prod.fst fallback a) := by
    exact traceSeedLaw_expectation (rectangleDistribution (α := S) thresholds) n
      (localDiagonal (rectangleAccept thresholds (L x).weight)
        (rectangleAccept thresholds (R y).weight) Prod.fst fallback a)
  rw [he]
  simp only [Information.totalVariation, CorrelatedSampling.totalVariation] at h ⊢
  simpa only [mul_div_assoc, mul_assoc] using
    mul_le_mul_of_nonneg_left h (μ.nonnegative (x,y))

theorem samplerOutputLaw_uniform_diagonal
    (μ : FiniteDistribution (X × Y))
    (L : X → FiniteDistribution S) (R : Y → FiniteDistribution S)
    (fallback : S) (n : Nat) (x : X) (y : Y) (a : S) :
    μ.weight (x,y) * min ((L x).weight a) ((R y).weight a) /
        (1 + Information.totalVariation (L x).weight (R y).weight) *
        (1 - uniformRejectionRate S ^ n) ≤
      (samplerOutputLaw (sharedProfileThresholds L R) μ L R fallback n).weight ((x,y),a,a) := by
  have hp := pow_le_pow_left₀ (sharedProfileReject_nonnegative L R x y)
    (sharedProfile_reject_le_rate L R x y) n
  have hco : 0 ≤ μ.weight (x,y) * min ((L x).weight a) ((R y).weight a) /
      (1 + Information.totalVariation (L x).weight (R y).weight) :=
    div_nonneg (mul_nonneg (μ.nonnegative (x,y))
      (le_min ((L x).nonnegative a) ((R y).nonnegative a)))
      (by linarith [Information.totalVariation_nonneg (L x).weight (R y).weight])
  have h := samplerOutputLaw_diagonal (sharedProfileThresholds L R) μ L R
    (left_mem_sharedProfileThresholds L R) (right_mem_sharedProfileThresholds L R)
    fallback n x y a
  exact (mul_le_mul_of_nonneg_left (sub_le_sub_left hp 1) hco).trans h

theorem samplerOutputLaw_totalVariation
    (p : FiniteDistribution ((X × Y) × S)) (μ : FiniteDistribution (X × Y))
    (L : X → FiniteDistribution S) (R : Y → FiniteDistribution S)
    (fallback : S) (n : Nat) :
    Information.totalVariation
      (samplerOutputLaw (sharedProfileThresholds L R) μ L R fallback n).weight
      (diagonalWeights p.weight) ≤
      2 * Information.totalVariation p.weight (fun z => μ.weight z.1 * (L z.1.1).weight z.2) +
      2 * Information.totalVariation p.weight (fun z => μ.weight z.1 * (R z.1.2).weight z.2) +
      uniformRejectionRate S ^ n := by
  apply diagonal_sampler_totalVariation_le p.weight μ.weight
    (fun xy => (L xy.1).weight) (fun xy => (R xy.2).weight) _
    (Information.gameLaw_isProbability p) (Information.gameLaw_isProbability μ)
    (fun xy => Information.gameLaw_isProbability (L xy.1))
    (fun xy => Information.gameLaw_isProbability (R xy.2))
    (Information.gameLaw_isProbability _)
    (pow_nonneg (uniformRejectionRate_nonnegative S) n)
  · exact pow_le_one₀ (uniformRejectionRate_nonnegative S)
      (uniformRejectionRate_lt_one S).le
  · intro xy a
    exact samplerOutputLaw_uniform_diagonal μ L R fallback n xy.1 xy.2 a

theorem samplerOutputLaw_arbitrarily_close
    (p : FiniteDistribution ((X × Y) × S)) (μ : FiniteDistribution (X × Y))
    (L : X → FiniteDistribution S) (R : Y → FiniteDistribution S)
    (fallback : S) (η : ℝ) (hη : 0 < η) : ∃ n : Nat,
    Information.totalVariation
      (samplerOutputLaw (sharedProfileThresholds L R) μ L R fallback n).weight
      (diagonalWeights p.weight) ≤
      2 * Information.totalVariation p.weight (fun z => μ.weight z.1 * (L z.1.1).weight z.2) +
      2 * Information.totalVariation p.weight (fun z => μ.weight z.1 * (R z.1.2).weight z.2) + η := by
  obtain ⟨n,hn⟩ := exists_uniformRejectionRate_pow_lt S η hη
  exact ⟨n, (samplerOutputLaw_totalVariation p μ L R fallback n).trans
    (by linarith)⟩

end
end IndependentSetsGames.Foundations.Repetition

end OAI