lean/OAI/Computability/DirectedFeedback/Probability/Mixture.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Games.ProductLaw

namespace OAI

namespace DFVSGames.Foundations.Games.FiniteDistribution

open scoped BigOperators
noncomputable section

variable {Seed Ω : Type*} [Fintype Seed] [Fintype Ω]

def mixture (seedLaw : FiniteDistribution Seed)
    (laws : Seed → FiniteDistribution Ω) : FiniteDistribution Ω where
  weight x := ∑ seed, seedLaw.weight seed * (laws seed).weight x
  nonnegative x := Finset.sum_nonneg fun seed _ =>
    mul_nonneg (seedLaw.nonnegative seed) ((laws seed).nonnegative x)
  normalized := by
    rw [Finset.sum_comm]
    simp_rw [← Finset.mul_sum, (laws _).normalized, mul_one]
    exact seedLaw.normalized

theorem probability_mixture (seedLaw : FiniteDistribution Seed)
    (laws : Seed → FiniteDistribution Ω) (event : Ω → Bool) :
    (seedLaw.mixture laws).probability event =
      ∑ seed, seedLaw.weight seed * (laws seed).probability event := by
  unfold probability mixture
  calc
    _ = ∑ x, ∑ seed,
        seedLaw.weight seed * (if event x then (laws seed).weight x else 0) := by
      apply Finset.sum_congr rfl
      intro x _
      cases event x <;> simp
    _ = _ := by
      rw [Finset.sum_comm]
      apply Finset.sum_congr rfl
      intro seed _
      rw [Finset.mul_sum]

theorem probability_mixture_le (seedLaw : FiniteDistribution Seed)
    (laws : Seed → FiniteDistribution Ω) (event : Ω → Bool) (bound : ℝ)
    (h : ∀ seed, (laws seed).probability event ≤ bound) :
    (seedLaw.mixture laws).probability event ≤ bound := by
  rw [probability_mixture]
  calc
    _ ≤ ∑ seed, seedLaw.weight seed * bound :=
      Finset.sum_le_sum fun seed _ =>
        mul_le_mul_of_nonneg_left (h seed) (seedLaw.nonnegative seed)
    _ = bound := by rw [← Finset.sum_mul, seedLaw.normalized, one_mul]

end
end DFVSGames.Foundations.Games.FiniteDistribution

namespace DFVSGames.Foundations.Games.FiniteDistribution
open scoped BigOperators
noncomputable section
variable {A B C : Type*} [Fintype A] [Fintype B] [Fintype C]

theorem weight_eq_probability_singleton [DecidableEq A] (μ : FiniteDistribution A) (a : A) :
    μ.weight a = μ.probability (fun x => decide (x = a)) := by
  classical
  simp [probability]

theorem pushforward_comp (μ : FiniteDistribution A) (f : A → B) (g : B → C) :
    (μ.pushforward f).pushforward g = μ.pushforward (fun a => g (f a)) := by
  classical
  apply eq_of_weight_eq
  intro c
  rw [weight_eq_probability_singleton, weight_eq_probability_singleton]
  simp only [probability_pushforward]

theorem transport_eq_pushforward (μ : FiniteDistribution A) (e : A ≃ B) :
    μ.transport e = μ.pushforward e := by
  classical
  apply eq_of_weight_eq
  intro b
  rw [weight_eq_probability_singleton, weight_eq_probability_singleton,
    probability_transport, probability_pushforward]

theorem expectation_pushforward (μ : FiniteDistribution A) (f : A → B) (h : B → ℝ) :
    (μ.pushforward f).expectation h = μ.expectation (fun a => h (f a)) := by
  classical
  simp only [expectation, pushforward, Finset.sum_mul]
  rw [Finset.sum_comm]
  apply Finset.sum_congr rfl
  intro a _
  simp [ite_mul]

theorem pushforward_mixture (μ : FiniteDistribution A) (ν : A → FiniteDistribution B)
    (f : B → C) :
    (μ.mixture ν).pushforward f = μ.mixture (fun a => (ν a).pushforward f) := by
  classical
  apply eq_of_weight_eq
  intro c
  rw [weight_eq_probability_singleton, weight_eq_probability_singleton]
  simp only [probability_pushforward, probability_mixture]

end
end DFVSGames.Foundations.Games.FiniteDistribution

namespace DFVSGames.Clean

open Foundations.Games
open scoped BigOperators

noncomputable section

def bernoulli (p : ℝ) (hp₀ : 0 ≤ p) (hp₁ : p ≤ 1) :
    FiniteDistribution Bool where
  weight b := if b then p else 1 - p
  nonnegative b := by cases b <;> simp [hp₀, sub_nonneg.mpr hp₁]
  normalized := by simp []

def maskCount {k : ℕ} (mask : Fin k → Bool) : ℕ :=
  (Finset.univ.filter fun i => mask i = true).card

theorem product_mask {k : ℕ} (mask : Fin k → Bool) (ρ : ℝ) :
    (∏ i, if mask i then ρ else 1) = ρ ^ maskCount mask := by
  classical
  simp [maskCount, Finset.prod_ite]

theorem iid_product_moment {Ω : Type*} [Fintype Ω]
    (μ : FiniteDistribution Ω) (f : Ω → ℝ) (k : ℕ) :
    (μ.iid k).expectation (fun x => ∏ i, f (x i)) = μ.expectation f ^ k := by
  classical
  unfold FiniteDistribution.expectation FiniteDistribution.iid
  rw [Fintype.sum_pow]
  apply Finset.sum_congr rfl
  intro x _
  rw [Finset.prod_mul_distrib]

theorem bernoulli_mask_moment (p : ℝ) (hp₀ : 0 ≤ p) (hp₁ : p ≤ 1)
    (k : ℕ) (ρ : ℝ) :
    ((bernoulli p hp₀ hp₁).iid k).expectation (fun mask => ρ ^ maskCount mask) =
      (1 - p * (1 - ρ)) ^ k := by
  calc
    _ = ((bernoulli p hp₀ hp₁).iid k).expectation
        (fun mask => ∏ i, if mask i then ρ else 1) :=
      FiniteDistribution.expectation_congr _ (fun mask => (product_mask mask ρ).symm)
    _ = ((bernoulli p hp₀ hp₁).expectation (fun b => if b then ρ else 1)) ^ k :=
      iid_product_moment (bernoulli p hp₀ hp₁) (fun b : Bool => if b then ρ else 1) k
    _ = _ := by
      congr 1
      simp [FiniteDistribution.expectation, bernoulli]
      ring

variable {E : Type*} [Fintype E] [Nonempty E]

def cleanProbability (β : ℝ) (E : Type*) [Fintype E] : ℝ :=
  β / (Fintype.card E : ℝ)

omit [Nonempty E] in
theorem cleanProbability_nonneg {β : ℝ} (hβ₀ : 0 ≤ β) :
    0 ≤ cleanProbability β E :=
  div_nonneg hβ₀ (Nat.cast_nonneg _)

theorem cleanProbability_le_one {β : ℝ} (hβ₁ : β ≤ 1) :
    cleanProbability β E ≤ 1 := by
  have hc : (1 : ℝ) ≤ Fintype.card E := by
    exact_mod_cast Fintype.card_pos (α := E)
  apply (div_le_iff₀ (lt_of_lt_of_le zero_lt_one hc)).mpr
  simpa using hβ₁.trans hc

def singletonSlopeLaw (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1) :
    FiniteDistribution (Bool × E) :=
  (bernoulli β hβ₀ hβ₁).product (FiniteDistribution.uniform E)

def cleanBit (origin : E) (sample : Bool × E) : Bool := by
  classical
  exact sample.1 && decide (sample.2 = origin)

theorem clean_pushforward (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1) (origin : E) :
    (singletonSlopeLaw (E := E) β hβ₀ hβ₁).pushforward (cleanBit origin) =
      bernoulli (cleanProbability β E)
        (cleanProbability_nonneg hβ₀) (cleanProbability_le_one hβ₁) := by
  classical
  let μ := (singletonSlopeLaw (E := E) β hβ₀ hβ₁).pushforward (cleanBit origin)
  have ht : μ.weight true = cleanProbability β E := by
    simp [μ, FiniteDistribution.pushforward, singletonSlopeLaw,
      FiniteDistribution.product, Fintype.sum_prod_type,
      bernoulli, cleanBit, FiniteDistribution.uniform, cleanProbability, div_eq_mul_inv]
  apply FiniteDistribution.eq_of_weight_eq
  intro b
  cases b with
  | false =>
      change μ.weight false = 1 - cleanProbability β E
      have hn := μ.normalized
      simp only [Fintype.sum_bool] at hn
      linarith
  | true => exact ht

theorem clean_mask_law (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
    (origin : E) (k : ℕ) :
    ((singletonSlopeLaw (E := E) β hβ₀ hβ₁).iid k).pushforward
        (fun sample i => cleanBit origin (sample i)) =
      (bernoulli (cleanProbability β E)
        (cleanProbability_nonneg hβ₀) (cleanProbability_le_one hβ₁)).iid k := by
  rw [← FiniteDistribution.iid_pushforward, clean_pushforward]

theorem clean_mask_moment (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
    (origin : E) (k : ℕ) (ρ : ℝ) :
    ((singletonSlopeLaw (E := E) β hβ₀ hβ₁).iid k).expectation
        (fun sample => ρ ^ maskCount (fun i => cleanBit origin (sample i))) =
      (1 - cleanProbability β E * (1 - ρ)) ^ k := by
  rw [← FiniteDistribution.expectation_pushforward
    ((singletonSlopeLaw (E := E) β hβ₀ hβ₁).iid k)
    (fun sample i => cleanBit origin (sample i)) (fun mask => ρ ^ maskCount mask)]
  rw [clean_mask_law, bernoulli_mask_moment]

theorem cleanProbability_ge_of_card_le {β : ℝ} (hβ₀ : 0 ≤ β) {N : ℝ}
    (hcard : (Fintype.card E : ℝ) ≤ N) :
    β / N ≤ cleanProbability β E := by
  exact div_le_div_of_nonneg_left hβ₀
    (Nat.cast_pos.mpr Fintype.card_pos) hcard

theorem cleanProbability_ge_dimension {β : ℝ} (hβ₀ : 0 ≤ β) (dimW rs : ℕ)
    (hcard : Fintype.card E ≤ 2 ^ (dimW + rs)) :
    β * (2 : ℝ) ^ (-((dimW + rs : ℕ) : ℤ)) ≤ cleanProbability β E := by
  rw [zpow_neg, zpow_natCast, ← div_eq_mul_inv]
  apply cleanProbability_ge_of_card_le hβ₀
  exact_mod_cast hcard

theorem mask_rate_antitone {p q ρ : ℝ} (hp₀ : 0 ≤ p) (hp₁ : p ≤ 1)
    (hqp : q ≤ p) (hρ₀ : 0 ≤ ρ) (hρ₁ : ρ ≤ 1) (k : ℕ) :
    (1 - p * (1 - ρ)) ^ k ≤ (1 - q * (1 - ρ)) ^ k := by
  have hl : 0 ≤ 1 - p * (1 - ρ) := by
    nlinarith [mul_nonneg hp₀ hρ₀]
  apply pow_le_pow_left₀ hl
  exact sub_le_sub_left (mul_le_mul_of_nonneg_right hqp (sub_nonneg.mpr hρ₁)) 1

theorem clean_mask_moment_le (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
    (origin : E) (k : ℕ) {N : ℝ} (hcard : (Fintype.card E : ℝ) ≤ N) :
    ((singletonSlopeLaw (E := E) β hβ₀ hβ₁).iid k).expectation
        (fun sample => (1 - (1 : ℝ) / 3600) ^
          maskCount (fun i => cleanBit origin (sample i))) ≤
      (1 - (β / N) / 3600) ^ k := by
  rw [clean_mask_moment]
  have h := mask_rate_antitone (cleanProbability_nonneg (E := E) hβ₀)
    (cleanProbability_le_one (E := E) hβ₁)
    (cleanProbability_ge_of_card_le hβ₀ hcard)
    (ρ := 1 - (1 : ℝ) / 3600) (by norm_num) (by norm_num) k
  convert h using 1 ; congr 1 ; ring

theorem clean_mask_moment_le_dimension (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
    (origin : E) (k dimW rs : ℕ) (hcard : Fintype.card E ≤ 2 ^ (dimW + rs)) :
    ((singletonSlopeLaw (E := E) β hβ₀ hβ₁).iid k).expectation
        (fun sample => (1 - (1 : ℝ) / 3600) ^
          maskCount (fun i => cleanBit origin (sample i))) ≤
      (1 - (β / (2 : ℝ) ^ (dimW + rs)) / 3600) ^ k := by
  apply clean_mask_moment_le β hβ₀ hβ₁ origin k
  exact_mod_cast hcard

theorem mask_rate_le_exp {x : ℝ} (hx : x ≤ 1) (k : ℕ) :
    (1 - x) ^ k ≤ Real.exp (-(k : ℝ) * x) := by
  calc
    (1 - x) ^ k ≤ Real.exp (-x) ^ k :=
      pow_le_pow_left₀ (sub_nonneg.mpr hx) (Real.one_sub_le_exp_neg x) k
    _ = Real.exp (-(k : ℝ) * x) := by
      rw [← Real.exp_nat_mul]
      congr 1
      ring

theorem clean_mask_moment_le_exp (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
    (origin : E) (k : ℕ) {N : ℝ} (hcard : (Fintype.card E : ℝ) ≤ N) :
    ((singletonSlopeLaw (E := E) β hβ₀ hβ₁).iid k).expectation
        (fun sample => (1 - (1 : ℝ) / 3600) ^
          maskCount (fun i => cleanBit origin (sample i))) ≤
      Real.exp (-(k : ℝ) * (β / N) / 3600) := by
  have hN : 1 ≤ N := by
    have hc : (1 : ℝ) ≤ Fintype.card E := by
      exact_mod_cast Fintype.card_pos (α := E)
    exact hc.trans hcard
  have hq : β / N ≤ 1 := (div_le_iff₀ (by linarith)).mpr (by simpa using hβ₁.trans hN)
  have hx : β / N / 3600 ≤ 1 := (div_le_iff₀ (by norm_num)).mpr (by linarith)
  have h := (clean_mask_moment_le β hβ₀ hβ₁ origin k hcard).trans
    (mask_rate_le_exp hx k)
  simpa only [mul_div_assoc] using h

theorem cube_singleton_probability_nonneg (n : ℕ) : 0 ≤ (1 / (n : ℝ)) ^ 2 :=
  sq_nonneg _

theorem cube_singleton_probability_le_one {n : ℕ} (hn : 1 ≤ n) :
    (1 / (n : ℝ)) ^ 2 ≤ 1 := by
  have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn
  have hq₀ : 0 ≤ 1 / (n : ℝ) := div_nonneg zero_le_one (Nat.cast_nonneg _)
  have hq₁ : 1 / (n : ℝ) ≤ 1 :=
    (div_le_iff₀ (by linarith)).mpr (by simpa using hn')
  nlinarith [mul_nonneg hq₀ (sub_nonneg.mpr hq₁)]

theorem cube_mask_rate_le_exp {n : ℕ} (hn : 1 ≤ n) {N : ℝ} (hN : 1 ≤ N) :
    (1 - ((1 / (n : ℝ)) ^ 2 / N) / 3600) ^ (n ^ 3) ≤
      Real.exp (-(n : ℝ) / (3600 * N)) := by
  have hn₀ : (n : ℝ) ≠ 0 := by
    have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn
    linarith
  have hN₀ : N ≠ 0 := by linarith
  have hq : (1 / (n : ℝ)) ^ 2 / N ≤ 1 :=
    (div_le_iff₀ (by linarith)).mpr
      (by simpa using (cube_singleton_probability_le_one hn).trans hN)
  have hx : (1 / (n : ℝ)) ^ 2 / N / 3600 ≤ 1 :=
    (div_le_iff₀ (by norm_num)).mpr (by linarith)
  have he : -((n ^ 3 : ℕ) : ℝ) * ((1 / (n : ℝ)) ^ 2 / N / 3600) =
      -(n : ℝ) / (3600 * N) := by
    simp only [Nat.cast_pow]
    field_simp [hn₀, hN₀]

  calc
    _ ≤ Real.exp (-((n ^ 3 : ℕ) : ℝ) * ((1 / (n : ℝ)) ^ 2 / N / 3600)) :=
      mask_rate_le_exp hx (n ^ 3)
    _ = _ := congrArg Real.exp he

theorem exists_cube_exponential_lt {N error : ℝ} (hN : 0 < N) (herror : 0 < error) :
    ∃ n : ℕ, 1 ≤ n ∧ Real.exp (-(n : ℝ) / (3600 * N)) < error := by
  let rate := Real.exp (-(1 / (3600 * N)))
  have hrate₀ : 0 ≤ rate := (Real.exp_pos _).le
  have hrate₁ : rate < 1 := Real.exp_lt_one_iff.mpr
    (neg_lt_zero.mpr (one_div_pos.mpr (mul_pos (by norm_num) hN)))
  have ht := tendsto_pow_atTop_nhds_zero_of_lt_one hrate₀ hrate₁
  obtain ⟨m, hm⟩ := (ht.eventually (gt_mem_nhds herror)).exists
  refine ⟨m + 1, Nat.succ_le_succ (Nat.zero_le m), ?_⟩
  have hp : rate ^ (m + 1) < error :=
    lt_of_le_of_lt (by
      rw [pow_succ]
      exact mul_le_of_le_one_right (pow_nonneg hrate₀ m) hrate₁.le) hm
  have he : Real.exp (-((m + 1 : ℕ) : ℝ) / (3600 * N)) = rate ^ (m + 1) := by
    rw [← Real.exp_nat_mul]
    congr 1
    ring
  rwa [he]

theorem exists_cube_mask_rate_lt {N error : ℝ} (hN : 1 ≤ N) (herror : 0 < error) :
    ∃ n : ℕ, 1 ≤ n ∧
      (1 - ((1 / (n : ℝ)) ^ 2 / N) / 3600) ^ (n ^ 3) < error := by
  obtain ⟨n, hn, he⟩ := exists_cube_exponential_lt (by linarith : 0 < N) herror
  exact ⟨n, hn, (cube_mask_rate_le_exp hn hN).trans_lt he⟩

end
end DFVSGames.Clean

namespace DFVSGames.Clean.Experiment

open scoped BigOperators
open Foundations.Games
open Soundness
open Soundness.ConditionalIncidences Soundness.ConditionalSimulation
open Soundness.ConditionalGameLaw

noncomputable section

variable {R O N : Type} [Fintype R] [DecidableEq R]
  [Fintype O] [DecidableEq O] [Fintype N] [DecidableEq N]

abbrev Coefficient (R : Type) := ZeroInformation.Bits R
abbrev Additional (k : ℕ) (R : Type) := Coefficient R × (Fin k → Coefficient R × Coefficient R)
abbrev SparseSeed (k : ℕ) (R : Type) := Fin k → Bool × Coefficient R

def singletonSet {k : ℕ} (seed : SparseSeed k R) : Finset (Fin k) :=
  Finset.univ.filter fun i => (seed i).1 = true

def coefficients {k : ℕ} (extra : Additional k R) (seed : SparseSeed k R) :
    RawCoefficients (singletonSet seed) (Coefficient R) :=
  fun slot => match slot with
  | none => extra.1
  | some (.inl (i, j)) => if j = 0 then (extra.2 i.val).1 else (extra.2 i.val).2
  | some (.inr i) => (seed i.val).2

omit [DecidableEq R] in

theorem zeroSet_eq_cleanMask {k : ℕ} (extra : Additional k R) (seed : SparseSeed k R) :
    zeroSet (singletonSet seed) (coefficients extra seed) =
      Finset.univ.filter (fun i => cleanBit (ZeroInformation.zero : Coefficient R) (seed i) = true) := by
  classical
  ext i
  rw [mem_zeroSet]
  simp [singletonSet, coefficients, cleanBit]
  intro _
  rfl

omit [DecidableEq R] in
theorem zeroSet_card {k : ℕ} (extra : Additional k R) (seed : SparseSeed k R) :
    (zeroSet (singletonSet seed) (coefficients extra seed)).card =
      maskCount (fun i => cleanBit (ZeroInformation.zero : Coefficient R) (seed i)) := by
  rw [zeroSet_eq_cleanMask]
  rfl

abbrev LocalStrategies (k : ℕ) (R O N : Type) :=
  OuterStrategy (P := Fin k) (R := R) (O := O) (N := N)

def success (k : ℕ) (μ : FiniteDistribution (O × Fin 3))
    (g : IncidenceExtraction.Incidence O N)
    (extraLaw : FiniteDistribution (Additional k R))
    (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
    (strategy : Additional k R → SparseSeed k R → LocalStrategies k R O N) : ℝ :=
  extraLaw.expectation fun extra =>
    ((singletonSlopeLaw (E := Coefficient R) β hβ₀ hβ₁).iid k).expectation fun seed =>
      UpperBound.fixedAdviceSuccess μ (singletonSet seed) g (coefficients extra seed)
        (strategy extra seed)

private theorem expectation_mono_inline_Experiment {Ω : Type*} [Fintype Ω]
    (μ : FiniteDistribution Ω) {f g : Ω → ℝ} (h : ∀ x, f x ≤ g x) :
    μ.expectation f ≤ μ.expectation g := by
  unfold FiniteDistribution.expectation
  exact Finset.sum_le_sum fun x _ => mul_le_mul_of_nonneg_left (h x) (μ.nonnegative x)

private theorem expectation_const_inline_Experiment {Ω : Type*} [Fintype Ω]
    (μ : FiniteDistribution Ω) (c : ℝ) : μ.expectation (fun _ => c) = c := by
  rw [FiniteDistribution.expectation, ← Finset.sum_mul, μ.normalized, one_mul]

theorem success_le_of_repetition (k : ℕ) (μ : FiniteDistribution (O × Fin 3))
    (g : IncidenceExtraction.Incidence O N)
    (extraLaw : FiniteDistribution (Additional k R))
    (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
    (strategy : Additional k R → SparseSeed k R → LocalStrategies k R O N)
    (distinct : ∀ o i j, g.name o i = g.name o j → i = j)
    (ρ : ℝ)
    (rate : ∀ n, ((incidenceGame g (μ.pushforward (incidence g.name))).repetition n).value ≤ ρ ^ n) :
    success k μ g extraLaw β hβ₀ hβ₁ strategy ≤
      (1 - (β / (Fintype.card (Coefficient R) : ℝ)) * (1 - ρ)) ^ k := by
  unfold success
  calc
    _ ≤ extraLaw.expectation (fun _ =>
      (1 - (β / (Fintype.card (Coefficient R) : ℝ)) * (1 - ρ)) ^ k) := by
      apply expectation_mono_inline_Experiment
      intro extra
      calc
        _ ≤ ((singletonSlopeLaw (E := Coefficient R) β hβ₀ hβ₁).iid k).expectation
            (fun seed => ρ ^ maskCount (fun i => cleanBit (ZeroInformation.zero : Coefficient R) (seed i))) := by
          apply expectation_mono_inline_Experiment
          intro seed
          have bound := (UpperBound.fixed_advice_success_le_repeated_value μ
            (singletonSet seed) g (coefficients extra seed) (strategy extra seed) distinct).trans (rate _)
          simpa only [zeroSet_card] using bound
        _ = _ := clean_mask_moment β hβ₀ hβ₁ (ZeroInformation.zero : Coefficient R) k ρ
    _ = _ := expectation_const_inline_Experiment _ _

theorem randomized_success_le_of_repetition {Seed : Type*} [Fintype Seed]
    (k : ℕ) (μ : FiniteDistribution (O × Fin 3))
    (g : IncidenceExtraction.Incidence O N)
    (extraLaw : FiniteDistribution (Additional k R))
    (β : ℝ) (hβ₀ : 0 ≤ β) (hβ₁ : β ≤ 1)
    (seedLaw : FiniteDistribution Seed)
    (strategy : Seed → Additional k R → SparseSeed k R → LocalStrategies k R O N)
    (distinct : ∀ o i j, g.name o i = g.name o j → i = j)
    (ρ : ℝ)
    (rate : ∀ n, ((incidenceGame g (μ.pushforward (incidence g.name))).repetition n).value ≤ ρ ^ n) :
    seedLaw.expectation (fun seed => success k μ g extraLaw β hβ₀ hβ₁ (strategy seed)) ≤
      (1 - (β / (Fintype.card (Coefficient R) : ℝ)) * (1 - ρ)) ^ k := by
  calc
    _ ≤ seedLaw.expectation (fun _ =>
      (1 - (β / (Fintype.card (Coefficient R) : ℝ)) * (1 - ρ)) ^ k) :=
      expectation_mono_inline_Experiment _ (fun seed =>
        success_le_of_repetition k μ g extraLaw β hβ₀ hβ₁ (strategy seed) distinct ρ rate)
    _ = _ := expectation_const_inline_Experiment _ _

end
end DFVSGames.Clean.Experiment

namespace DFVSGames.Soundness.SubsetIntersection

open scoped BigOperators

def subsets (k t : ℕ) : Finset (Finset (Fin k)) :=
  (Finset.univ : Finset (Fin k)).powersetCard t

def uniformMean (k t : ℕ) (f : Finset (Fin k) → ℚ) : ℚ :=
  (∑ a ∈ subsets k t, f a) / (subsets k t).card

def independentMean (k t : ℕ) (f : Finset (Fin k) → Finset (Fin k) → ℚ) : ℚ :=
  uniformMean k t fun a => uniformMean k t fun b => f a b

def memberIndicator {k : ℕ} (i : Fin k) (a : Finset (Fin k)) : ℚ :=
  if i ∈ a then 1 else 0

def overlapProbability (k t : ℕ) : ℚ :=
  independentMean k t fun a b => if (a ∩ b).Nonempty then 1 else 0

theorem independentMean_eq_double_sum (k t : ℕ)
    (f : Finset (Fin k) → Finset (Fin k) → ℚ) :
    independentMean k t f =
      (∑ a ∈ subsets k t, ∑ b ∈ subsets k t, f a b) / ((subsets k t).card : ℚ)^2 := by
  unfold independentMean uniformMean
  simp only [div_eq_mul_inv, ← Finset.sum_mul, pow_two, mul_inv_rev]
  ring

theorem uniformMean_mul_left (k t : ℕ) (c : ℚ) (f : Finset (Fin k) → ℚ) :
    uniformMean k t (fun a => c * f a) = c * uniformMean k t f := by
  unfold uniformMean
  rw [← Finset.mul_sum]
  ring

theorem uniformMean_mul_right (k t : ℕ) (c : ℚ) (f : Finset (Fin k) → ℚ) :
    uniformMean k t (fun a => f a * c) = uniformMean k t f * c := by
  unfold uniformMean
  rw [← Finset.sum_mul]
  ring

theorem independentMean_product (k t : ℕ) (f g : Finset (Fin k) → ℚ) :
    independentMean k t (fun a b => f a * g b) = uniformMean k t f * uniformMean k t g := by
  simp only [independentMean, uniformMean_mul_left, uniformMean_mul_right]

theorem subsets_card (k t : ℕ) : (subsets k t).card = k.choose t := by
  simp [subsets]

theorem subsets_nonempty {k t : ℕ} (htk : t ≤ k) : (subsets k t).Nonempty := by
  apply Finset.powersetCard_nonempty.mpr
  simpa using htk

theorem subsets_card_ne_zero {k t : ℕ} (htk : t ≤ k) :
    ((subsets k t).card : ℚ) ≠ 0 := by
  exact_mod_cast (Finset.card_ne_zero.mpr (subsets_nonempty htk))

theorem count_containing {k t : ℕ} (ht : 0 < t) (i : Fin k) :
    ((subsets k t).filter (fun a => i ∈ a)).card =
      (k - 1).choose (t - 1) := by
  simpa [subsets] using
    (Finset.card_filter_powersetCard_subset ({i} : Finset (Fin k)) Finset.univ t
      (by simp) (by simpa using Nat.succ_le_of_lt ht))

theorem choose_predecessor_ratio {k t : ℕ} (hk : 0 < k) (ht : 0 < t) (htk : t ≤ k) :
    ((k - 1).choose (t - 1) : ℚ) / k.choose t = (t : ℚ) / k := by
  have hk0 : (k : ℚ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hk)
  have hc0 : (k.choose t : ℚ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt (Nat.choose_pos htk))
  have hrel : k * (k - 1).choose (t - 1) = k.choose t * t := by
    simpa [Nat.sub_add_cancel hk, Nat.sub_add_cancel ht] using
      (Nat.add_one_mul_choose_eq (k - 1) (t - 1))
  apply (div_eq_div_iff hc0 hk0).2
  exact_mod_cast (by simpa [Nat.mul_comm] using hrel)

theorem inclusion_probability {k t : ℕ} (hk : 0 < k) (htk : t ≤ k) (i : Fin k) :
    uniformMean k t (memberIndicator i) = (t : ℚ) / k := by
  by_cases ht : t = 0
  · subst t
    simp [uniformMean, subsets, memberIndicator]
  · have htpos : 0 < t := Nat.pos_of_ne_zero ht
    unfold uniformMean memberIndicator
    rw [Finset.sum_boole, count_containing htpos i, subsets_card]
    exact choose_predecessor_ratio hk htpos htk

theorem independent_inclusion_probability {k t : ℕ} (hk : 0 < k) (htk : t ≤ k) (i : Fin k) :
    independentMean k t (fun a b => memberIndicator i a * memberIndicator i b) =
      ((t : ℚ) / k)^2 := by
  rw [independentMean_product, inclusion_probability hk htk]
  ring

theorem uniformMean_congr {k t : ℕ} {f g : Finset (Fin k) → ℚ}
    (h : ∀ a ∈ subsets k t, f a = g a) : uniformMean k t f = uniformMean k t g := by
  unfold uniformMean
  rw [Finset.sum_congr rfl h]

theorem uniformMean_const {k t : ℕ} (htk : t ≤ k) (c : ℚ) :
    uniformMean k t (fun _ => c) = c := by
  have hn := subsets_card_ne_zero htk
  simp only [uniformMean, Finset.sum_const, nsmul_eq_mul]
  field_simp

theorem uniformMean_finset_sum {k t : ℕ} {α : Type*} (s : Finset α)
    (f : α → Finset (Fin k) → ℚ) :
    uniformMean k t (fun a => ∑ i ∈ s, f i a) =
      ∑ i ∈ s, uniformMean k t (f i) := by
  unfold uniformMean
  rw [Finset.sum_comm]
  simp only [div_eq_mul_inv, Finset.sum_mul]

theorem uniformMean_mono {k t : ℕ} {f g : Finset (Fin k) → ℚ}
    (h : ∀ a ∈ subsets k t, f a ≤ g a) : uniformMean k t f ≤ uniformMean k t g := by
  apply div_le_div_of_nonneg_right (Finset.sum_le_sum h)
  exact Nat.cast_nonneg _

theorem independentMean_mono {k t : ℕ} {f g : Finset (Fin k) → Finset (Fin k) → ℚ}
    (h : ∀ a ∈ subsets k t, ∀ b ∈ subsets k t, f a b ≤ g a b) :
    independentMean k t f ≤ independentMean k t g := by
  apply uniformMean_mono
  intro a ha
  apply uniformMean_mono
  exact h a ha

theorem intersection_card_indicator {k : ℕ} (a b : Finset (Fin k)) :
    ((a ∩ b).card : ℚ) = ∑ i ∈ a, memberIndicator i b := by
  simp [memberIndicator]

theorem expected_intersection_fixed {k t : ℕ} (hk : 0 < k) (htk : t ≤ k)
    (a : Finset (Fin k)) :
    uniformMean k t (fun b => ((a ∩ b).card : ℚ)) = a.card * ((t : ℚ) / k) := by
  simp_rw [intersection_card_indicator]
  rw [uniformMean_finset_sum]
  simp [inclusion_probability hk htk, nsmul_eq_mul]

theorem expected_intersection {k t : ℕ} (hk : 0 < k) (htk : t ≤ k) :
    independentMean k t (fun a b => ((a ∩ b).card : ℚ)) = (t : ℚ)^2 / k := by
  unfold independentMean
  simp_rw [expected_intersection_fixed hk htk]
  calc
    uniformMean k t (fun a => a.card * ((t : ℚ) / k)) =
        uniformMean k t (fun _ => (t : ℚ) * ((t : ℚ) / k)) := by
      apply uniformMean_congr
      intro a ha
      have hcard : a.card = t := (Finset.mem_powersetCard.mp ha).2
      rw [hcard]
    _ = (t : ℚ) * ((t : ℚ) / k) := uniformMean_const htk _
    _ = (t : ℚ)^2 / k := by ring

theorem nonempty_indicator_le_card {k : ℕ} (a b : Finset (Fin k)) :
    (if (a ∩ b).Nonempty then (1 : ℚ) else 0) ≤ (a ∩ b).card := by
  split_ifs with h
  · exact_mod_cast (Finset.card_pos.mpr h)
  · exact Nat.cast_nonneg _

theorem overlap_probability_le {k t : ℕ} (hk : 0 < k) (htk : t ≤ k) :
    overlapProbability k t ≤ (t : ℚ)^2 / k := by
  calc
    overlapProbability k t ≤ independentMean k t (fun a b => ((a ∩ b).card : ℚ)) :=
      independentMean_mono (fun a _ b _ => nonempty_indicator_le_card a b)
    _ = (t : ℚ)^2 / k := expected_intersection hk htk

end DFVSGames.Soundness.SubsetIntersection

namespace DFVSGames.Soundness.PartnerSampling

open scoped BigOperators
open SubsetIntersection (subsets)

abbrev ActiveSet (k t : ℕ) := ↥(subsets k t)

abbrev Partner (k t : ℕ) := Σ a : ActiveSet k t, (↥a.val → Fin 3)

theorem activeSet_card {k t : ℕ} (a : ActiveSet k t) : a.val.card = t :=
  (Finset.mem_powersetCard.mp a.property).2

theorem slotTuples_card {k t : ℕ} (a : ActiveSet k t) :
    Fintype.card (↥a.val → Fin 3) = 3 ^ t := by
  simp [activeSet_card a]

theorem partner_card (k t : ℕ) :
    Fintype.card (Partner k t) = (subsets k t).card * 3 ^ t := by
  rw [Fintype.card_sigma]
  simp only [slotTuples_card, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
    Fintype.card_coe, Nat.cast_id]

theorem partner_nonempty {k t : ℕ} (htk : t ≤ k) : Nonempty (Partner k t) := by
  obtain ⟨a, ha⟩ := SubsetIntersection.subsets_nonempty htk
  exact ⟨⟨⟨a, ha⟩, fun _ => 0⟩⟩

theorem uniformMean_real_cast (k t : ℕ) (f : Finset (Fin k) → ℚ) :
    (SubsetIntersection.uniformMean k t f : ℝ) =
      (𝔼 a ∈ subsets k t, (f a : ℝ)) := by
  rw [Finset.expect_eq_sum_div_card]
  simp [SubsetIntersection.uniformMean]

theorem independentMean_real_cast (k t : ℕ)
    (f : Finset (Fin k) → Finset (Fin k) → ℚ) :
    (SubsetIntersection.independentMean k t f : ℝ) =
      (𝔼 a ∈ subsets k t, 𝔼 b ∈ subsets k t, (f a b : ℝ)) := by
  simp only [SubsetIntersection.independentMean, uniformMean_real_cast]

theorem subset_overlap_probability_le {k t : ℕ} (hk : 0 < k) (htk : t ≤ k) :
    (𝔼 a ∈ subsets k t, 𝔼 b ∈ subsets k t,
      if (a ∩ b).Nonempty then (1 : ℝ) else 0) ≤ (t : ℝ)^2 / k := by
  have h : (SubsetIntersection.overlapProbability k t : ℝ) ≤ (t : ℝ)^2 / k := by
    have h := SubsetIntersection.overlap_probability_le hk htk
    have hh := (Rat.cast_le (K := ℝ)).mpr h
    simpa using hh
  simpa only [SubsetIntersection.overlapProbability, independentMean_real_cast,
    apply_ite, Rat.cast_one, Rat.cast_zero] using h

theorem activeSet_expect (k t : ℕ) (f : Finset (Fin k) → ℝ) :
    (𝔼 a : ActiveSet k t, f a.val) = (𝔼 a ∈ subsets k t, f a) := by
  simp only [Finset.expect_eq_sum_div_card, Finset.card_univ, Fintype.card_coe,
    Finset.sum_coe_sort]

theorem partner_expect_activeSet (k t : ℕ) (f : ActiveSet k t → ℝ) :
    (𝔼 p : Partner k t, f p.1) = (𝔼 a : ActiveSet k t, f a) := by
  have hsum : (∑ p : Partner k t, f p.1) = (3 : ℝ)^t * ∑ a : ActiveSet k t, f a := by
    rw [Fintype.sum_sigma]
    simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, slotTuples_card,
      Nat.cast_pow, Nat.cast_ofNat]
    rw [Finset.mul_sum]
  rw [Fintype.expect_eq_sum_div_card, Fintype.expect_eq_sum_div_card, hsum, partner_card]
  simp only [Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat, Fintype.card_coe]
  have hc : (3 : ℝ)^t ≠ 0 := pow_ne_zero _ (by norm_num)
  simpa only [mul_comm] using
    (mul_div_mul_left (∑ a : ActiveSet k t, f a) ((subsets k t).card : ℝ) hc)

theorem partner_expect_active (k t : ℕ) (f : Finset (Fin k) → ℝ) :
    (𝔼 p : Partner k t, f p.1.val) = (𝔼 a ∈ subsets k t, f a) := by
  rw [partner_expect_activeSet k t (fun a => f a.val), activeSet_expect]

theorem partner_pair_expect_active (k t : ℕ)
    (f : Finset (Fin k) → Finset (Fin k) → ℝ) :
    (𝔼 p : Partner k t, 𝔼 q : Partner k t, f p.1.val q.1.val) =
      (𝔼 a ∈ subsets k t, 𝔼 b ∈ subsets k t, f a b) := by
  calc
    _ = 𝔼 p : Partner k t, 𝔼 b ∈ subsets k t, f p.1.val b := by
      apply Finset.expect_congr rfl
      intro p _
      exact partner_expect_active k t (fun b => f p.1.val b)
    _ = _ := partner_expect_active k t (fun a => 𝔼 b ∈ subsets k t, f a b)

theorem partner_active_overlap_probability_le {k t : ℕ} (hk : 0 < k) (htk : t ≤ k) :
    (𝔼 p : Partner k t, 𝔼 q : Partner k t,
      if (p.1.val ∩ q.1.val).Nonempty then (1 : ℝ) else 0) ≤ (t : ℝ)^2 / k := by
  rw [partner_pair_expect_active k t (fun a b => if (a ∩ b).Nonempty then (1 : ℝ) else 0)]
  exact subset_overlap_probability_le hk htk

def active {k t : ℕ} (p : Partner k t) (j : Fin k) : Bool :=
  decide (j ∈ p.1.val)

def decodeSlot (i : Fin 3) : PartnerProjection.Slot :=
  if i.val = 0 then .first else if i.val = 1 then .second else .third

def slots {k t : ℕ} (p : Partner k t) (j : Fin k) : PartnerProjection.Slot :=
  if hj : j ∈ p.1.val then decodeSlot (p.2 ⟨j, hj⟩) else .first

noncomputable def kernel {k t : ℕ} (rhs : Fin k → Bool) (p : Partner k t) :
    Submodule (ZMod 2) (PartnerProjection.SourcePoint rhs) :=
  (PartnerLinear.projection rhs (active p) (slots p)).ker

def activeSubtypeEquiv {k t : ℕ} (p : Partner k t) :
    PartnerLinear.Active (active p) ≃ ↥p.1.val :=
  Equiv.subtypeEquivRight (fun j => by simp [active])

theorem kernel_finrank {k t : ℕ} (rhs : Fin k → Bool) (p : Partner k t) :
    Module.finrank (ZMod 2) (kernel rhs p) = t := by
  rw [kernel, PartnerLinear.projection_ker_finrank]
  calc
    Fintype.card (PartnerLinear.Active (active p)) = Fintype.card ↥p.1.val :=
      Fintype.card_congr (activeSubtypeEquiv p)
    _ = t := by simpa using activeSet_card p.1

theorem kernel_inf_finrank_le {k t : ℕ} (rhs : Fin k → Bool) (p q : Partner k t) :
    Module.finrank (ZMod 2) ↥(kernel rhs p ⊓ kernel rhs q : Submodule (ZMod 2) (PartnerProjection.SourcePoint rhs)) ≤ t := by
  calc
    Module.finrank (ZMod 2) ↥(kernel rhs p ⊓ kernel rhs q : Submodule (ZMod 2) (PartnerProjection.SourcePoint rhs)) ≤
        Module.finrank (ZMod 2) (kernel rhs p) := Submodule.finrank_mono inf_le_left
    _ = t := kernel_finrank rhs p

theorem kernel_inf_eq_bot_of_no_overlap {k t : ℕ}
    (rhs : Fin k → Bool) (p q : Partner k t)
    (h : ¬ (p.1.val ∩ q.1.val).Nonempty) :
    kernel rhs p ⊓ kernel rhs q = ⊥ := by
  apply PartnerLinear.disjoint_projection_kernels
    rhs (active p) (slots p) (active q) (slots q)
  intro j hj
  have hp : j ∈ p.1.val := by simpa [active] using hj
  have hq : j ∉ q.1.val := by
    intro hq
    exact h ⟨j, Finset.mem_inter.mpr ⟨hp, hq⟩⟩
  simp [active, hq]

def activeOverlap {k t : ℕ} (p q : Partner k t) : ℝ :=
  if (p.1.val ∩ q.1.val).Nonempty then 1 else 0

noncomputable def kernelCollision {k t : ℕ}
    (rhs : Fin k → Bool) (p q : Partner k t) : ℝ := by
  classical
  exact if kernel rhs p ⊓ kernel rhs q ≠ ⊥ then 1 else 0

theorem kernelCollision_le_activeOverlap {k t : ℕ}
    (rhs : Fin k → Bool) (p q : Partner k t) :
    kernelCollision rhs p q ≤ activeOverlap p q := by
  classical
  by_cases h : (p.1.val ∩ q.1.val).Nonempty
  · simp only [activeOverlap, ite_eq_left h, kernelCollision]
    split_ifs <;> norm_num
  · have hb := kernel_inf_eq_bot_of_no_overlap rhs p q h
    simp [kernelCollision, activeOverlap, h, hb]

theorem kernel_collision_probability_le {k t : ℕ}
    (hk : 0 < k) (htk : t ≤ k) (rhs : Fin k → Bool) :
    (𝔼 p : Partner k t, 𝔼 q : Partner k t,
      kernelCollision rhs p q) ≤ (t : ℝ)^2 / k := by
  calc
    (𝔼 p : Partner k t, 𝔼 q : Partner k t, kernelCollision rhs p q) ≤
        (𝔼 p : Partner k t, 𝔼 q : Partner k t, activeOverlap p q) := by
      apply Finset.expect_le_expect
      intro p _
      apply Finset.expect_le_expect
      intro q _
      exact kernelCollision_le_activeOverlap rhs p q
    _ ≤ (t : ℝ)^2 / k := partner_active_overlap_probability_le hk htk

theorem kernel_overlap_weight_pointwise {k t : ℕ}
    (rhs : Fin k → Bool) (m : ℕ) (p q : Partner k t) :
    (2 : ℝ) ^ (m * Module.finrank (ZMod 2) ↥(kernel rhs p ⊓ kernel rhs q : Submodule (ZMod 2) (PartnerProjection.SourcePoint rhs))) ≤
      1 + activeOverlap p q * (2 : ℝ) ^ (m * t) := by
  have hcap :
      (2 : ℝ) ^ (m * Module.finrank (ZMod 2) ↥(kernel rhs p ⊓ kernel rhs q : Submodule (ZMod 2) (PartnerProjection.SourcePoint rhs))) ≤
        (2 : ℝ) ^ (m * t) := by
    apply pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2)
    exact Nat.mul_le_mul_left m (kernel_inf_finrank_le rhs p q)
  by_cases h : (p.1.val ∩ q.1.val).Nonempty
  · simp only [activeOverlap, ite_eq_left h, one_mul]
    linarith
  · have hb := kernel_inf_eq_bot_of_no_overlap rhs p q h
    rw [hb]
    simp [activeOverlap, h]

theorem kernel_overlap_weight_le {k t : ℕ}
    (hk : 0 < k) (htk : t ≤ k) (rhs : Fin k → Bool) (m : ℕ) :
    (𝔼 p : Partner k t, 𝔼 q : Partner k t,
      (2 : ℝ) ^ (m * Module.finrank (ZMod 2) ↥(kernel rhs p ⊓ kernel rhs q : Submodule (ZMod 2) (PartnerProjection.SourcePoint rhs)))) ≤
      1 + ((t : ℝ)^2 / k) * (2 : ℝ) ^ (m * t) := by
  let : Nonempty (Partner k t) := partner_nonempty htk
  have hmean :
      (𝔼 p : Partner k t, 𝔼 q : Partner k t,
        (1 + activeOverlap p q * (2 : ℝ) ^ (m * t))) =
      1 + (𝔼 p : Partner k t, 𝔼 q : Partner k t, activeOverlap p q) *
        (2 : ℝ) ^ (m * t) := by
    simp_rw [Finset.expect_add_distrib, ← Finset.expect_mul, Fintype.expect_const]
  calc
    (𝔼 p : Partner k t, 𝔼 q : Partner k t,
      (2 : ℝ) ^ (m * Module.finrank (ZMod 2) ↥(kernel rhs p ⊓ kernel rhs q : Submodule (ZMod 2) (PartnerProjection.SourcePoint rhs)))) ≤
        (𝔼 p : Partner k t, 𝔼 q : Partner k t,
          (1 + activeOverlap p q * (2 : ℝ) ^ (m * t))) := by
      apply Finset.expect_le_expect
      intro p _
      apply Finset.expect_le_expect
      intro q _
      exact kernel_overlap_weight_pointwise rhs m p q
    _ = 1 + (𝔼 p : Partner k t, 𝔼 q : Partner k t, activeOverlap p q) *
        (2 : ℝ) ^ (m * t) := hmean
    _ ≤ 1 + ((t : ℝ)^2 / k) * (2 : ℝ) ^ (m * t) := by
      apply add_le_add le_rfl
      apply mul_le_mul_of_nonneg_right
      · exact partner_active_overlap_probability_le hk htk
      · exact pow_nonneg (by norm_num) _

end DFVSGames.Soundness.PartnerSampling

end OAI