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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import OAI.Computability.PerfectCompleteness.Foundations.OddLists
import OAI.Computability.PerfectCompleteness.Foundations.RecursiveSpaceEquivLemmas
import OAI.Computability.PerfectCompleteness.Repetition.RepetitionRateLemmas
import OAI.Computability.PerfectCompleteness.Sampling.FiniteListSampling
namespace OAI
section
namespace PerfectCompleteness.OddListGame
open scoped BigOperators
open UniqueGamesTheorem.Foundations.Games
open OddLists FiniteListSampling
noncomputable section
variable {E Q₁ Q₂ A₁ A₂ : Type*}
[Fintype E] [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
[DecidableEq Q₁] [DecidableEq Q₂] [DecidableEq A₁] [DecidableEq A₂]
def base (μ : FiniteDistribution E) (left : E → Q₁) (right : E → Q₂)
(projection : E → A₁ → A₂) : OccurrenceGame E Q₁ Q₂ A₁ A₂ :=
OccurrenceGame.ofProjection μ left right projection
def game (s : Nat) (μ : FiniteDistribution E) (left : E → Q₁) (right : E → Q₂)
(projection : E → A₁ → A₂) :
OccurrenceGame E Q₁ Q₂ (OddList s A₁) (OddList s A₂) :=
OccurrenceGame.ofProjection μ left right (fun e => OddList.map (projection e))
def responses {Q A : Type*} [Fintype A] [DecidableEq A] {s : Nat}
(lists : Q → OddList s A) (q : Q) : FiniteDistribution A :=
uniformList (lists q).val (lists q).nonempty
omit [DecidableEq Q₁] [DecidableEq Q₂] [DecidableEq A₁] in
theorem accepted_pair {s : Nat} (μ : FiniteDistribution E)
(left : E → Q₁) (right : E → Q₂) (projection : E → A₁ → A₂)
(strategy : Strategy Q₁ Q₂ (OddList s A₁) (OddList s A₂)) (e : E)
(accepted : (game s μ left right projection).wins strategy e = true) :
∃ a ∈ (strategy.1 (left e)).val, ∃ b ∈ (strategy.2 (right e)).val,
(base μ left right projection).accepts e a b = true := by
have hmap : OddList.map (projection e) (strategy.1 (left e)) = strategy.2 (right e) := by
simpa only [game, OccurrenceGame.wins, OccurrenceGame.ofProjection, decide_eq_true_eq]
using accepted
obtain ⟨a, ha, b, hb, hab⟩ := OddList.exists_accepting_pair_of_eq
(projection e) (strategy.1 (left e)) (strategy.2 (right e)) hmap
exact ⟨a, ha, b, hb, by simpa [base, OccurrenceGame.ofProjection] using hab⟩
omit [DecidableEq Q₁] [DecidableEq Q₂] in
theorem success_le_randomized {s : Nat} [NeZero s] (μ : FiniteDistribution E)
(left : E → Q₁) (right : E → Q₂) (projection : E → A₁ → A₂)
(strategy : Strategy Q₁ Q₂ (OddList s A₁) (OddList s A₂)) :
(game s μ left right projection).success strategy ≤
(s : ℝ) ^ 2 * (base μ left right projection).stochasticSuccess
(responses strategy.1) (responses strategy.2) := by
let G := base μ left right projection
let pair := fun e => pairAcceptance (responses strategy.1 (left e))
(responses strategy.2 (right e)) (G.accepts e)
have hs : 0 < (s : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr (Nat.pos_of_neZero s)) 2
have hpoint (e : E) :
(if (game s μ left right projection).wins strategy e then (1 : ℝ) else 0) ≤
(s : ℝ) ^ 2 * pair e := by
by_cases hwin : (game s μ left right projection).wins strategy e = true
· simp only [hwin, ite_true]
have h := bounded_uniform_pair_lower
(strategy.1 (left e)).val (strategy.2 (right e)).val
(strategy.1 (left e)).nonempty (strategy.2 (right e)).nonempty s
(strategy.1 (left e)).card_le (strategy.2 (right e)).card_le (G.accepts e)
(accepted_pair μ left right projection strategy e hwin)
have hscaled := (div_le_iff₀ hs).mp h
change 1 ≤ (s : ℝ) ^ 2 *
pairAcceptance (uniformList (strategy.1 (left e)).val (strategy.1 (left e)).nonempty)
(uniformList (strategy.2 (right e)).val (strategy.2 (right e)).nonempty) (G.accepts e)
exact hscaled.trans_eq (mul_comm _ _)
· simp only [hwin]
exact mul_nonneg (sq_nonneg _) (pairAcceptance_nonnegative _ _ _)
have hpair : μ.expectation pair = G.stochasticSuccess
(responses strategy.1) (responses strategy.2) := by
unfold OccurrenceGame.stochasticSuccess
apply FiniteDistribution.expectation_congr
intro e
exact pairAcceptance_eq_expectation _ _ _
calc
_ = μ.expectation
(fun e => if (game s μ left right projection).wins strategy e then 1 else 0) :=
(game s μ left right projection).success_eq_occurrence_expectation strategy
_ ≤ μ.expectation (fun e => (s : ℝ) ^ 2 * pair e) :=
SmallBias.expectation_mono μ hpoint
_ = (s : ℝ) ^ 2 * μ.expectation pair := by
simp only [FiniteDistribution.expectation, Finset.mul_sum]
apply Finset.sum_congr rfl
intro e _
ring
_ = _ := by rw [hpair]
theorem value_le {s : Nat} [NeZero s] [Nonempty A₁] [Nonempty A₂]
(μ : FiniteDistribution E) (left : E → Q₁) (right : E → Q₂)
(projection : E → A₁ → A₂) :
(game s μ left right projection).value ≤
(s : ℝ) ^ 2 * (base μ left right projection).value := by
apply ((game s μ left right projection).value_le_iff _).mpr
intro strategy
exact (success_le_randomized μ left right projection strategy).trans
(mul_le_mul_of_nonneg_left
((base μ left right projection).stochasticSuccess_le_value
(responses strategy.1) (responses strategy.2)) (sq_nonneg _))
omit [DecidableEq Q₁] [DecidableEq Q₂] [DecidableEq A₁] in
theorem endpointDetermined {s : Nat} (μ : FiniteDistribution E)
(left : E → Q₁) (right : E → Q₂) (projection : E → A₁ → A₂)
(determined : (base μ left right projection).EndpointDetermined) :
(game s μ left right projection).EndpointDetermined := by
intro e e' hl hr L R
have hprojection : projection e = projection e' := by
funext a
have h := determined e e' hl hr a (projection e a)
have htrue : (base μ left right projection).accepts e' a (projection e a) = true := by
rw [← h]
simp [base, OccurrenceGame.ofProjection]
have heq : projection e' a = projection e a := by
simpa only [base, OccurrenceGame.ofProjection, decide_eq_true_eq] using htrue
exact heq.symm
simp only [game, OccurrenceGame.ofProjection, hprojection]
by_cases h : OddList.map (projection e') L = R <;> simp only [h, decide_true, decide_false]
theorem repetition_halfRate {s L : Nat} [NeZero s] [Nonempty A₁] [Nonempty A₂]
(μ : FiniteDistribution E) (left : E → Q₁) (right : E → Q₂)
(projection : E → A₁ → A₂)
(determined : (base μ left right projection).EndpointDetermined)
(gap : (s : ℝ) ^ 2 * (base μ left right projection).value ≤ 1 / 2)
(hL : 0 < L)
(alphabet : UniqueGamesTheorem.Foundations.Repetition.logTwo
((Fintype.card (OddList s A₁) : ℝ) * (Fintype.card (OddList s A₂) : ℝ)) ≤ L)
(n : Nat) :
((game s μ left right projection).repetition n).value ≤
(RepetitionRate.halfRate L : ℝ) ^ n := by
exact RepetitionRate.occurrence_repetition_halfRate (game s μ left right projection)
((game s μ left right projection).presentationOfDetermined
(endpointDetermined μ left right projection determined)) hL alphabet
((value_le μ left right projection).trans gap) n
end
end PerfectCompleteness.OddListGame
end
end OAI
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