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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 | import Mathlib.Algebra.BigOperators.Field
import OAI.Computability.PerfectCompleteness.Foundations.OccurrenceStochasticLemmas
namespace OAI
section
namespace PerfectCompleteness.FiniteListSampling
open scoped BigOperators
open UniqueGamesTheorem.Foundations.Games
noncomputable section
variable {A B : Type*} [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B]
def uniformList (L : Finset A) (nonempty : L.Nonempty) : FiniteDistribution A where
weight a := if a ∈ L then 1 / (L.card : ℝ) else 0
nonnegative a := by
split
· exact one_div_nonneg.mpr (Nat.cast_nonneg _)
· exact le_rfl
normalized := by
have hcard : (L.card : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt nonempty.card_pos)
simp only [Finset.sum_ite_mem_eq, Finset.sum_const, nsmul_eq_mul, mul_one_div]
exact div_self hcard
@[simp] theorem uniformList_weight_mem (L : Finset A) (hne : L.Nonempty)
(a : A) (ha : a ∈ L) : (uniformList L hne).weight a = 1 / (L.card : ℝ) := by
simp [uniformList, ha]
omit [DecidableEq A] in
theorem probability_ge_weight (μ : FiniteDistribution A) (event : A → Bool)
(a : A) (ha : event a = true) : μ.weight a ≤ μ.probability event := by
change μ.weight a ≤ ∑ x, if event x then μ.weight x else 0
have h := Finset.single_le_sum
(f := fun x : A => if event x then μ.weight x else 0)
(fun x _ => by split; exact μ.nonnegative x; exact le_rfl) (Finset.mem_univ a)
simpa only [ha, ite_true] using h
def pairAcceptance (μ : FiniteDistribution A) (ν : FiniteDistribution B)
(accepts : A → B → Bool) : ℝ :=
μ.expectation (fun a => ν.probability (accepts a))
omit [DecidableEq A] [DecidableEq B] in
theorem pairAcceptance_nonnegative (μ : FiniteDistribution A) (ν : FiniteDistribution B)
(accepts : A → B → Bool) : 0 ≤ pairAcceptance μ ν accepts := by
apply Finset.sum_nonneg
intro a _
exact mul_nonneg (μ.nonnegative a) (ν.probability_nonnegative _)
omit [DecidableEq A] [DecidableEq B] in
theorem pair_weight_le_acceptance (μ : FiniteDistribution A) (ν : FiniteDistribution B)
(accepts : A → B → Bool) (a : A) (b : B) (accepted : accepts a b = true) :
μ.weight a * ν.weight b ≤ pairAcceptance μ ν accepts := by
have hinner := probability_ge_weight ν (accepts a) b accepted
calc
_ ≤ μ.weight a * ν.probability (accepts a) :=
mul_le_mul_of_nonneg_left hinner (μ.nonnegative a)
_ ≤ _ := Finset.single_le_sum
(fun x _ => mul_nonneg (μ.nonnegative x) (ν.probability_nonnegative _))
(Finset.mem_univ a)
theorem uniform_pair_lower (L : Finset A) (R : Finset B)
(hL : L.Nonempty) (hR : R.Nonempty) (accepts : A → B → Bool)
(witness : ∃ a ∈ L, ∃ b ∈ R, accepts a b = true) :
1 / ((L.card : ℝ) * (R.card : ℝ)) ≤
pairAcceptance (uniformList L hL) (uniformList R hR) accepts := by
obtain ⟨a, ha, b, hb, hab⟩ := witness
have h := pair_weight_le_acceptance (uniformList L hL) (uniformList R hR) accepts a b hab
simpa only [uniformList_weight_mem L hL a ha, uniformList_weight_mem R hR b hb,
← one_div_mul_one_div] using h
theorem bounded_uniform_pair_lower (L : Finset A) (R : Finset B)
(hL : L.Nonempty) (hR : R.Nonempty) (s : Nat)
(hLs : L.card ≤ s) (hRs : R.card ≤ s) (accepts : A → B → Bool)
(witness : ∃ a ∈ L, ∃ b ∈ R, accepts a b = true) :
1 / (s : ℝ) ^ 2 ≤
pairAcceptance (uniformList L hL) (uniformList R hR) accepts := by
have hLc : 0 < (L.card : ℝ) := Nat.cast_pos.mpr hL.card_pos
have hRc : 0 < (R.card : ℝ) := Nat.cast_pos.mpr hR.card_pos
have hLs' : (L.card : ℝ) ≤ s := by exact_mod_cast hLs
have hRs' : (R.card : ℝ) ≤ s := by exact_mod_cast hRs
have hprod : (L.card : ℝ) * (R.card : ℝ) ≤ (s : ℝ) ^ 2 := by
simpa only [pow_two] using
mul_le_mul hLs' hRs' (Nat.cast_nonneg R.card) (Nat.cast_nonneg s)
exact (one_div_le_one_div_of_le (mul_pos hLc hRc) hprod).trans
(uniform_pair_lower L R hL hR accepts witness)
omit [DecidableEq A] [DecidableEq B] in
theorem pairAcceptance_eq_expectation (μ : FiniteDistribution A) (ν : FiniteDistribution B)
(accepts : A → B → Bool) :
pairAcceptance μ ν accepts = μ.expectation
(fun a => ν.expectation (fun b => if accepts a b then 1 else 0)) := by
unfold pairAcceptance
apply FiniteDistribution.expectation_congr
intro a
simp [FiniteDistribution.probability, FiniteDistribution.expectation, mul_ite]
end
end PerfectCompleteness.FiniteListSampling
end
end OAI
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