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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | import Mathlib.Algebra.Field.ZMod
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.LinearIndependent.Defs
import Mathlib.Tactic.Positivity
import OAI.Computability.PerfectCompleteness.Algebra.UniformLinearImage
import OAI.Computability.PerfectCompleteness.Sampling.UniformLatent
import OAI.Computability.UniqueGames.Reduction.BinaryLinear
namespace OAI
section
namespace PerfectCompleteness.AffineFunctionalDependence
open scoped BigOperators Classical
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Integration.BinaryLinear (F2)
noncomputable section
private theorem probability_cover_le_sum {Ω I : Type*} [Fintype Ω] [Fintype I]
(μ : FiniteDistribution Ω) (event : Ω → Bool) (cover : I → Ω → Bool)
(hcover : ∀ x, event x = true → ∃ i, cover i x = true) :
μ.probability event ≤ ∑ i, μ.probability (cover i) := by
unfold FiniteDistribution.probability
rw [Finset.sum_comm]
apply Finset.sum_le_sum
intro x _
by_cases hx : event x = true
· obtain ⟨i, hi⟩ := hcover x hx
have h := Finset.single_le_sum (s := Finset.univ)
(f := fun j => if cover j x then μ.weight x else 0) (a := i)
(fun j _ => by split <;> first | exact μ.nonnegative x | exact le_rfl)
(Finset.mem_univ i)
simpa only [hx, hi, ite_true] using h
· simp only [hx]
apply Finset.sum_nonneg
intro i _
split
· exact μ.nonnegative x
· exact le_rfl
variable {N V : Type*} [AddCommGroup N] [Module F2 N]
[AddCommGroup V] [Module F2 V] [Fintype N] [Fintype V]
theorem affine_zero_probability_le (f : N →ₗ[F2] V) (b : V) :
(FiniteDistribution.uniform N).probability (fun x => decide (b + f x = 0)) ≤
1 / (Fintype.card (LinearMap.range f) : ℝ) := by
have hlaw := UniformLinearImage.uniform_pushforward_linearMap
f.rangeRestrict f.surjective_rangeRestrict
have hp := congrArg (fun μ : FiniteDistribution (LinearMap.range f) =>
μ.probability (fun y => decide (b + (y : V) = 0))) hlaw
rw [FiniteDistribution.probability_pushforward] at hp
change (FiniteDistribution.uniform N).probability
(fun x => decide (b + f x = 0)) = _ at hp
rw [hp]
by_cases hzero : ∃ y : LinearMap.range f, b + (y : V) = 0
· obtain ⟨z, hz⟩ := hzero
have hevent : (fun y : LinearMap.range f => decide (b + (y : V) = 0)) =
(fun y => decide (y = z)) := by
funext y
apply Bool.eq_iff_iff.mpr
simp only [decide_eq_true_eq]
constructor
· intro hy
apply Subtype.ext
exact add_left_cancel (hy.trans hz.symm)
· rintro rfl
exact hz
rw [hevent, UniformLatent.probability_uniform_singleton]
· have hz : ∀ y : LinearMap.range f, b + (y : V) ≠ 0 := by
simpa only [not_exists] using hzero
simp [FiniteDistribution.probability, hz]
variable {m : Nat}
def coefficientSum (c : Fin m → F2) : (Fin m → N) →ₗ[F2] N where
toFun x := ∑ i, c i • x i
map_add' x y := by simp only [Pi.add_apply, smul_add, Finset.sum_add_distrib]
map_smul' a x := by
simp only [Pi.smul_apply, Finset.smul_sum, smul_smul, mul_comm, RingHom.id_apply]
@[simp] theorem coefficientSum_apply (c : Fin m → F2) (x : Fin m → N) :
coefficientSum c x = ∑ i, c i • x i := rfl
theorem coefficientSum_surjective (c : Fin m → F2) (hc : c ≠ 0) :
Function.Surjective (coefficientSum (N := N) c) := by
have hcoord : ∃ i, c i ≠ 0 := by
by_contra h
push Not at h
exact hc (funext h)
obtain ⟨i, hi⟩ := hcoord
have hi1 : c i = 1 :=
(UniqueGamesTheorem.Integration.BinaryLinear.scalar_cases (c i)).resolve_left hi
intro n
refine ⟨Function.update 0 i n, ?_⟩
rw [coefficientSum_apply, Finset.sum_eq_single i]
· simp [hi1]
· intro j _ hji
simp [Function.update_of_ne hji]
· intro h
exact (h (Finset.mem_univ i)).elim
theorem relation_probability_le (f : N →ₗ[F2] V) (offsets : Fin m → V)
(c : Fin m → F2) (hc : c ≠ 0) :
(FiniteDistribution.uniform (Fin m → N)).probability
(fun x => decide ((∑ i, c i • (offsets i + f (x i))) = 0)) ≤
1 / (Fintype.card (LinearMap.range f) : ℝ) := by
have hlaw := UniformLinearImage.uniform_pushforward_linearMap
(coefficientSum (N := N) c) (coefficientSum_surjective c hc)
have hp := congrArg (fun μ : FiniteDistribution N =>
μ.probability (fun n => decide ((∑ i, c i • offsets i) + f n = 0))) hlaw
rw [FiniteDistribution.probability_pushforward] at hp
have hevent :
(fun x : Fin m → N => decide ((∑ i, c i • (offsets i + f (x i))) = 0)) =
(fun x => decide ((∑ i, c i • offsets i) + f (coefficientSum c x) = 0)) := by
funext x
simp only [coefficientSum_apply, smul_add, Finset.sum_add_distrib,
map_sum, map_smul]
rw [hevent, hp]
exact affine_zero_probability_le f (∑ i, c i • offsets i)
theorem dependence_probability_le (f : N →ₗ[F2] V) (offsets : Fin m → V) :
(FiniteDistribution.uniform (Fin m → N)).probability
(fun x => decide (¬ LinearIndependent F2 (fun i => offsets i + f (x i)))) ≤
(2 : ℝ) ^ m / (2 : ℝ) ^ Module.finrank F2 (LinearMap.range f) := by
let μ := FiniteDistribution.uniform (Fin m → N)
let event := fun x : Fin m → N =>
decide (¬ LinearIndependent F2 (fun i => offsets i + f (x i)))
let cover := fun (c : Fin m → F2) (x : Fin m → N) =>
decide (c ≠ 0 ∧ (∑ i, c i • (offsets i + f (x i))) = 0)
have hcover : ∀ x, event x = true → ∃ c, cover c x = true := by
intro x hx
have hdep : ¬ LinearIndependent F2 (fun i => offsets i + f (x i)) :=
of_decide_eq_true hx
obtain ⟨c, hc, i, hi⟩ := Fintype.not_linearIndependent_iff.mp hdep
have hc0 : c ≠ 0 := by
intro hz
exact hi (congrFun hz i)
refine ⟨c, ?_⟩
exact decide_eq_true_eq.mpr ⟨hc0, hc⟩
have hbound (c : Fin m → F2) :
μ.probability (cover c) ≤ 1 / (Fintype.card (LinearMap.range f) : ℝ) := by
by_cases hc : c = 0
· subst c
simp [cover, FiniteDistribution.probability]
· simpa [μ, cover, hc] using relation_probability_le f offsets c hc
calc
μ.probability event ≤ ∑ c, μ.probability (cover c) :=
probability_cover_le_sum μ event cover hcover
_ ≤ ∑ _c : Fin m → F2, 1 / (Fintype.card (LinearMap.range f) : ℝ) :=
Finset.sum_le_sum (fun c _ => hbound c)
_ = (2 : ℝ) ^ m / (2 : ℝ) ^ Module.finrank F2 (LinearMap.range f) := by
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
Fintype.card_fun, Fintype.card_fin,
Module.card_eq_pow_finrank (K := F2) (V := LinearMap.range f),
show Fintype.card F2 = 2 by decide, Nat.cast_pow, Nat.cast_ofNat,
mul_one_div]
end
end PerfectCompleteness.AffineFunctionalDependence
end
end OAI
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