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16a4018 | 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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators
namespace BinaryCoordinateSweeps
/-- The positions of a binary deck of dimension d. -/
abbrev Slot (d : ℕ) := Fin d → Bool
/-- Independent switches on the edges parallel to one coordinate. -/
def coordinateLayer (d : ℕ) (j : Fin d)
(c : (({i : Fin d // i ≠ j} → Bool)) → Bool) : Equiv.Perm (Slot d) :=
let e := Equiv.piSplitAt j (fun _ : Fin d => Bool)
let sw : Equiv.Perm (Bool × ({i : Fin d // i ≠ j} → Bool)) :=
{ toFun := fun x => (x.1 ^^ c x.2, x.2)
invFun := fun x => (x.1 ^^ c x.2, x.2)
left_inv := fun x => by simp
right_inv := fun x => by simp }
e.trans (sw.trans e.symm)
abbrev SweepCoins (d : ℕ) :=
(j : Fin d) → ({i : Fin d // i ≠ j} → Bool) → Bool
/-- The coordinate layers are applied in increasing coordinate order. -/
def binarySweep (d : ℕ) (c : SweepCoins d) : Equiv.Perm (Slot d) :=
(List.ofFn (fun j => coordinateLayer d j (c j))).reverse.prod
def finiteLaw {Ω G : Type*} [Fintype Ω] [Fintype G] (f : Ω → G) (g : G) : ℝ := by
classical
exact ∑ ω, if f ω = g then (Fintype.card Ω : ℝ)⁻¹ else 0
/-- Half the unnormalized sum of absolute probability-mass differences. -/
def totalVariation {G : Type*} [Fintype G] (p q : G → ℝ) : ℝ :=
(1 / ((2 : ℕ) : ℝ)) * ∑ g, |p g - q g|
def uniformLaw (G : Type*) [Fintype G] : G → ℝ :=
fun _ => (Fintype.card G : ℝ)⁻¹
def binaryLaw (d : ℕ) : Equiv.Perm (Slot d) → ℝ :=
finiteLaw (binarySweep d)
abbrev RepSpace (D : ℕ) := EuclideanSpace ℂ (Fin D)
def IsUnitaryRep {G : Type*} [Monoid G]
{D : ℕ} (ρ : Representation ℂ G (RepSpace D)) : Prop :=
∀ g x, ‖ρ g x‖ = ‖x‖
def averageOperator {G : Type*} [Fintype G] [Monoid G] {D : ℕ}
(p : G → ℝ) (ρ : Representation ℂ G (RepSpace D)) :
RepSpace D →L[ℂ] RepSpace D :=
LinearMap.toContinuousLinearMap (∑ g, (p g : ℂ) • ρ g)
def BinaryContractionTarget : Prop :=
∃ g : ℝ, 0 < g ∧ ∃ d₀ : ℕ, ∀ d ≥ d₀, ∀ D : ℕ,
∀ ρ : Representation ℂ (Equiv.Perm (Slot d)) (RepSpace D),
ρ.IsIrreducible → IsUnitaryRep ρ →
‖averageOperator (binaryLaw d) ρ‖ ≤ (D : ℝ) ^ (-g)
def realSign {α : Type*} [Fintype α] [DecidableEq α] : Equiv.Perm α →* ℝ :=
(Int.castRingHom ℝ).toMonoidHom.comp ((Units.coeHom ℤ).comp Equiv.Perm.sign)
section FiniteLaws
variable {G : Type*} [Fintype G] [Group G]
def convolution (p q : G → ℝ) (g : G) : ℝ := ∑ x, p x * q (x⁻¹ * g)
def pointMassOne (g : G) : ℝ := by
classical
exact if g = 1 then 1 else 0
/-- The law of independent repetitions, with the empty product at the identity. -/
def convolutionPower (p : G → ℝ) : ℕ → G → ℝ
| 0 => pointMassOne
| n + 1 => convolution p (convolutionPower p n)
end FiniteLaws
def sweepLaw (d t : ℕ) : Equiv.Perm (Slot d) → ℝ :=
convolutionPower (binaryLaw d) t
/-- A single number of sweeps works uniformly over deterministic initial decks. -/
def UniformSweepMixingTarget : Prop :=
∃ w : ℕ, ∀ ε : ℝ, 0 < ε → ∃ d₀ : ℕ, ∀ d ≥ d₀,
∀ τ : Equiv.Perm (Slot d),
totalVariation (fun g => sweepLaw d w (g * τ⁻¹))
(uniformLaw (Equiv.Perm (Slot d))) ≤ ε
end BinaryCoordinateSweeps
end
open scoped BigOperators
theorem binary_sweep_contraction_and_mixing :
BinaryCoordinateSweeps.BinaryContractionTarget ∧
(∀ d : ℕ, 0 < d → ∑ g : Equiv.Perm (BinaryCoordinateSweeps.Slot d),
BinaryCoordinateSweeps.binaryLaw d g * BinaryCoordinateSweeps.realSign g = 0) ∧
BinaryCoordinateSweeps.UniformSweepMixingTarget := by
sorry
end OAI
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