File size: 3,733 Bytes
9dbc64e | 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 | import OAI.Computability.DepthThree.AlgebraQuotient
import OAI.Computability.DepthThree.InterpolationSigns
namespace OAI
universe uDepth1 uDepth2
noncomputable section
namespace DepthThreeLowerBound
namespace Interpolation
open BinaryAlgebra
def coefficientEncoding {r : ℕ} (p : Fin r → F2) (t : ℕ) :
(Fin t → Fin r → Bool) ≃ (Fin t → InputQuotient p) :=
Equiv.piCongrRight (fun _ =>
(bitsEquivF2 (Fin r)).trans (coordinates p).toEquiv.symm)
@[simp] theorem coefficientEncoding_apply {r t : ℕ} (p : Fin r → F2)
(β : Fin t → Fin r → Bool) (j : Fin t) :
coefficientEncoding p t β j = encode p (fun i => bitValue (β j i)) := rfl
theorem finiteAvg_coefficientEncoding {r t : ℕ} (p : Fin r → F2)
(Φ : (Fin t → InputQuotient p) → ℝ) :
finiteAvg (fun β : Fin t → Fin r → Bool =>
Φ (fun j => encode p (fun i => bitValue (β j i)))) = finiteAvg Φ :=
finiteAvg_equiv (coefficientEncoding p t) Φ
theorem finiteAvg_degreeZero {r : ℕ} (p : Fin r → F2) (hr : 0 < r)
(Φ : F2 → ℝ) :
finiteAvg (fun z : InputQuotient p => Φ (degreeZero (inputPolynomial_monic p) z)) =
finiteAvg Φ := by
classical
exact AlgebraFiniteFibers.finiteAvg_comp
(degreeZero (inputPolynomial_monic p)).toAddMonoidHom
(degreeZero_surjective (inputPolynomial_monic p)
(by simpa only [inputPolynomial_natDegree] using hr)) Φ
section Irreducible
variable {r t : ℕ} (p : Fin r → F2)
variable [Fact (Irreducible (inputPolynomial p))]
variable (hr : 0 < r)
variable {ι : Type uDepth1} [Fintype ι] [DecidableEq ι]
variable (a : ι → InputQuotient p) (hsize : Fintype.card ι ≤ t)
variable (ha : Function.Injective a)
include hr hsize ha
theorem finiteAvg_quotient_acceptEval (Φ : (ι → Bool) → ℝ) :
finiteAvg (fun β : Fin t → InputQuotient p =>
Φ (acceptEval (degreeZero (inputPolynomial_monic p)) t a β)) = finiteAvg Φ := by
exact finiteAvg_acceptEval (degreeZero (inputPolynomial_monic p))
(degreeZero_one (inputPolynomial_monic p)
(by simpa only [inputPolynomial_natDegree] using hr)) a hsize ha Φ
theorem finiteAvg_encoded_acceptEval (Φ : (ι → Bool) → ℝ) :
finiteAvg (fun β : Fin t → Fin r → Bool =>
Φ (acceptEval (degreeZero (inputPolynomial_monic p)) t a
(fun j => encode p (fun i => bitValue (β j i))))) = finiteAvg Φ := by
calc
finiteAvg (fun β : Fin t → Fin r → Bool =>
Φ (acceptEval (degreeZero (inputPolynomial_monic p)) t a
(fun j => encode p (fun i => bitValue (β j i))))) =
finiteAvg (fun β : Fin t → InputQuotient p =>
Φ (acceptEval (degreeZero (inputPolynomial_monic p)) t a β)) :=
finiteAvg_coefficientEncoding p
(fun β : Fin t → InputQuotient p =>
Φ (acceptEval (degreeZero (inputPolynomial_monic p)) t a β))
_ = finiteAvg Φ := finiteAvg_quotient_acceptEval p hr a hsize ha Φ
end Irreducible
theorem finiteAvg_encoded_acceptEval_finset {r t : ℕ} (p : Fin r → F2)
[Fact (Irreducible (inputPolynomial p))] (hr : 0 < r) {Z : Type uDepth2} [DecidableEq Z]
(a : Z → InputQuotient p) (s : Finset Z) (hsize : s.card ≤ t)
(ha : Set.InjOn a (↑s : Set Z)) (Φ : (s → Bool) → ℝ) :
finiteAvg (fun β : Fin t → Fin r → Bool =>
Φ (fun z : s => acceptEval (degreeZero (inputPolynomial_monic p)) t a
(fun j => encode p (fun i => bitValue (β j i))) z)) = finiteAvg Φ := by
classical
apply finiteAvg_encoded_acceptEval p hr (fun z : s => a z)
(by simpa only [Fintype.card_coe] using hsize)
intro x y hxy
exact Subtype.ext (ha x.property y.property hxy)
end Interpolation
end DepthThreeLowerBound
end
end OAI
|