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import OAI.Computability.DepthThree.InterpolationEvaluation
namespace OAI
universe uDepth1 uDepth2 uDepth3 uDepth4
noncomputable section
open scoped BigOperators
namespace DepthThreeLowerBound
namespace Interpolation
open BinaryAlgebra
variable {F : Type uDepth1} {ι : Type uDepth2}
variable [Field F] [Module F2 F]
def bitEval (ell : F →ₗ[F2] F2) (t : ℕ) (a : ι → F) :
(Fin t → F) →+ (ι → F2) where
toFun β i := ell (coeffEval t a β i)
map_zero' := by
funext i
simp only [map_zero, Pi.zero_apply]
map_add' β γ := by
funext i
simp only [map_add, Pi.add_apply]
@[simp] theorem bitEval_apply (ell : F →ₗ[F2] F2) (t : ℕ) (a : ι → F)
(β : Fin t → F) (i : ι) :
bitEval ell t a β i = ell (∑ j : Fin t, β j * a i ^ (j : ℕ)) := rfl
theorem functional_surjective (ell : F →ₗ[F2] F2) (hOne : ell 1 = 1) :
Function.Surjective ell := by
intro b
refine ⟨b • (1 : F), ?_⟩
simp only [map_smul, hOne, smul_eq_mul, mul_one]
theorem bitEval_surjective [Fintype ι] (ell : F →ₗ[F2] F2) (hOne : ell 1 = 1)
{t : ℕ} (a : ι → F) (hsize : Fintype.card ι ≤ t)
(ha : Function.Injective a) : Function.Surjective (bitEval ell t a) := by
intro b
obtain ⟨β, hβ⟩ := coeffEval_surjective a hsize ha (fun i => b i • (1 : F))
refine ⟨β, ?_⟩
funext i
change ell (coeffEval t a β i) = b i
rw [hβ]
simp only [map_smul, hOne, smul_eq_mul, mul_one]
def coefficientBits (ell : F →ₗ[F2] F2) (t : ℕ) (a : ι → F)
(β : Fin t → F) : ι → Bool :=
fun i => fieldBit (bitEval ell t a β i)
def acceptEval (ell : F →ₗ[F2] F2) (t : ℕ) (a : ι → F)
(β : Fin t → F) : ι → Bool :=
fun i => !(coefficientBits ell t a β i)
@[simp] theorem acceptEval_eq_true (ell : F →ₗ[F2] F2) (t : ℕ) (a : ι → F)
(β : Fin t → F) (i : ι) :
acceptEval ell t a β i = true ↔
ell (∑ j : Fin t, β j * a i ^ (j : ℕ)) = 0 := by
simp [acceptEval, coefficientBits]
def complementEquiv (ι : Type uDepth3) : (ι → Bool) ≃ (ι → Bool) where
toFun b i := !(b i)
invFun b i := !(b i)
left_inv b := by funext i; exact Bool.not_not (b i)
right_inv b := by funext i; exact Bool.not_not (b i)
section Finite
variable [Fintype F] [Fintype ι] [DecidableEq ι]
variable (ell : F →ₗ[F2] F2) (hOne : ell 1 = 1)
variable {t : ℕ} (a : ι → F) (hsize : Fintype.card ι ≤ t)
variable (ha : Function.Injective a)
include hOne hsize ha
theorem finiteAvg_bitEval (Φ : (ι → F2) → ℝ) :
finiteAvg (fun β : Fin t → F => Φ (bitEval ell t a β)) = finiteAvg Φ := by
classical
exact AlgebraFiniteFibers.finiteAvg_comp (bitEval ell t a)
(bitEval_surjective ell hOne a hsize ha) Φ
theorem finiteAvg_coefficientBits (Φ : (ι → Bool) → ℝ) :
finiteAvg (fun β : Fin t → F => Φ (coefficientBits ell t a β)) =
finiteAvg Φ := by
calc
finiteAvg (fun β : Fin t → F => Φ (coefficientBits ell t a β)) =
finiteAvg (fun b : ι → F2 => Φ ((bitsEquivF2 ι).symm b)) :=
finiteAvg_bitEval ell hOne a hsize ha
(fun b : ι → F2 => Φ ((bitsEquivF2 ι).symm b))
_ = finiteAvg Φ := finiteAvg_equiv (bitsEquivF2 ι).symm Φ
theorem finiteAvg_acceptEval (Φ : (ι → Bool) → ℝ) :
finiteAvg (fun β : Fin t → F => Φ (acceptEval ell t a β)) =
finiteAvg Φ := by
calc
finiteAvg (fun β : Fin t → F => Φ (acceptEval ell t a β)) =
finiteAvg (fun b : ι → Bool => Φ (complementEquiv ι b)) :=
finiteAvg_coefficientBits ell hOne a hsize ha
(fun b : ι → Bool => Φ (complementEquiv ι b))
_ = finiteAvg Φ := finiteAvg_equiv (complementEquiv ι) Φ
end Finite
section FiberCounts
variable [Fintype F] [DecidableEq F] [Fintype ι]
variable (ell : F →ₗ[F2] F2) (hOne : ell 1 = 1)
variable {t : ℕ} (a : ι → F) (hsize : Fintype.card ι ≤ t)
variable (ha : Function.Injective a)
include hOne hsize ha
theorem card_bitEval_fiber_mul (b : ι → F2) :
2 ^ Fintype.card ι *
Fintype.card {β : Fin t → F // bitEval ell t a β = b} =
Fintype.card F ^ t := by
classical
have hL := bitEval_surjective ell hOne a hsize ha
have hcard := AlgebraFiniteFibers.card_eq_mul_card_kernel (bitEval ell t a) hL
rw [← AlgebraFiniteFibers.card_fiber_eq (bitEval ell t a) hL b] at hcard
simpa only [Fintype.card_fun, Fintype.card_fin, ZMod.card] using hcard.symm
theorem card_bitEval_fiber {r : ℕ} (hr : 1 ≤ r)
(hcard : Fintype.card F = 2 ^ r) (b : ι → F2) :
Fintype.card {β : Fin t → F // bitEval ell t a β = b} =
2 ^ (r * t - Fintype.card ι) := by
have hn : Fintype.card ι ≤ r * t := calc
Fintype.card ι ≤ t := hsize
_ = 1 * t := (one_mul t).symm
_ ≤ r * t := Nat.mul_le_mul_right t hr
apply Nat.eq_of_mul_eq_mul_left (pow_pos (by decide : 0 < (2 : ℕ)) _)
calc
2 ^ Fintype.card ι *
Fintype.card {β : Fin t → F // bitEval ell t a β = b} =
Fintype.card F ^ t := card_bitEval_fiber_mul ell hOne a hsize ha b
_ = 2 ^ (r * t) := by rw [hcard, pow_mul]
_ = 2 ^ (Fintype.card ι + (r * t - Fintype.card ι)) := by
rw [Nat.add_sub_of_le hn]
_ = 2 ^ Fintype.card ι * 2 ^ (r * t - Fintype.card ι) := pow_add _ _ _
end FiberCounts
theorem finiteAvg_acceptEval_finset [Fintype F] {Z : Type uDepth4} [DecidableEq Z]
(ell : F →ₗ[F2] F2) (hOne : ell 1 = 1) {t : ℕ} (a : Z → F)
(s : Finset Z) (hsize : s.card ≤ t)
(ha : Set.InjOn a (↑s : Set Z)) (Φ : (s → Bool) → ℝ) :
finiteAvg (fun β : Fin t → F => Φ (fun z : s => acceptEval ell t a β z)) =
finiteAvg Φ := by
classical
apply finiteAvg_acceptEval ell hOne (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
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