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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 95 96 97 98 99 100 | import Mathlib.LinearAlgebra.Lagrange
import OAI.Computability.DepthThree.AlgebraFiniteFibers
namespace OAI
universe uDepth1 uDepth2
noncomputable section
open scoped BigOperators Polynomial
namespace DepthThreeLowerBound
namespace Interpolation
variable {F : Type uDepth1} {ι : Type uDepth2}
def coeffEval [CommSemiring F] (t : ℕ) (a : ι → F) :
(Fin t → F) →ₗ[F] (ι → F) where
toFun β i := ∑ j : Fin t, β j * a i ^ (j : ℕ)
map_add' β γ := by
funext i
simp only [Pi.add_apply, add_mul, Finset.sum_add_distrib]
map_smul' c β := by
funext i
simp only [Pi.smul_apply, smul_eq_mul, Finset.mul_sum, mul_assoc,
RingHom.id_apply]
@[simp] theorem coeffEval_apply [CommSemiring F] (t : ℕ) (a : ι → F)
(β : Fin t → F) (i : ι) :
coeffEval t a β i = ∑ j : Fin t, β j * a i ^ (j : ℕ) := rfl
theorem coeffEval_surjective [Field F] [Fintype ι] {t : ℕ} (a : ι → F)
(hsize : Fintype.card ι ≤ t) (ha : Function.Injective a) :
Function.Surjective (coeffEval t a) := by
classical
intro v
let p : F[X] := Lagrange.interpolate Finset.univ a v
have hinj : Set.InjOn a (↑(Finset.univ : Finset ι) : Set ι) :=
fun _ _ _ _ hij => ha hij
have hp : p ∈ Polynomial.degreeLT F (Fintype.card ι) := by
apply Polynomial.mem_degreeLT.mpr
simpa only [Finset.card_univ] using Lagrange.degree_interpolate_lt v hinj
have hpt : p ∈ Polynomial.degreeLT F t := Polynomial.degreeLT_mono hsize hp
refine ⟨Polynomial.degreeLTEquiv F t ⟨p, hpt⟩, ?_⟩
funext i
change (∑ j : Fin t,
Polynomial.degreeLTEquiv F t ⟨p, hpt⟩ j * a i ^ (j : ℕ)) = v i
rw [← Polynomial.eval_eq_sum_degreeLTEquiv hpt]
exact Lagrange.eval_interpolate_at_node v hinj (Finset.mem_univ i)
section FiniteField
variable [Field F] [Fintype F] [DecidableEq F] [Fintype ι] [DecidableEq ι]
variable {t : ℕ} (a : ι → F)
variable (hsize : Fintype.card ι ≤ t) (ha : Function.Injective a)
include hsize ha
theorem card_coeffEval_fiber_mul (v : ι → F) :
Fintype.card F ^ Fintype.card ι *
Fintype.card {β : Fin t → F // coeffEval t a β = v} =
Fintype.card F ^ t := by
classical
let L := (coeffEval t a).toAddMonoidHom
have hL : Function.Surjective L := coeffEval_surjective a hsize ha
have hcard := AlgebraFiniteFibers.card_eq_mul_card_kernel L hL
rw [← AlgebraFiniteFibers.card_fiber_eq L hL v] at hcard
simp only [Fintype.card_fun, Fintype.card_fin] at hcard
simp only [Fintype.card_eq_nat_card] at hcard ⊢
simpa only [L, LinearMap.toAddMonoidHom_coe] using hcard.symm
theorem card_coeffEval_fiber (v : ι → F) :
Fintype.card {β : Fin t → F // coeffEval t a β = v} =
Fintype.card F ^ (t - Fintype.card ι) := by
apply Nat.eq_of_mul_eq_mul_left
(pow_pos (Fintype.card_pos (α := F)) (Fintype.card ι))
calc
Fintype.card F ^ Fintype.card ι *
Fintype.card {β : Fin t → F // coeffEval t a β = v} =
Fintype.card F ^ t := card_coeffEval_fiber_mul a hsize ha v
_ = Fintype.card F ^ (Fintype.card ι + (t - Fintype.card ι)) := by
rw [Nat.add_sub_of_le hsize]
_ = Fintype.card F ^ Fintype.card ι *
Fintype.card F ^ (t - Fintype.card ι) := pow_add _ _ _
theorem finiteAvg_coeffEval (Φ : (ι → F) → ℝ) :
finiteAvg (fun β : Fin t → F => Φ (coeffEval t a β)) = finiteAvg Φ := by
classical
exact AlgebraFiniteFibers.finiteAvg_comp (coeffEval t a).toAddMonoidHom
(coeffEval_surjective a hsize ha) Φ
end FiniteField
end Interpolation
end DepthThreeLowerBound
end
end OAI
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