| 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 |
|
|
