| import Mathlib |
| import OAI.Computability.DeterministicSum.Interpolation |
|
|
| namespace OAI |
|
|
| /-! Individual-degree bounds and Newton polynomial conversion. -/ |
|
|
| namespace DeterministicThreeSum.Newton |
| open Finset |
| open scoped BigOperators |
| noncomputable section |
| variable {R : Type*} [CommRing R] {q d : ℕ} |
|
|
| noncomputable def univariate (node : Fin q → R) (j : Fin q) : Polynomial R := |
| ∏ a ∈ Iio j, (Polynomial.X-Polynomial.C (node a)) |
|
|
| lemma univariate_degree (node : Fin q → R) (j : Fin q) : |
| (univariate node j).natDegree ≤ j.val := by |
| classical |
| rcases subsingleton_or_nontrivial R with hR | hR |
| · have := hR |
| have hz : univariate node j = 0 := Subsingleton.elim _ _ |
| simp [hz] |
| have := hR |
| apply (Polynomial.natDegree_prod_le _ _).trans |
| simp |
|
|
| noncomputable def conversion (node : Fin q → R) (j v : Fin q) : R := |
| (univariate node j).coeff v.val |
|
|
| lemma conversion_zero (node : Fin q → R) (j v : Fin q) (hjv : j < v) : |
| conversion node j v = 0 := |
| Polynomial.coeff_eq_zero_of_natDegree_lt ((univariate_degree node j).trans_lt hjv) |
|
|
| lemma univariate_factor (node : Fin q → R) (j : Fin q) (k : Fin d) : |
| (∏ a ∈ Iio j, (MvPolynomial.X k-MvPolynomial.C (node a))) = |
| ∑ v : Fin q, MvPolynomial.C (conversion node j v)*MvPolynomial.X k^v.val := by |
| classical |
| have hpoly := Polynomial.as_sum_range' (univariate node j) q |
| ((univariate_degree node j).trans_lt j.isLt) |
| have hm := congrArg (Polynomial.toMvPolynomial (R:=R) k) hpoly |
| simpa only [univariate,map_prod,map_sub,Polynomial.toMvPolynomial_X, |
| Polynomial.toMvPolynomial_C,map_sum,← Polynomial.C_mul_X_pow_eq_monomial,map_mul,map_pow, |
| ← Fin.sum_univ_eq_sum_range,conversion] using hm |
| |
| lemma basis_expansion (node : Fin q → R) (j : Fin d → Fin q) : |
| basisPolynomial node j = ∑ v : Fin d → Fin q, |
| MvPolynomial.C (∏ i, conversion node (j i) (v i)) * |
| ∏ i : Fin d, MvPolynomial.X i^(v i).val := by |
| classical |
| unfold basisPolynomial |
| simp_rw [univariate_factor] |
| rw [Fintype.prod_sum (fun i : Fin d => fun v : Fin q => |
| MvPolynomial.C (conversion node (j i) v)*MvPolynomial.X i^v.val)] |
| apply sum_congr rfl |
| intro v hv |
| rw [prod_mul_distrib,← map_prod] |
| |
| noncomputable def exponentVector (v : Fin d → Fin q) : Fin d →₀ ℕ := |
| Finsupp.equivFunOnFinite.symm (fun i => (v i).val) |
| @[simp] lemma exponentVector_apply (v : Fin d → Fin q) (i : Fin d) : |
| exponentVector v i = (v i).val := by simp [exponentVector] |
| |
| lemma indicator_exponentVector (v : Fin d → Fin q) : |
| Finsupp.indicator univ (fun i _ => (v i).val) = exponentVector v := by |
| ext i |
| simp |
| |
| lemma exponentVector_injective : Function.Injective (exponentVector (d:=d) (q:=q)) := by |
| intro v w h |
| funext i |
| apply Fin.ext |
| simpa using congrArg (fun z : Fin d →₀ ℕ => z i) h |
| |
| lemma coefficient_basis (node : Fin q → R) (j v : Fin d → Fin q) : |
| (basisPolynomial node j).coeff (exponentVector v) = |
| ∏ i, conversion node (j i) (v i) := by |
| classical |
| rw [basis_expansion,MvPolynomial.coeff_sum] |
| simp only [MvPolynomial.coeff_C_mul,MvPolynomial.coeff_prod_X_pow, |
| indicator_exponentVector,exponentVector_injective.eq_iff] |
| simp |
| |
| lemma coefficient_truncated (node : Fin q → R) (f : (Fin d → Fin q) → R) |
| (D : ℕ) (v : Fin d → Fin q) : |
| (truncatedPolynomial node f D).coeff (exponentVector v) = |
| ∑ j : Fin d → Fin q, (if degree j ≤ D then coefficients node f j else 0)* |
| ∏ i, conversion node (j i) (v i) := by |
| classical |
| simp only [truncatedPolynomial,MvPolynomial.coeff_sum,sum_filter] |
| apply sum_congr rfl |
| intro j hj |
| split_ifs <;> simp_all [coefficient_basis] |
| |
| lemma basis_individual_degree (node : Fin q → R) (j : Fin d → Fin q) (k : Fin d) : |
| (basisPolynomial node j).degreeOf k ≤ (j k).val := by |
| classical |
| rcases subsingleton_or_nontrivial R with hR | hR |
| · have := hR |
| have hz : basisPolynomial node j = 0 := Subsingleton.elim _ _ |
| simp [hz] |
| have := hR |
| unfold basisPolynomial |
| apply (MvPolynomial.degreeOf_prod_le k _ _).trans |
| calc |
| (∑ i : Fin d, (∏ a ∈ Iio (j i), (MvPolynomial.X i-MvPolynomial.C (node a))).degreeOf k) ≤ |
| ∑ i : Fin d, if i=k then (j i).val else 0 := by |
| apply sum_le_sum |
| intro i hi |
| apply (MvPolynomial.degreeOf_prod_le k _ _).trans |
| calc |
| _ ≤ ∑ _a ∈ Iio (j i), if k=i then 1 else 0 := sum_le_sum (by |
| intro a ha |
| exact (MvPolynomial.degreeOf_sub_le k _ _).trans (by simp [MvPolynomial.degreeOf_X])) |
| _ = _ := by by_cases hik : i=k <;> simp [hik,Ne.symm] |
| _ = _ := by simp |
| |
| lemma truncated_individual_degree (node : Fin q → R) (f : (Fin d → Fin q) → R) |
| (D : ℕ) (k : Fin d) (hq : 0 < q) : |
| (truncatedPolynomial node f D).degreeOf k < q := by |
| classical |
| apply (MvPolynomial.degreeOf_sum_le k _ _).trans_lt |
| apply Finset.sup_lt_iff hq |>.mpr |
| intro j hj |
| exact (MvPolynomial.degreeOf_C_mul_le _ _ _).trans_lt |
| ((basis_individual_degree node j k).trans_lt (j k).isLt) |
| end |
| end DeterministicThreeSum.Newton |
| |
| end OAI |
| |
