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ea76aac | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | 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
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