| import Mathlib.Algebra.MvPolynomial.PDeriv |
| import Mathlib.Basic.Complex.Basic |
|
|
| namespace OAI |
|
|
| namespace PiExponentApprox |
|
|
| noncomputable section |
|
|
| open MvPolynomial |
|
|
| def monomialWeight {m : ℕ} (w : Fin (m + 1) → ℝ) |
| (d : Fin (m + 1) →₀ ℕ) : ℝ := |
| ∑ i, (d i : ℝ) * w i |
|
|
| def HasWeightedDegreeLE {m : ℕ} (w : Fin (m + 1) → ℝ) (N : ℝ) |
| (p : MvPolynomial (Fin (m + 1)) ℂ) : Prop := |
| ∀ d ∈ p.support, monomialWeight w d ≤ N |
|
|
| theorem monomialWeight_add {m : ℕ} (w : Fin (m + 1) → ℝ) |
| (d e : Fin (m + 1) →₀ ℕ) : |
| monomialWeight w (d + e) = monomialWeight w d + monomialWeight w e := by |
| simp [monomialWeight, Finsupp.add_apply, Nat.cast_add, add_mul, Finset.sum_add_distrib] |
|
|
| theorem monomialWeight_single {m : ℕ} (w : Fin (m + 1) → ℝ) |
| (i : Fin (m + 1)) : monomialWeight w (Finsupp.single i 1) = w i := by |
| classical |
| simp [monomialWeight, Finsupp.single_apply] |
|
|
| theorem HasWeightedDegreeLE.zero {m : ℕ} (w : Fin (m + 1) → ℝ) (N : ℝ) : |
| HasWeightedDegreeLE w N (0 : MvPolynomial (Fin (m + 1)) ℂ) := by |
| simp [HasWeightedDegreeLE] |
|
|
| theorem HasWeightedDegreeLE.mono {m : ℕ} {w : Fin (m + 1) → ℝ} {N M : ℝ} |
| {p : MvPolynomial (Fin (m + 1)) ℂ} (hp : HasWeightedDegreeLE w N p) |
| (hNM : N ≤ M) : HasWeightedDegreeLE w M p := |
| fun d hd => (hp d hd).trans hNM |
|
|
| theorem HasWeightedDegreeLE.add {m : ℕ} {w : Fin (m + 1) → ℝ} {N : ℝ} |
| {p q : MvPolynomial (Fin (m + 1)) ℂ} |
| (hp : HasWeightedDegreeLE w N p) (hq : HasWeightedDegreeLE w N q) : |
| HasWeightedDegreeLE w N (p + q) := by |
| intro d hd |
| rcases Finset.mem_union.mp (MvPolynomial.support_add hd) with hd | hd |
| · exact hp d hd |
| · exact hq d hd |
|
|
| theorem HasWeightedDegreeLE.sum {m : ℕ} {α : Type*} {w : Fin (m + 1) → ℝ} |
| {N : ℝ} (s : Finset α) (p : α → MvPolynomial (Fin (m + 1)) ℂ) |
| (hp : ∀ a ∈ s, HasWeightedDegreeLE w N (p a)) : |
| HasWeightedDegreeLE w N (∑ a ∈ s, p a) := by |
| classical |
| induction s using Finset.induction_on with |
| | empty => simpa using HasWeightedDegreeLE.zero w N |
| | @insert a s ha ih => |
| rw [Finset.sum_insert ha] |
| exact (hp a (Finset.mem_insert_self a s)).add |
| (ih (fun b hb => hp b (Finset.mem_insert_of_mem hb))) |
|
|
| theorem pderiv_support_shift {m : ℕ} (i : Fin (m + 1)) |
| (p : MvPolynomial (Fin (m + 1)) ℂ) (d : Fin (m + 1) →₀ ℕ) |
| (hd : d ∈ (MvPolynomial.pderiv i p).support) : d + Finsupp.single i 1 ∈ p.support := by |
| apply MvPolynomial.mem_support_iff.mpr |
| intro hz |
| have hcoeff := MvPolynomial.mem_support_iff.mp hd |
| apply hcoeff |
| rw [MvPolynomial.coeff_pderiv, hz, zero_mul] |
|
|
| theorem HasWeightedDegreeLE.pderiv {m : ℕ} {w : Fin (m + 1) → ℝ} {N : ℝ} |
| {p : MvPolynomial (Fin (m + 1)) ℂ} (hp : HasWeightedDegreeLE w N p) |
| (i : Fin (m + 1)) : HasWeightedDegreeLE w (N - w i) (MvPolynomial.pderiv i p) := by |
| intro d hd |
| apply (le_sub_iff_add_le).mpr |
| have h := hp (d + Finsupp.single i 1) (pderiv_support_shift i p d hd) |
| simpa only [monomialWeight_add, monomialWeight_single] using h |
|
|
| theorem HasWeightedDegreeLE.X_mul {m : ℕ} {w : Fin (m + 1) → ℝ} {N : ℝ} |
| {p : MvPolynomial (Fin (m + 1)) ℂ} (hp : HasWeightedDegreeLE w N p) |
| (i : Fin (m + 1)) : HasWeightedDegreeLE w (N + w i) (MvPolynomial.X i * p) := by |
| intro d hd |
| have hsupp := MvPolynomial.support_X_mul i p |
| rw [hsupp] at hd |
| obtain ⟨e, he, hed⟩ := Finset.mem_map.mp hd |
| have heq : Finsupp.single i 1 + e = d := hed |
| rw [← heq, monomialWeight_add, monomialWeight_single, add_comm (w i)] |
| exact add_le_add (hp e he) (le_refl (w i)) |
|
|
| theorem HasWeightedDegreeLE.X_mul_pderiv {m : ℕ} {w : Fin (m + 1) → ℝ} {N : ℝ} |
| {p : MvPolynomial (Fin (m + 1)) ℂ} (hp : HasWeightedDegreeLE w N p) |
| (i : Fin (m + 1)) : HasWeightedDegreeLE w N (MvPolynomial.X i * MvPolynomial.pderiv i p) := by |
| simpa only [sub_add_cancel] using (hp.pderiv i).X_mul i |
|
|
| theorem HasWeightedDegreeLE.pderiv_preserves {m : ℕ} {w : Fin (m + 1) → ℝ} |
| {N : ℝ} {p : MvPolynomial (Fin (m + 1)) ℂ} |
| (hp : HasWeightedDegreeLE w N p) (i : Fin (m + 1)) (hi : 0 ≤ w i) : |
| HasWeightedDegreeLE w N (MvPolynomial.pderiv i p) := |
| (hp.pderiv i).mono (sub_le_self N hi) |
|
|
| end |
|
|
| end PiExponentApprox |
|
|
| end OAI |
|
|
