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b797a5b | 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 | import Mathlib.Algebra.Polynomial.Degree.SmallDegree
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.Tactic
namespace OAI
namespace PiExponentJets.W27
open Polynomial
theorem difference_taylor_coefficient (p : Polynomial ℚ) (s : ℕ)
(hp : p.natDegree = s+1) (a : ℚ) :
(p - Polynomial.taylor a p).coeff s = -a * (s+1 : ℚ) * p.leadingCoeff := by
have hd : (Polynomial.hasseDeriv s p).natDegree ≤ 1 := by
simpa [hp] using Polynomial.natDegree_hasseDeriv_le p s
have hlin := Polynomial.eq_X_add_C_of_natDegree_le_one hd
have h0 : (Polynomial.hasseDeriv s p).coeff 0 = p.coeff s := by
simp [Polynomial.hasseDeriv_coeff]
have h1 : (Polynomial.hasseDeriv s p).coeff 1 = (s+1 : ℚ) * p.leadingCoeff := by
rw [Polynomial.hasseDeriv_coeff]
simp only [Nat.add_comm 1 s, Nat.choose_succ_self_right, Nat.cast_add, Nat.cast_one]
rw [← hp, Polynomial.coeff_natDegree]
rw [Polynomial.coeff_sub, Polynomial.taylor_coeff, hlin]
simp only [Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X]
rw [h0, h1]
ring
theorem difference_taylor_natDegree_le (p : Polynomial ℚ) (s : ℕ)
(hp : p.natDegree = s+1) (a : ℚ) :
(p - Polynomial.taylor a p).natDegree ≤ s := by
apply Polynomial.natDegree_le_iff_coeff_eq_zero.mpr
intro n hn
rw [Polynomial.coeff_sub]
by_cases he : n = s+1
· subst n
rw [← hp, Polynomial.coeff_natDegree, Polynomial.coeff_taylor_natDegree, sub_self]
· have hlt : p.natDegree < n := by omega
rw [Polynomial.coeff_eq_zero_of_natDegree_lt hlt,
Polynomial.coeff_eq_zero_of_natDegree_lt (by simpa using hlt), sub_self]
theorem backward_difference_degree_and_leading
(p : Polynomial ℚ) (s : ℕ) (hp : p.natDegree = s+1)
(d : ℚ) (hd : d ≠ 0) :
(p - Polynomial.taylor (-d) p).natDegree = s ∧
(p - Polynomial.taylor (-d) p).leadingCoeff = d * (s+1 : ℚ) * p.leadingCoeff := by
have hp0 : p ≠ 0 := by
intro hz
simp [hz] at hp
have hs : (s+1 : ℚ) ≠ 0 := by positivity
have hc : (p - Polynomial.taylor (-d) p).coeff s =
d * (s+1 : ℚ) * p.leadingCoeff := by
simpa using difference_taylor_coefficient p s hp (-d)
have hcn : (p - Polynomial.taylor (-d) p).coeff s ≠ 0 := by
rw [hc]
exact mul_ne_zero (mul_ne_zero hd hs) (Polynomial.leadingCoeff_ne_zero.mpr hp0)
have hdeg := Polynomial.natDegree_eq_of_le_of_coeff_ne_zero
(difference_taylor_natDegree_le p s hp (-d)) hcn
refine ⟨hdeg, ?_⟩
simpa only [Polynomial.leadingCoeff, hdeg] using hc
theorem backward_difference_multiplicity
(p : Polynomial ℚ) (s : ℕ) (hp : p.natDegree = s+1)
(d : ℚ) (hd : d ≠ 0) :
(p - Polynomial.taylor (-d) p).leadingCoeff *
((p - Polynomial.taylor (-d) p).natDegree.factorial : ℚ) =
d * (p.leadingCoeff * (p.natDegree.factorial : ℚ)) := by
obtain ⟨hdeg, hl⟩ := backward_difference_degree_and_leading p s hp d hd
rw [hl, hdeg, hp, Nat.factorial_succ]
push_cast
ring
theorem backward_difference_eq_comp (p : Polynomial ℚ) (d : ℚ) :
p - Polynomial.taylor (-d) p = p - p.comp (Polynomial.X - Polynomial.C d) := by
simp only [Polynomial.taylor_apply, Polynomial.C_neg, sub_eq_add_neg]
end PiExponentJets.W27
end OAI
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