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59d1c80 | 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 | import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
import Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Tactic.FunProp
namespace OAI
noncomputable section
open Function Set
open scoped ContDiff Topology BigOperators
namespace SmoothLocal.Taylor
def normalizedTimePower (a : ℝ) (m : ℕ) (t : ℝ) : ℝ :=
(t - a)^m / (m.factorial : ℝ)
theorem normalizedTimePower_contDiff (a : ℝ) (m : ℕ) :
ContDiff ℝ ∞ (normalizedTimePower a m) := by
unfold normalizedTimePower
fun_prop
theorem normalizedTimePower_hasDerivAt (a t : ℝ) (m : ℕ) :
HasDerivAt (normalizedTimePower a (m + 1)) (normalizedTimePower a m t) t := by
have h := (((hasDerivAt_id t).sub_const a).pow (m + 1)).div_const ((m + 1).factorial : ℝ)
change HasDerivAt (normalizedTimePower a (m + 1))
((m + 1 : ℕ) * (t - a)^(m + 1 - 1) * 1 / ((m + 1).factorial : ℝ)) t at h
simp only [Nat.add_sub_cancel, mul_one] at h
have he : (m + 1 : ℕ) * (t - a)^m / ((m + 1).factorial : ℝ) = normalizedTimePower a m t := by
simp only [Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, normalizedTimePower]
have hm : (m.factorial : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero m)
have hm1 : (m : ℝ) + 1 ≠ 0 := by positivity
field_simp [hm, hm1]
rw [he] at h
exact h
theorem normalizedTimePower_deriv (a : ℝ) (m : ℕ) :
deriv (normalizedTimePower a (m + 1)) = normalizedTimePower a m := by
funext t
exact (normalizedTimePower_hasDerivAt a t m).deriv
theorem normalizedTimePower_iteratedDeriv_at_center (a : ℝ) (m k : ℕ) :
iteratedDeriv k (normalizedTimePower a m) a = if k = m then 1 else 0 := by
change iteratedDeriv k (fun t : ℝ => (t - a)^m / (m.factorial : ℝ)) a = _
rw [iteratedDeriv_div_const]
rw [iteratedDeriv_comp_sub_const (f := fun t : ℝ => t^m)]
change iteratedDeriv k (fun t : ℝ => t^m) (a - a) / (m.factorial : ℝ) = _
rw [sub_self, iteratedDeriv_fun_pow_zero]
split_ifs with h
· exact div_self (Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero m) : (m.factorial : ℝ) ≠ 0)
· simpa only [Nat.cast_zero] using (zero_div (m.factorial : ℝ))
theorem coefficient_timePower_jet (a C : ℝ) (m k : ℕ) :
iteratedDeriv k (fun t => C * normalizedTimePower a m t) a =
if k = m then C else 0 := by
rw [iteratedDeriv_const_mul_field, normalizedTimePower_iteratedDeriv_at_center]
split_ifs <;> simp
section Composition
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
theorem iteratedDeriv_comp_eq_of_jets {Q : E → F} {f g : ℝ → E} {a : ℝ} {n : ℕ}
(hf : ContDiffAt ℝ ∞ f a) (hg : ContDiffAt ℝ ∞ g a)
(hQ : ContDiffAt ℝ ∞ Q (f a))
(hjets : ∀ j ≤ n, iteratedDeriv j f a = iteratedDeriv j g a) :
iteratedDeriv n (Q ∘ f) a = iteratedDeriv n (Q ∘ g) a := by
have hvalue : f a = g a := by simpa only [iteratedDeriv_zero] using hjets 0 (Nat.zero_le n)
have hQg : ContDiffAt ℝ ∞ Q (g a) := by rw [← hvalue]; exact hQ
rw [iteratedDeriv_vcomp_eq_sum_orderedFinpartition hQ hf (WithTop.coe_le_coe.mpr le_top),
iteratedDeriv_vcomp_eq_sum_orderedFinpartition hQg hg (WithTop.coe_le_coe.mpr le_top)]
apply Finset.sum_congr rfl
intro c hc
rw [hvalue]
congr 1
funext j
exact hjets (c.partSize j) (c.partSize_le j)
theorem iteratedDeriv_add_eq_of_zero_jets {f h : ℝ → E} {a : ℝ} {n : ℕ}
(hf : ContDiffAt ℝ ∞ f a) (hh : ContDiffAt ℝ ∞ h a)
(hzero : ∀ j ≤ n, iteratedDeriv j h a = 0) :
∀ j ≤ n, iteratedDeriv j (fun t => f t + h t) a = iteratedDeriv j f a := by
intro j hj
rw [iteratedDeriv_fun_add (hf.of_le (WithTop.coe_le_coe.mpr le_top))
(hh.of_le (WithTop.coe_le_coe.mpr le_top)), hzero j hj, add_zero]
end Composition
end SmoothLocal.Taylor
end
end OAI
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