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[Merged by Bors] - feat: add HasFTaylorSeriesUpToOn.congr_series #19722

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[Merged by Bors] - feat: add HasFTaylorSeriesUpToOn.congr_series #19722

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4a5e66f
feat: add `HasFTaylorSeriesUpToOn.congr_series`
urkud Dec 4, 2024
426017c
Golf as suggested
urkud Dec 5, 2024
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10 changes: 10 additions & 0 deletions Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean
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Expand Up @@ -146,6 +146,16 @@ theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
rw [h₁ x hx]
exact h.zero_eq x hx

theorem HasFTaylorSeriesUpToOn.congr_series {q} (hp : HasFTaylorSeriesUpToOn n f p s)
(hpq : ∀ m : ℕ, m ≤ n → EqOn (p · m) (q · m) s) :
HasFTaylorSeriesUpToOn n f q s where
zero_eq x hx := by simp only [← (hpq 0 (zero_le n) hx), hp.zero_eq x hx]
fderivWithin m hm x hx := by
refine ((hp.fderivWithin m hm x hx).congr' (hpq m hm.le).symm hx).congr_fderiv ?_
refine congrArg _ (hpq (m + 1) ?_ hx)
exact ENat.add_one_natCast_le_withTop_of_lt hm
cont m hm := (hp.cont m hm).congr (hpq m hm).symm

theorem HasFTaylorSeriesUpToOn.mono (h : HasFTaylorSeriesUpToOn n f p s) {t : Set E} (hst : t ⊆ s) :
HasFTaylorSeriesUpToOn n f p t :=
⟨fun x hx => h.zero_eq x (hst hx), fun m hm x hx => (h.fderivWithin m hm x (hst hx)).mono hst,
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