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urkud committed Jan 3, 2025
1 parent 7be0b06 commit c812665
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2 changes: 1 addition & 1 deletion Mathlib/Analysis/Analytic/IteratedFDeriv.lean
Original file line number Diff line number Diff line change
Expand Up @@ -196,7 +196,7 @@ theorem HasFPowerSeriesWithinOnBall.iteratedFDerivWithin_eq_sum
apply HasFPowerSeriesWithinOnBall.iteratedFDerivWithin_eq_sum_of_subset
· exact h.mono inter_subset_left
· exact h'.mono inter_subset_left
· exact hs.inter EMetric.isOpen_ball
· exact hs.inter_isOpen EMetric.isOpen_ball
· exact ⟨hx, EMetric.mem_ball_self h.r_pos⟩
· exact inter_subset_right

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5 changes: 2 additions & 3 deletions Mathlib/Analysis/Calculus/Taylor.lean
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Expand Up @@ -305,8 +305,7 @@ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (h
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (mod_cast n.lt_succ_self)
(uniqueDiffOn_Icc h)
hf.differentiableOn_iteratedDerivWithin (mod_cast n.lt_succ_self) (.Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : ∀ y ∈ Ico a x,
‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤
Expand Down Expand Up @@ -350,4 +349,4 @@ theorem exists_taylor_mean_remainder_bound {f : ℝ → E} {a b : ℝ} {n : ℕ}
intro x hx
rw [div_mul_eq_mul_div₀]
refine taylor_mean_remainder_bound hab hf hx fun y => ?_
exact (hf.continuousOn_iteratedDerivWithin rfl.le <| uniqueDiffOn_Icc h).norm.le_sSup_image_Icc
exact (hf.continuousOn_iteratedDerivWithin le_rfl <| .Icc h).norm.le_sSup_image_Icc

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