state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case e_a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (x - x₀) ^ (n + 1) • (↑(n + 1)!)⁻¹ • iteratedDerivWithin (n + 1) f s x₀ =
((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev] | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, Polyno... | Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ x : ℝ
⊢ taylorWithinEval f 0 s x₀ x = f x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | dsimp only [taylorWithinEval] | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ := by
| Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ x : ℝ
⊢ (PolynomialModule.eval x) (taylorWithin f 0 s x₀) = f x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | dsimp only [taylorWithin] | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ := by
dsimp only [taylorWithinEval]
| Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ x : ℝ
⊢ (PolynomialModule.eval x)
(∑ k in Finset.range (0 + 1),
(PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
((PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | dsimp only [taylorCoeffWithin] | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
| Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ x : ℝ
⊢ (PolynomialModule.eval x)
(∑ k in Finset.range (0 + 1),
(PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
((PolynomialModule.single ℝ k) ((↑k !)⁻¹ • iteratedDerivWi... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
| Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ : ℝ
⊢ taylorWithinEval f n s x₀ x₀ = f x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | induction' n with k hk | /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ := by
| Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK | /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ | Mathlib_Analysis_Calculus_Taylor |
case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ : ℝ
⊢ taylorWithinEval f Nat.zero s x₀ x₀ = f x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact taylor_within_zero_eval _ _ _ _ | /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ := by
induction' n with k hk
· | Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK | /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ : ℝ
k : ℕ
hk : taylorWithinEval f k s x₀ x₀ = f x₀
⊢ taylorWithinEval f (Nat.succ k) s x₀ x₀ = f x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp [hk] | /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ := by
induction' n with k hk
· exact taylor_within_zero_eval _ _ _ _
| Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK | /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | induction' n with k hk | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by
| Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ | Mathlib_Analysis_Calculus_Taylor |
case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ x : ℝ
⊢ taylorWithinEval f Nat.zero s x₀ x =
∑ k in Finset.range (Nat.zero + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by
induction' n with k hk
· | Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ x : ℝ
k : ℕ
hk : taylorWithinEval f k s x₀ x = ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀
⊢ taylorWithinEval f (Nat.succ k) s x₀ x =
∑ k in Finset... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by
induction' n with k hk
· simp
| Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
s : Set ℝ
x₀ x : ℝ
k : ℕ
hk : taylorWithinEval f k s x₀ x = ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀
⊢ ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • i... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp [Nat.factorial] | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by
induction' n with k hk
· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
| Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
hf : ContDiffOn ℝ (↑n) f s
⊢ ContinuousOn (fun t => taylorWithinEval f n s t x) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp_rw [taylor_within_apply] | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
hf : ContDiffOn ℝ (↑n) f s
⊢ ContinuousOn (fun t => ∑ k in Finset.range (n + 1), ((↑k !)⁻¹ * (x - t) ^ k) • iteratedDerivWithin k f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
hf : ContDiffOn ℝ (↑n) f s
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ ContinuousOn (fun t => ((↑i !)⁻¹ * (x - t) ^ i) • iteratedDerivWithin i f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
hf : ContDiffOn ℝ (↑n) f s
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ ContinuousOn (fun t => iteratedDerivWithin i f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
hf :
(∀ (m : ℕ), ↑m ≤ ↑n → ContinuousOn (iteratedDerivWithin m f s) s) ∧
∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s
i : ℕ
hi : i ∈ Finset.r... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | cases' hf with hf_left | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
i : ℕ
hi : i ∈ Finset.range (n + 1)
hf_left : ∀ (m : ℕ), ↑m ≤ ↑n → ContinuousOn (iteratedDerivWithin m f s) s
right✝ : ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (i... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | specialize hf_left i | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
i : ℕ
hi : i ∈ Finset.range (n + 1)
right✝ : ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s
hf_left : ↑i ≤ ↑n → ContinuousOn (iteratedDeri... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [Finset.mem_range] at hi | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
i : ℕ
right✝ : ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s
hf_left : ↑i ≤ ↑n → ContinuousOn (iteratedDerivWithin i f s) s
hi : i < n + ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | refine' hf_left _ | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
n : ℕ
s : Set ℝ
hs : UniqueDiffOn ℝ s
i : ℕ
right✝ : ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s
hf_left : ↑i ≤ ↑n → ContinuousOn (iteratedDerivWithin i f s) s
hi : i < n + ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t x : ℝ
n : ℕ
⊢ HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(↑n + 1) * (x - t) ^ n) t | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp_rw [sub_eq_neg_add] | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
| Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t x : ℝ
n : ℕ
⊢ HasDerivAt (fun y => (-y + x) ^ (n + 1)) (-(↑n + 1) * (-t + x) ^ n) t | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
| Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t x : ℝ
n : ℕ
⊢ HasDerivAt (fun y => (-y + x) ^ (n + 1)) ((↑n + 1) * (-t + x) ^ n * -1) t | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 ... | Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7.h.e'_5.h.e'_5
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t x : ℝ
n : ℕ
⊢ ↑n + 1 = ↑(n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [Nat.cast_add, Nat.cast_one] | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 ... | Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • it... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ iteratedDerivWithin (k + 2) f s y = derivWithin (iteratedDeri... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ derivWithin (iteratedDerivWithin (k + 1) f s) s y = derivWith... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact (derivWithin_of_mem hs ht hf).symm | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1))
(-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_au... | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | field_simp | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ -((x - y) ^ k * ((↑k + 1) * ↑k !)) = (-1 + -↑... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | ring | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [this] | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | convert this.smul hf using 1 | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => (... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | field_simp | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => (... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [neg_div, neg_smul, sub_eq_add_neg] | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
n : ℕ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
hf : ContDiffOn ℝ (↑n) f s
hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | induction' n with k hk | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
hf : ContDiffOn ℝ (↑Nat.zero) f s
hf' : DifferentiableWithinAt ℝ (iteratedD... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul] | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
hf : ContDiffOn ℝ (↑Nat.zero) f s
hf' : DifferentiableWithinAt ℝ (iteratedD... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [iteratedDerivWithin_zero] at hf' | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
hf : ContDiffOn ℝ (↑Nat.zero) f s
hf' : DifferentiableWithinAt ℝ f s y
⊢ Ha... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [iteratedDerivWithin_one (hs_unique _ (h hy))] | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
hf : ContDiffOn ℝ (↑Nat.zero) f s
hf' : DifferentiableWithinAt ℝ f s y
⊢ Ha... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | norm_num | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
hf : ContDiffOn ℝ (↑Nat.zero) f s
hf' : DifferentiableWithinAt ℝ f s y
⊢ Ha... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact hf'.hasDerivWithinAt.mono h | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
k : ℕ
hk :
ContDiffOn ℝ (↑k) f s →
DifferentiableWithinAt ℝ (iterated... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp_rw [Nat.add_succ, taylorWithinEval_succ] | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
k : ℕ
hk :
ContDiffOn ℝ (↑k) f s →
DifferentiableWithinAt ℝ (iterated... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
k : ℕ
hk :
ContDiffOn ℝ (↑k) f s →
DifferentiableWithinAt ℝ (iterated... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
k : ℕ
hk :
ContDiffOn ℝ (↑k) f s →
DifferentiableWithinAt ℝ (iterated... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
k : ℕ
hk :
ContDiffOn ℝ (↑k) f s →
DifferentiableWithinAt ℝ (iterated... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
k : ℕ
hf : ContDiffOn ℝ (↑(Nat.succ k)) f s
hf' : DifferentiableWithinAt ℝ ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
s s' : Set ℝ
hs'_unique : UniqueDiffWithinAt ℝ s' y
hs_unique : UniqueDiffOn ℝ s
hs' : s' ∈ 𝓝[s] y
hy : y ∈ s'
h : s' ⊆ s
k : ℕ
hf : ContDiffOn ℝ (↑(Nat.succ k)) f s
hf' : DifferentiableWithinAt ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact (add_sub_cancel'_right _ _).symm | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x)
(fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g'... | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib_Analysis_Calculus_Taylor |
case intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | use y, hy | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasD... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [taylorWithinEval_self] at h | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasD... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasD... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [← h] | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasD... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | field_simp [g'_ne y hy] | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasD... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | ring | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there e... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
⊢ ∃ x' ∈ Ioo x₀ x,
f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
iterat... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
⊢ ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | refine' Continuous.continuousOn _ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
⊢ Continuous fun t => (x - t) ^ (n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact (continuous_const.sub continuous_id').pow _ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
⊢ ∃ x' ∈ Ioo x₀ x,
... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne' | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
⊢ ∀ y ∈ Ioo x₀ x, (... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | intro y hy | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
y : ℝ
hy : y ∈ Ioo ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | refine' pow_ne_zero _ _ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
y : ℝ
hy : y ∈ Ioo ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [mem_Ioo] at hy | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
y : ℝ
hy : x₀ < y ∧... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [sub_ne_zero] | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
y : ℝ
hy : x₀ < y ∧... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact hy.2.ne' | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
xy_ne : ∀ y ∈ Ioo x... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
xy_ne : ∀ y ∈ Ioo x... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
⟨y, hy, h⟩ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
case intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
xy... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | use y, hy | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
xy_ne : ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
xy_ne : ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [h, neg_div, ← div_neg, neg_mul, neg_neg] | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
xy_ne : ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | field_simp [xy_ne y hy, Nat.factorial] | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)
xy_ne : ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | ring | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
⊢ ∃ x' ∈ Ioo x₀ x,
f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
iterat... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
⊢ Continuous id | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | continuity | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn id (Icc x₀ x)
⊢ ∃ x' ∈ Ioo x₀ x,
f x - taylorWithinEval... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _ | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn id (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn id (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
case intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn id (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, Ha... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | use y, hy | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn id (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDeriv... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [h] | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn id (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDeriv... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | field_simp [n.factorial_ne_zero] | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
case right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ < x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn id (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDeriv... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | ring | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such
that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
⊢ ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rcases eq_or_lt_of_le hab with (rfl | h) | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a C x : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx : x ∈ Icc a a
hC : ∀ y ∈ Icc a a, ‖iteratedDerivWithin (n + 1) f (Icc a a) y‖ ≤ C
⊢ ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ C * (x -... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [Icc_self, mem_singleton_iff] at hx | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a C x : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx : x = a
hC : ∀ y ∈ Icc a a, ‖iteratedDerivWithin (n + 1) f (Icc a a) y‖ ≤ C
⊢ ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ C * (x - a) ^ ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp [hx] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
⊢ ‖f x - taylorWithinEval f n (Icc a b) a x... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h) | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWith... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have h' : ∀ y ∈ Ico a x,
‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤
(n ! : ℝ)⁻¹ * |x - a| ^ n * C := by
rintro y ⟨hay, hyx⟩
rw [norm_smul, Real.norm_eq_abs]
gcongr
· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_n... | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (I... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rintro y ⟨hay, hyx⟩ | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWi... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [norm_smul, Real.norm_eq_abs] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWi... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | gcongr | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro.h₁
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDeri... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro.h₁
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDeri... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | gcongr | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro.h₁.h.hab
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iterat... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [abs_of_nonneg, abs_of_nonneg] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro.h₁.h.hab
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iterat... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | linarith | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro.h₁.h.hab
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iterat... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | linarith | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro.h₁.h.hab
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iterat... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | linarith | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case intro.h₂
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDeri... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact hC y ⟨hay, hyx.le.trans hx.2⟩ | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWith... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have A : ∀ t ∈ Icc a x, HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t := by
intro t ht
have I : Icc a x ⊆ Icc a b := Icc_subset_Icc_right hx.2
exact (has_deriv_within_taylorWithinEval_at_Icc x h (I ht) ... | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (I... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | intro t ht | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
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