mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
f g : 𝕜 → F
hf : ContDiffAt 𝕜 (↑n) f x
hg : ContDiffAt 𝕜 (↑n) g x
⊢ iteratedDeriv n (f + g) x = iteratedDeriv n f x + iteratedDeriv n g x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
f g : 𝕜 → F
hf : ContDiffAt 𝕜 (↑n) f x
hg : ContDiffAt 𝕜 (↑n) g x
⊢ iteratedDeriv n (f + g) x = iteratedDeriv n f x + iteratedDeriv n g x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
f : 𝕜 → F
hn : 0 < n
c : F
⊢ iteratedDeriv n (fun z ↦ c + f z) x = iteratedDeriv n f x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
f : 𝕜 → F
hn : 0 < n
c : F
⊢ iteratedDerivWithin n (fun z ↦ c + f z) Set.univ x = iteratedDerivWithin n f Set.univ x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ι : Type u_3
f : ι → E → F
u : Finset ι
i : ℕ
h : ∀ j ∈ u, ContDiff 𝕜 (↑i) (f j)
x : E
⊢ iteratedFDeriv 𝕜 i (fun x ↦ ∑ j ∈ u, f j x) x = (∑ j ∈ u, iteratedFDeriv 𝕜 i (f j)) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ι : Type u_3
f : ι → E → F
u : Finset ι
i : ℕ
h : ∀ j ∈ u, ContDiff 𝕜 (↑i) (f j)
x : E
⊢ iteratedFDeriv 𝕜 i (fun x ↦ ∑ j ∈ u, f j x) x = ∑ c ∈ u, iteratedFDeriv 𝕜 i (f c) x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
a : 𝕜
⊢ iteratedDeriv n (-f) a = -iteratedDeriv n f a

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
a : 𝕜
⊢ iteratedDerivWithin n (-f) Set.univ a = -iteratedDerivWithin n f Set.univ a
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
f g : 𝕜 → F
hf : ContDiffAt 𝕜 (↑n) f x
hg : ContDiffAt 𝕜 (↑n) g x
⊢ iteratedDeriv n (f - g) x = iteratedDeriv n f x - iteratedDeriv n g x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
f g : 𝕜 → F
hf : ContDiffAt 𝕜 (↑n) f x
hg : ContDiffAt 𝕜 (↑n) g x
⊢ iteratedDeriv n (f - g) x = iteratedDeriv n f x - iteratedDeriv n g x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
s : Set E
x : E
n : ℕ∞ω
𝔸' : Type u_4
ι : Type u_5
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
t : Finset ι
f : ι → E → 𝔸'
h : ∀ i ∈ t, ContDiffWithinAt 𝕜 n (f i) s x
⊢ ContDiffWithinAt 𝕜 n (fun y ↦ ∏ i ∈ t, f i y) s x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
s : Set E
x : E
n : ℕ∞ω
𝔸' : Type u_4
ι : Type u_5
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
t : Finset ι
f : ι → E → 𝔸'
h : ∀ i ∈ t, ContDiffWithinAt 𝕜 n (f i) s x
⊢ ContDiffWithinAt 𝕜 n (fun y ↦ (∏ c ∈ t, f c) y) s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
x : 𝕜
R : Type u_3
inst✝² : DistribSMul R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
n : ℕ
f : 𝕜 → F
h : ContDiffAt 𝕜 (↑n) f x
c : R
⊢ iteratedDeriv n (c • f) x = c • iteratedDeriv n f x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
x : 𝕜
R : Type u_3
inst✝² : DistribSMul R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
n : ℕ
f : 𝕜 → F
h : ContDiffAt 𝕜 (↑n) f x
c : R
⊢ iteratedDeriv n (c • f) x = c • iteratedDeriv n f x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
n : ℕ∞ω
𝔸 : Type u_3
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
f : E → 𝔸
hf : ContDiff 𝕜 n f
⊢ ContDiff 𝕜 n fun x ↦ f x ^ 0

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
n : ℕ∞ω
𝔸 : Type u_3
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
f : E → 𝔸
hf : ContDiff 𝕜 n f
⊢ ContDiff 𝕜 n fun x ↦ 1
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
n : ℕ∞ω
𝔸 : Type u_3
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
f : E → 𝔸
hf : ContDiff 𝕜 n f
⊢ ContDiff 𝕜 n fun x ↦ f x ^ 0

Tactic:
simpa using contDiff_const

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
𝕝 : Type u_4
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
n : ℕ
c : 𝕝
f : 𝕜 → F
⊢ iteratedDeriv n (c • f) x = c • iteratedDeriv n f x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
𝕝 : Type u_4
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
n : ℕ
c : 𝕝
f : 𝕜 → F
⊢ iteratedDeriv n (c • f) x = c • iteratedDeriv n f x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
n : ℕ∞ω
𝔸 : Type u_3
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
f : E → 𝔸
hf : ContDiff 𝕜 n f
m : ℕ
⊢ ContDiff 𝕜 n fun x ↦ f x ^ (m + 1)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
n : ℕ∞ω
𝔸 : Type u_3
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
f : E → 𝔸
hf : ContDiff 𝕜 n f
m : ℕ
⊢ ContDiff 𝕜 n fun x ↦ f x ^ m * f x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
𝕝 : Type u_4
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
n : ℕ
c : 𝕝
f : 𝕜 → F
⊢ iteratedDeriv n (fun x ↦ c • f x) x = c • iteratedDeriv n f x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
𝕝 : Type u_4
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
n : ℕ
c : 𝕝
f : 𝕜 → F
⊢ iteratedDeriv n (fun x ↦ c • f x) x = c • iteratedDeriv n f x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
s : Set E
x : E
𝕜' : Type u_6
inst✝¹ : NormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : E → 𝕜'
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
c : 𝕜'
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ f x / c) s x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
s : Set E
x : E
𝕜' : Type u_6
inst✝¹ : NormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : E → 𝕜'
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
c : 𝕜'
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ f x * c⁻¹) s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝔸 : Type u_5
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
n : ℕ
f : 𝕜 → 𝔸
c : 𝔸
hf : ContDiffAt 𝕜 (↑n) f x
⊢ iteratedDeriv n (fun x ↦ c * f x) x = c * iteratedDeriv n f x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝔸 : Type u_5
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
n : ℕ
f : 𝕜 → 𝔸
c : 𝔸
hf : ContDiffAt 𝕜 (↑n) f x
⊢ iteratedDeriv n (fun x ↦ c * f x) x = c * iteratedDeriv n f x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕜' : Type u_6
inst✝¹ : NormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : E → 𝕜'
n : ℕ∞ω
hf : ContDiff 𝕜 n f
c : 𝕜'
⊢ ContDiff 𝕜 n fun x ↦ f x / c

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕜' : Type u_6
inst✝¹ : NormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : E → 𝕜'
n : ℕ∞ω
hf : ContDiff 𝕜 n f
c : 𝕜'
⊢ ContDiff 𝕜 n fun x ↦ f x * c⁻¹
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_6
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
n : ℕ
c : 𝕜'
f : 𝕜 → 𝕜'
⊢ iteratedDeriv n (fun x ↦ c * f x) x = c * iteratedDeriv n f x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_6
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
n : ℕ
c : 𝕜'
f : 𝕜 → 𝕜'
⊢ iteratedDeriv n (fun x ↦ c * f x) x = c * iteratedDeriv n f x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_6
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
n : ℕ
f : 𝕜 → 𝕜'
c : 𝕜'
⊢ iteratedDeriv n (fun x ↦ f x * c) x = iteratedDeriv n f x * c

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_6
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
n : ℕ
f : 𝕜 → 𝕜'
c : 𝕜'
⊢ iteratedDeriv n (fun x ↦ f x * c) x = iteratedDeriv n f x * c
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
h : ContDiff 𝕜 (↑n) f
c x : 𝕜
⊢ iteratedDeriv n (fun x ↦ f (c * x)) x = c ^ n • iteratedDeriv n f (c * x)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
h : ContDiff 𝕜 (↑n) f
c x : 𝕜
⊢ iteratedDeriv n (fun x ↦ f (c * x)) x = c ^ n • iteratedDeriv n f (c * x)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
f : 𝕜 → 𝕜
h : ContDiff 𝕜 (↑n) f
c : 𝕜
⊢ (iteratedDeriv n fun x ↦ f (c * x)) = fun x ↦ c ^ n * iteratedDeriv n f (c * x)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
f : 𝕜 → 𝕜
h : ContDiff 𝕜 (↑n) f
c : 𝕜
⊢ (iteratedDeriv n fun x ↦ f (c * x)) = fun x ↦ c ^ n * iteratedDeriv n f (c * x)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
𝔸 : Type u_5
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
f g : 𝕜 → 𝔸
hf : ContDiffAt 𝕜 (↑n) f x
hg : ContDiffAt 𝕜 (↑n) g x
⊢ iteratedDeriv n (f * g) x =
    ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * iteratedDeriv i f x * iteratedDeriv (n - i) g x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
𝔸 : Type u_5
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
f g : 𝕜 → 𝔸
hf : ContDiffAt 𝕜 (↑n) f x
hg : ContDiffAt 𝕜 (↑n) g x
⊢ iteratedDeriv n (f * g) x =
    ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * iteratedDeriv i f x * iteratedDeriv (n - i) g x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
m k : ℕ
⊢ iteratedDeriv k (fun x ↦ x ^ m) x = ↑(m.descFactorial k) * x ^ (m - k)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
m k : ℕ
⊢ iteratedDeriv k (fun x ↦ x ^ m) x = ↑(m.descFactorial k) * x ^ (m - k)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n✝ : ℕ
f : 𝕜 → F
s : 𝕜
n : ℕ
IH : (iteratedDeriv n fun z ↦ f (s + z)) = fun t ↦ iteratedDeriv n f (s + t)
⊢ (iteratedDeriv (n + 1) fun z ↦ f (s + z)) = fun t ↦ iteratedDeriv (n + 1) f (s + t)

Tactic:
simpa using

Goal state after:
case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n✝ : ℕ
f : 𝕜 → F
s : 𝕜
n : ℕ
IH : (iteratedDeriv n fun z ↦ f (s + z)) = fun t ↦ iteratedDeriv n f (s + t)
⊢ (deriv fun t ↦ iteratedDeriv n f (s + t)) = fun t ↦ deriv (iteratedDeriv n f) (s + t)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n✝ : ℕ
f : 𝕜 → F
s : 𝕜
n : ℕ
IH : (iteratedDeriv n fun z ↦ f (z + s)) = fun t ↦ iteratedDeriv n f (t + s)
⊢ (iteratedDeriv (n + 1) fun z ↦ f (z + s)) = fun t ↦ iteratedDeriv (n + 1) f (t + s)

Tactic:
simpa using

Goal state after:
case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n✝ : ℕ
f : 𝕜 → F
s : 𝕜
n : ℕ
IH : (iteratedDeriv n fun z ↦ f (z + s)) = fun t ↦ iteratedDeriv n f (t + s)
⊢ (deriv fun t ↦ iteratedDeriv n f (t + s)) = fun t ↦ deriv (iteratedDeriv n f) (t + s)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : 𝕜
⊢ (iteratedDeriv n fun z ↦ f (s - z)) = fun t ↦ (-1) ^ n • iteratedDeriv n f (s - t)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : 𝕜
⊢ ∀ (x : 𝕜), iteratedDeriv n (fun z ↦ f (s - z)) x = (-1) ^ n • iteratedDeriv n f (s - x)
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

Tactic:
obtain ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, ∀ x ∈ Icc a b, f c ≤ f x :=
    isCompact_Icc.exists_isMinOn ne hfc

Goal state after:
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ι : Type u_7
n : ℕ
x : 𝕜
f : ι → 𝕜 → F
I : Finset ι
s : Set 𝕜
hx : x ∈ s
hs : UniqueDiffOn 𝕜 s
hf : ∀ i ∈ I, ContDiffWithinAt 𝕜 (↑n) (f i) s x
⊢ iteratedDerivWithin n (fun x ↦ ∑ i ∈ I, f i x) s x = ∑ i ∈ I, iteratedDerivWithin n (f i) s x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ι : Type u_7
n : ℕ
x : 𝕜
f : ι → 𝕜 → F
I : Finset ι
s : Set 𝕜
hx : x ∈ s
hs : UniqueDiffOn 𝕜 s
hf : ∀ i ∈ I, ContDiffWithinAt 𝕜 (↑n) (f i) s x
⊢ iteratedDerivWithin n (fun x ↦ ∑ i ∈ I, f i x) s x = ∑ i ∈ I, iteratedDerivWithin n (f i) s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ι : Type u_7
n : ℕ
x : 𝕜
f : ι → 𝕜 → F
I : Finset ι
hf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x
⊢ iteratedDeriv n (∑ i ∈ I, f i) x = ∑ i ∈ I, iteratedDeriv n (f i) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ι : Type u_7
n : ℕ
x : 𝕜
f : ι → 𝕜 → F
I : Finset ι
hf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x
⊢ iteratedDeriv n (∑ i ∈ I, f i) x = ∑ i ∈ I, iteratedDeriv n (f i) x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ι : Type u_7
n : ℕ
x : 𝕜
f : ι → 𝕜 → F
I : Finset ι
hf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x
⊢ iteratedDeriv n (fun z ↦ ∑ i ∈ I, f i z) x = ∑ i ∈ I, iteratedDeriv n (f i) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ι : Type u_7
n : ℕ
x : 𝕜
f : ι → 𝕜 → F
I : Finset ι
hf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x
⊢ iteratedDeriv n (fun z ↦ ∑ i ∈ I, f i z) x = ∑ i ∈ I, iteratedDeriv n (f i) x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
n : ℕ∞ω
E' : Type u_3
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
F' : Type u_4
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜 F'
f : E → F
g : E' → F'
x : E
y : E'
hf : ContDiffWithinAt 𝕜 n f univ x
hg : ContDiffWithinAt 𝕜 n g univ y
⊢ ContDiffWithinAt 𝕜 n (Prod.map f g) univ (x, y)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
n : ℕ∞ω
E' : Type u_3
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
F' : Type u_4
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜 F'
f : E → F
g : E' → F'
x : E
y : E'
hf : ContDiffWithinAt 𝕜 n f univ x
hg : ContDiffWithinAt 𝕜 n g univ y
⊢ ContDiffWithinAt 𝕜 n (Prod.map f g) univ (x, y)
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type uE
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
f g : E → 𝕜
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hg : ContDiffWithinAt 𝕜 n g s x
hx : g x ≠ 0
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ f x / g x) s x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type uE
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
f g : E → 𝕜
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hg : ContDiffWithinAt 𝕜 n g s x
hx : g x ≠ 0
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ f x * (g x)⁻¹) s x
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
⊢ μ s ≠ ∞

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
⊢ ¬μ s = ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
ne_top : μ s ≠ ∞
this : 0 < ε / ↑c
⊢ μ s < μ s + ε / ↑c

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
ne_top : μ s ≠ ∞
this : 0 < ε / ↑c
⊢ μ s < μ s + ε / ↑c
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
ne_top : μ s ≠ ∞
this : μ s < μ s + ε / ↑c
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u < μ s + ε / ↑c
⊢ ↑c ≠ 0

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
ne_top : μ s ≠ ∞
this : μ s < μ s + ε / ↑c
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u < μ s + ε / ↑c
⊢ ¬c = 0
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
f' : ℝ → ℝ := f ∘ Neg.neg
b : ℝ
hb₁ : b < a
hb₂ : ∀ ⦃x : ℝ⦄, x ∈ Ioo b a → deriv f x ∈ Iic (-1)
x : ℝ
hx : x ∈ Ioo (-a) (-b)
this : deriv f' x = deriv f (-x) * deriv Neg.neg x
⊢ deriv f' x = -deriv f (-x)

Tactic:
simpa using

Goal state after:
f : ℝ → ℝ
a : ℝ
f' : ℝ → ℝ := f ∘ Neg.neg
b : ℝ
hb₁ : b < a
hb₂ : ∀ ⦃x : ℝ⦄, x ∈ Ioo b a → deriv f x ∈ Iic (-1)
x : ℝ
hx : x ∈ Ioo (-a) (-b)
this : deriv f' x = deriv f (-x) * deriv Neg.neg x
⊢ deriv f' x = -deriv f (-x)
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x ↦ ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
x : α
⊢ ↑(f x) < f' x

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x ↦ ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
x : α
⊢ ↑(f x) < f' x