mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
n : ℕ∞ω
ι : Type u_3
inst✝² : Fintype ι
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
Φ : E → (i : ι) → F' i
⊢ ContDiff 𝕜 n Φ ↔ ∀ (i : ι), ContDiff 𝕜 n fun x ↦ Φ x i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra 𝕂 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕂
htx : t • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ HasStrictDerivAt (fun u ↦ exp (u • x)) (exp (t • x) * x) t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra 𝕂 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕂
htx : t • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ HasStrictDerivAt (fun u ↦ exp (u • x)) (x * exp (t • x)) t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type u_3
inst✝³ : Fintype ι
F' : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (F' i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (F' i)
inst✝ : DecidableEq ι
k : ℕ∞ω
x : (i : ι) → F' i
i : ι
⊢ ContDiff 𝕜 k (update x i)

Tactic:
by

Goal state after:
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
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ∞ω
q : E → FormalMultilinearSeries 𝕜 E F
g : E → F
hf : HasFTaylorSeriesUpToOn n f p s
hg : HasFTaylorSeriesUpToOn n g q s
⊢ HasFTaylorSeriesUpToOn n (f + g) (p + q) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: ⊢ AnalyticOnNhd ℂ cexp univ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: s : Set ℂ
x : ℂ
⊢ AnalyticWithinAt ℂ cexp s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: x : ℂ
⊢ 1 < 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: x : ℂ
⊢ HasDerivAt cexp (cexp x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
i : Fin c.length
e : E
v : Fin m → E
j : Fin c.length
hij : j ≠ i
⊢ update c.partSize i (c.partSize i + 1) j = c.partSize j

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: n : ℕ
⊢ deriv^[n + 1] cexp = cexp

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n✝ : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ↑(n + 1) ≠ ↑⊤

Tactic:
by

Goal state after:
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
x : E
n : ℕ∞ω
f g : E → F
hf : ContDiffAt 𝕜 n f x
hg : ContDiffAt 𝕜 n g x
⊢ ContDiffAt 𝕜 n (fun x ↦ f x + g x) x

Tactic:
by

Goal state after:
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
s : Set E
x : E
i : ℕ
f g : E → F
hf : ContDiffWithinAt 𝕜 (↑i) f s x
hg : ContDiffWithinAt 𝕜 (↑i) g s x
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ ↑i ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: n : ℕ
c : ℂ
⊢ (iteratedDeriv n fun s ↦ cexp (c * s)) = fun s ↦ c ^ n * cexp (c * s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: ⊢ AnalyticOnNhd ℝ rexp univ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: n : ℕ
⊢ deriv^[n + 1] rexp = rexp

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
i : Fin c.length
⊢ (q.compAlongOrderedFinpartition p (c.extend (some i))).curryLeft =
    ((compAlongOrderedFinpartitionL 𝕜 E F G c) (q c.length)).toContinuousLinearMap (fun i ↦ p (c.partSize i)) i ∘SL
      (p (c.partSize i + 1)).curryLeft

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: n : ℕ
c : ℝ
⊢ (iteratedDeriv n fun s ↦ rexp (c * s)) = fun s ↦ c ^ n * rexp (c * s)

Tactic:
by

Goal state after:
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
s : Set E
x : E
i : ℕ
f g : E → F
hf : ContDiffWithinAt 𝕜 (↑i) f s x
hg : ContDiffWithinAt 𝕜 (↑i) g s x
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
this : ∀ᶠ (x : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 (↑i) f s x ∧ ContDiffWithinAt 𝕜 (↑i) g s x
t : Set E
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
hxt : x ∈ t
h : t ∩ insert x s ⊆ {x | (fun x ↦ ContDiffWithinAt 𝕜 (↑i) f s x ∧ ContDiffWithinAt 𝕜 (↑i) g s x) x}
a : E
ha : a ∈ s ∩ t
⊢ a ∈ t ∩ insert x s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : E
⊢ Monotone fun s ↦ posTangentConeAt s a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : ∃ᶠ (t : ℝ) in 𝓝[>] 0, x + t • y ∈ s
⊢ y ∈ posTangentConeAt s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : [x -[ℝ] x + y] ⊆ s
⊢ y ∈ posTangentConeAt s x

Tactic:
by

Goal state after:
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
s : Set E
x : E
i : ℕ
f g : E → F
hf : ContDiffWithinAt 𝕜 (↑i) f s x
hg : ContDiffWithinAt 𝕜 (↑i) g s x
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
this : ∀ᶠ (x : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 (↑i) f s x ∧ ContDiffWithinAt 𝕜 (↑i) g s x
t : Set E
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
hxt : x ∈ t
h : t ∩ insert x s ⊆ {x | (fun x ↦ ContDiffWithinAt 𝕜 (↑i) f s x ∧ ContDiffWithinAt 𝕜 (↑i) g s x) x}
hft : ContDiffOn 𝕜 (↑i) f (s ∩ t)
a : E
ha : a ∈ s ∩ t
⊢ a ∈ t ∩ insert x s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
c : OrderedFinpartition m
B : (F [×c.length]→L[𝕜] G) →L[𝕜] (E [×c.partSize i]→L[𝕜] F) [×c.length]→L[𝕜] E [×m]→L[𝕜] G :=
  compAlongOrderedFinpartitionL 𝕜 E F G c
cm : ↑c.length ≤ ↑m
i : Fin c.length
⊢ ↑(c.partSize i) ≤ ↑m

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
s : Set E
a y : E
h : IsLocalMaxOn f s a
hf : HasFDerivWithinAt f f' s a
hy : y ∈ posTangentConeAt s a
⊢ f' y ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMaxOn f s a
hy : y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ (fderivWithin ℝ f s a) y ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMaxOn f s a
hy : y ∈ posTangentConeAt s a
⊢ (fderivWithin ℝ f s a) y ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
s : Set E
a y : E
h : IsLocalMaxOn f s a
hf : HasFDerivWithinAt f f' s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
⊢ 0 ≤ f' y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMaxOn f s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ (fderivWithin ℝ f s a) y = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMaxOn f s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
⊢ (fderivWithin ℝ f s a) y = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
s : Set E
a y : E
h : IsLocalMinOn f s a
hf : HasFDerivWithinAt f f' s a
hy : y ∈ posTangentConeAt s a
⊢ 0 ≤ f' y

Tactic:
by

Goal state after:
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
s : Set E
x : E
i : ℕ
f g : E → F
hf : ContDiffWithinAt 𝕜 (↑i) f s x
hg : ContDiffWithinAt 𝕜 (↑i) g s x
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ iteratedFDerivWithin 𝕜 i (f + g) s x = iteratedFDerivWithin 𝕜 i f s x + iteratedFDerivWithin 𝕜 i g s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMinOn f s a
hy : y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ 0 ≤ (fderivWithin ℝ f s a) y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMinOn f s a
hy : y ∈ posTangentConeAt s a
⊢ 0 ≤ (fderivWithin ℝ f s a) y

Tactic:
by

Goal state after:
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
x : E
i : ℕ
f g : E → F
hf : ContDiffAt 𝕜 (↑i) f x
hg : ContDiffAt 𝕜 (↑i) g x
⊢ iteratedFDeriv 𝕜 i (f + g) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
s : Set E
a y : E
h : IsLocalMinOn f s a
hf : HasFDerivWithinAt f f' s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
⊢ f' y = 0

Tactic:
by

Goal state after:
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
x : E
n : ℕ∞ω
f : E → F
hf : ContDiffAt 𝕜 n f x
⊢ ContDiffAt 𝕜 n (fun x ↦ -f x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
c : OrderedFinpartition m
⊢ HasFDerivWithinAt (fun x ↦ (q (f x)).compAlongOrderedFinpartition (p x) c)
    (∑ i, ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMinOn f s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ (fderivWithin ℝ f s a) y = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMinOn f s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
⊢ (fderivWithin ℝ f s a) y = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑n) f s x
hg : ContDiffWithinAt 𝕜 (↑n) g s x
⊢ iteratedDerivWithin n (f • g) s x =
    ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x

Tactic:
by

Goal state after:
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
s : Set E
f : E → F
hu : UniqueDiffOn 𝕜 s
i : ℕ
hi : ∀ {x : E}, x ∈ s → iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
x : E
hx : x ∈ s
h : Fin (i + 1) → E
⊢ ((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (-f) s) s x) (h 0)) (tail h) =
    ((fderivWithin 𝕜 (-iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h)

Tactic:
by

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

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

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
a : E
h : IsLocalMin f a
hf : HasFDerivAt f f' a
⊢ f' = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
a : E
h : IsLocalMin f a
⊢ fderiv ℝ f a = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
a : E
h : IsLocalMax f a
⊢ fderiv ℝ f a = 0

Tactic:
by

Goal state after:
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
s : Set E
f : E → F
hu : UniqueDiffOn 𝕜 s
i : ℕ
hi : ∀ {x : E}, x ∈ s → iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
x : E
hx : x ∈ s
h : Fin (i + 1) → E
⊢ ((fderivWithin 𝕜 (-iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h) =
    -((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: s : Set ℝ
a : ℝ
h : 1 ∈ posTangentConeAt s a
ι : Type
l : Filter ι
hl : l.NeBot
c : ι → ℝ≥0
d : ι → ℝ
hd₀ : Tendsto d l (𝓝 0)
hd : ∀ᶠ (n : ι) in l, a + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 1)
⊢ Tendsto (fun x ↦ a + d x) l (𝓝 a)

Tactic:
by

Goal state after:
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
s : Set E
f : E → F
hu : UniqueDiffOn 𝕜 s
i : ℕ
hi : ∀ {x : E}, x ∈ s → iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
x : E
hx : x ∈ s
h : Fin (i + 1) → E
⊢ -((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h) = -(iteratedFDerivWithin 𝕜 (i + 1) f s x) h

Tactic:
by

Goal state after:
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
s : Set E
x : E
i : ℕ
f : E → F
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
⊢ ContDiffWithinAt 𝕜 (↑(k + 1)) (fun x ↦ x ^ i) s x

Tactic:
by

Goal state after:
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
x : E
i : ℕ
f : E → F
⊢ iteratedFDeriv 𝕜 i (-f) x = -iteratedFDeriv 𝕜 i f x

Tactic:
by

Goal state after:
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
x : E
n : ℕ∞ω
s : Set E
f g : E → F
hf : ContDiffWithinAt 𝕜 n f s x
hg : ContDiffWithinAt 𝕜 n g s x
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ f x - g x) s x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
⊢ ContDiffWithinAt 𝕜 (↑(k + 1)) (fun x ↦ x) s x

Tactic:
by

Goal state after:
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
x : E
n : ℕ∞ω
f g : E → F
hf : ContDiffAt 𝕜 n f x
hg : ContDiffAt 𝕜 n g x
⊢ ContDiffAt 𝕜 n (fun x ↦ f x - g x) x

Tactic:
by

Goal state after:
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
n : ℕ∞ω
s : Set E
f g : E → F
hf : ContDiffOn 𝕜 n f s
hg : ContDiffOn 𝕜 n g s
⊢ ContDiffOn 𝕜 n (fun x ↦ f x - g x) s

Tactic:
by

Goal state after:
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
n : ℕ∞ω
f g : E → F
hf : ContDiff 𝕜 n f
hg : ContDiff 𝕜 n g
⊢ ContDiff 𝕜 n fun x ↦ f x - g x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: s : Set ℝ
a : ℝ
⊢ 1 ∈ posTangentConeAt s a ↔ a ∈ closure (Ioi a ∩ s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: s : Set ℝ
a : ℝ
⊢ 1 ∈ posTangentConeAt s a ↔ ∃ᶠ (x : ℝ) in 𝓝[>] a, x ∈ s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
⊢ (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: f : ℝ → ℝ
f' a : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ f' = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
h : IsLocalMin f a
⊢ deriv f a = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
h : IsLocalMax f a
⊢ deriv f a = 0

Tactic:
by

Goal state after: