mathlib-initiative/mathlib-tactics huggingface dataset:
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
ι : Type u_3
f : ι → E → F
s : Finset ι
x : E
h : ∀ i ∈ s, ContDiffAt 𝕜 n (fun x ↦ f i x) x
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ ∑ i ∈ s, f i x) univ x

Tactic:
]

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
n : ℕ∞ω
ι : Type u_3
f : ι → E → F
s : Finset ι
x : E
h : ∀ i ∈ s, ContDiffAt 𝕜 n (fun x ↦ f i x) x
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ ∑ i ∈ s, f i x) univ x
Module: Mathlib.Topology.ExtendFrom

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A B : Set X
hB : B ⊆ closure[inst✝²] A
hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
⊢ ContinuousOn (extendFrom A f) B

Tactic:
by
  set φ := extendFrom A f
  intro x x_in
  suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝[B] x by
    simpa [ContinuousWithinAt, (closed_nhds_basis (φ x)).tendsto_right_iff]
  intro V' V'_in V'_closed
  obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ V ∩ A ⊆ f ⁻¹' V' := by
    have := tendsto_extendFrom (hf x x_in)
    rcases (nhdsWithin_basis_open x A).tendsto_left_iff.mp this V' V'_in with ⟨V, ⟨hxV, V_op⟩, hV⟩
    exact ⟨V, IsOpen.mem_nhds V_op hxV, V_op, hV⟩
  suffices ∀ y ∈ V ∩ B, φ y ∈ V' from
    mem_of_superset (inter_mem_inf V_in <| mem_principal_self B) this
  rintro y ⟨hyV, hyB⟩
  have := mem_closure_iff_nhdsWithin_neBot.mp (hB hyB)
  have limy : Tendsto f (𝓝[A] y) (𝓝 <| φ y) := tendsto_extendFrom (hf y hyB)
  have hVy : V ∈ 𝓝 y := IsOpen.mem_nhds V_op hyV
  have : V ∩ A ∈ 𝓝[A] y := by simpa only [inter_comm] using inter_mem_nhdsWithin A hVy
  exact V'_closed.mem_of_tendsto limy (mem_of_superset this hV)

Goal state after:
Module: Mathlib.Topology.ExtendFrom

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A : Set X
hA : Dense A
hf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
⊢ ContinuousOn (extendFrom A f) univ

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A : Set X
hA : Dense A
hf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
⊢ ContinuousOn (extendFrom A f) univ
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
ι : Type u_3
f : ι → E → F
u : Finset ι
i : ℕ
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
h : ∀ j ∈ u, ContDiffWithinAt 𝕜 (↑i) (f j) s x
⊢ iteratedFDerivWithin 𝕜 i (fun x ↦ ∑ j ∈ u, f j x) s x = ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x

Tactic:
]

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
s : Set E
ι : Type u_3
f : ι → E → F
u : Finset ι
i : ℕ
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
h : ∀ j ∈ u, ContDiffWithinAt 𝕜 (↑i) (f j) s x
⊢ iteratedFDerivWithin 𝕜 i (fun x ↦ ∑ j ∈ u, f j x) s x = ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x
Module: Mathlib.Topology.ExtendFrom

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A : Set X
hA : Dense A
hf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
⊢ Continuous[inst✝², inst✝¹] (extendFrom A f)

Tactic:
rw [← continuousOn_univ]
  exact continuousOn_extendFrom (fun x _ ↦ hA x) (by simpa using hf)

Goal state after:
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case neg.hB
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la lb : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
hab : a ≠ b
⊢ Icc a b ⊆ Icc a b

Tactic:
]

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

Goal state before:
pp: case fderivWithin
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
A :
  ∀ (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
B :
  HasFDerivWithinAt (fun x ↦ (q (f x)).taylorComp (p x) m)
    (∑ c, ∑ i, ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x
⊢ ∑ x_1 ∈ Finset.univ.sigma fun c ↦ Finset.univ, (q (f x)).compAlongOrderedFinpartition (p x) (x_1.fst.extend x_1.snd) =
    (q (f x)).taylorComp (p x) (m + 1)

Tactic:
]

Goal state after:
case fderivWithin
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
A :
  ∀ (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
B :
  HasFDerivWithinAt (fun x ↦ (q (f x)).taylorComp (p x) m)
    (∑ c, ∑ i, ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x
⊢ ∑ x_1 ∈ Finset.univ.sigma fun c ↦ Finset.univ, (q (f x)).compAlongOrderedFinpartition (p x) (x_1.fst.extend x_1.snd) =
    (q (f x)).taylorComp (p x) (m + 1)
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case cons
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
ι : Type u_3
f : ι → E → F
i : ℕ
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
a : ι
u : Finset ι
ha : a ∉ u
IH :
  (∀ j ∈ u, ContDiffWithinAt 𝕜 (↑i) (f j) s x) →
    iteratedFDerivWithin 𝕜 i (fun x ↦ ∑ j ∈ u, f j x) s x = ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x
h : ContDiffWithinAt 𝕜 (↑i) (f a) s x ∧ ∀ a ∈ u, ContDiffWithinAt 𝕜 (↑i) (f a) s x
⊢ iteratedFDerivWithin 𝕜 i (f a) s x + ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x =
    iteratedFDerivWithin 𝕜 i (f a) s x + ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x

Tactic:
]

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
ι : Type u_3
f : ι → E → F
u : Finset ι
i : ℕ
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
h : ∀ j ∈ u, ContDiffWithinAt 𝕜 (↑i) (f j) s x
x✝ : E
⊢ ∑ j ∈ u, f j x✝ = ∑ c ∈ u, f c x✝

Tactic:
]

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
ι : Type u_3
f : ι → E → F
u : Finset ι
n : ℕ
x : E
h : ∀ j ∈ u, ContDiffAt 𝕜 (↑n) (f j) x
x✝ : E
⊢ ∑ j ∈ u, f j x✝ = ∑ c ∈ u, f c x✝

Tactic:
]

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
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:
by simpa [iteratedFDerivWithin_univ] using
    iteratedFDerivWithin_fun_sum_apply uniqueDiffOn_univ (mem_univ x) (h · · |>.contDiffWithinAt)

Goal state after:
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case neg.hB
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hab : a < b
⊢ Ico a b ⊆ Icc a b

Tactic:
]

Goal state after:
case neg.hB
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hab : a < b
⊢ Ico a b ⊆ Icc a b
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type uF
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : E
n : ℕ∞ω
𝕜' : Type u_3
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
f : E → 𝕜'
g : E → F
hf : ContDiffWithinAt 𝕜 n f univ x
hg : ContDiffWithinAt 𝕜 n g univ x
⊢ ContDiffWithinAt 𝕜 n (f • g) univ x

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type uF
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : E
n : ℕ∞ω
𝕜' : Type u_3
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
f : E → 𝕜'
g : E → F
hf : ContDiffWithinAt 𝕜 n f univ x
hg : ContDiffWithinAt 𝕜 n g univ x
⊢ ContDiffWithinAt 𝕜 n (f • g) univ x
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
lb : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
this :
  ContinuousOn (f ∘ ⇑OrderDual.ofDual) (⇑OrderDual.ofDual ⁻¹' Ioo a b) →
    Tendsto (f ∘ ⇑OrderDual.ofDual) (𝓝[⇑OrderDual.ofDual ⁻¹' Iio b] OrderDual.toDual b) (𝓝 lb) →
      ContinuousOn (extendFrom (⇑OrderDual.ofDual ⁻¹' Ioo a b) (f ∘ ⇑OrderDual.ofDual)) (⇑OrderDual.ofDual ⁻¹' Ioc a b)
⊢ ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
lb : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
this :
  ContinuousOn (f ∘ ⇑OrderDual.ofDual) (⇑OrderDual.ofDual ⁻¹' Ioo a b) →
    Tendsto (f ∘ ⇑OrderDual.ofDual) (𝓝[⇑OrderDual.ofDual ⁻¹' Iio b] OrderDual.toDual b) (𝓝 lb) →
      ContinuousOn (extendFrom (⇑OrderDual.ofDual ⁻¹' Ioo a b) (f ∘ ⇑OrderDual.ofDual)) (⇑OrderDual.ofDual ⁻¹' Ioc a b)
⊢ ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b)
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:
by
  simpa only [smul_eq_mul] using iteratedDeriv_comp_const_smul h c

Goal state after:
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case hx
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : T2Space β
hab : a < b
ha : Tendsto f (𝓝[>] a) (𝓝 la)
⊢ a ∈ Icc a b

Tactic:
]

Goal state after:
case hx
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : T2Space β
hab : a < b
ha : Tendsto f (𝓝[>] a) (𝓝 la)
⊢ a ∈ Icc a b
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type uF
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
n : ℕ∞ω
R : Type u_3
inst✝² : DistribSMul R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
f : E → F
x : E
c : R
hf : ContDiffWithinAt 𝕜 n f univ x
⊢ ContDiffWithinAt 𝕜 n (fun y ↦ c • f y) univ x

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type uF
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
n : ℕ∞ω
R : Type u_3
inst✝² : DistribSMul R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
f : E → F
x : E
c : R
hf : ContDiffWithinAt 𝕜 n f univ x
⊢ ContDiffWithinAt 𝕜 n (fun y ↦ c • f y) univ x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type uF
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n : ℕ∞ω
A : Type u_4
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : Module A F
inst✝¹ : IsScalarTower 𝕜 A F
inst✝ : IsBoundedSMul A F
f : E → A
x : E
hf : ContDiffWithinAt 𝕜 n f univ x
v : F
⊢ ContDiffWithinAt 𝕜 n (fun y ↦ f y • v) univ x

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type uF
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n : ℕ∞ω
A : Type u_4
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : Module A F
inst✝¹ : IsScalarTower 𝕜 A F
inst✝ : IsBoundedSMul A F
f : E → A
x : E
hf : ContDiffWithinAt 𝕜 n f univ x
v : F
⊢ ContDiffWithinAt 𝕜 n (fun y ↦ f y • v) univ x
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case hx
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
lb : β
inst✝ : T2Space β
hab : a < b
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
⊢ b ∈ Icc a b

Tactic:
]

Goal state after:
case hx
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
lb : β
inst✝ : T2Space β
hab : a < b
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
⊢ b ∈ Icc a b
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case succ
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
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ ((⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 i fun z ↦ f (a • z))) v)
      x✝ =
    (a ^ (i + 1) •
        (⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 i f)) (a • v))
      x✝

Tactic:
]

Goal state after:
case succ
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
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ ((⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 i fun z ↦ f (a • z))) v)
      x✝ =
    (a ^ (i + 1) •
        (⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 i f)) (a • v))
      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 g : 𝕜 → F
x : 𝕜
hfg : f =ᶠ[𝓝 x] g
⊢ ((iteratedFDerivWithin 𝕜 n g Set.univ x) fun x ↦ 1) = (iteratedFDerivWithin 𝕜 n g Set.univ x) fun x ↦ 1

Tactic:
]

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

Goal state before:
pp: case succ
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
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ ((a ^ (i + 1) • fderiv 𝕜 (iteratedFDeriv 𝕜 i f) (a • v)) (x✝ 0)) (tail x✝) =
    a ^ (i + 1) • ((fderiv 𝕜 (iteratedFDeriv 𝕜 i f) (a • v)) (x✝ 0)) (tail x✝)

pp: case succ.h
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
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ DifferentiableAt 𝕜 (fun x ↦ iteratedFDeriv 𝕜 i f (a • x)) v

Tactic:
]

Goal state after:
case succ
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
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ ((a ^ (i + 1) • fderiv 𝕜 (iteratedFDeriv 𝕜 i f) (a • v)) (x✝ 0)) (tail x✝) =
    a ^ (i + 1) • ((fderiv 𝕜 (iteratedFDeriv 𝕜 i f) (a • v)) (x✝ 0)) (tail x✝)

case succ.h
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
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ DifferentiableAt 𝕜 (fun x ↦ iteratedFDeriv 𝕜 i f (a • x)) v
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case insert
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
i : ι
t : Finset ι
hi : i ∉ t
IH :
  (∀ i ∈ t, ContDiffWithinAt 𝕜 (↑n) (f i) s x) →
    iteratedDerivWithin n (∑ i ∈ t, f i) s x = ∑ i ∈ t, iteratedDerivWithin n (f i) s x
hf : ContDiffWithinAt 𝕜 (↑n) (f i) s x ∧ ∀ x_1 ∈ t, ContDiffWithinAt 𝕜 (↑n) (f x_1) s x
⊢ iteratedDerivWithin n (∑ i ∈ insert i t, f i) s x = ∑ i ∈ insert i t, iteratedDerivWithin n (f i) s x

Tactic:
]

Goal state after:
case insert
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
i : ι
t : Finset ι
hi : i ∉ t
IH :
  (∀ i ∈ t, ContDiffWithinAt 𝕜 (↑n) (f i) s x) →
    iteratedDerivWithin n (∑ i ∈ t, f i) s x = ∑ i ∈ t, iteratedDerivWithin n (f i) s x
hf : ContDiffWithinAt 𝕜 (↑n) (f i) s x ∧ ∀ x_1 ∈ t, ContDiffWithinAt 𝕜 (↑n) (f x_1) s x
⊢ iteratedDerivWithin n (∑ i ∈ insert i t, f i) s x = ∑ i ∈ insert i t, iteratedDerivWithin n (f i) s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case insert
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
i : ι
t : Finset ι
hi : i ∉ t
hf : ContDiffWithinAt 𝕜 (↑n) (f i) s x ∧ ∀ x_1 ∈ t, ContDiffWithinAt 𝕜 (↑n) (f x_1) s x
IH :
  (∀ i ∈ t, ContDiffWithinAt 𝕜 (↑n) (f i) s x) →
    iteratedDerivWithin n (fun a ↦ ∑ c ∈ t, f c a) s x = ∑ i ∈ t, iteratedDerivWithin n (f i) s x
⊢ iteratedDerivWithin n (f i) s x + ∑ i ∈ t, iteratedDerivWithin n (f i) s x =
    iteratedDerivWithin n (f i) s x + ∑ i ∈ t, iteratedDerivWithin n (f i) s x

Tactic:
]

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

Goal state before:
pp: case succ.h
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
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ DifferentiableAt 𝕜 (iteratedFDeriv 𝕜 i f ∘ fun x ↦ a • x) v

Tactic:
]

Goal state after:
case succ.h
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
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ DifferentiableAt 𝕜 (iteratedFDeriv 𝕜 i f ∘ fun x ↦ a • x) v
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 : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : f c = f a
hC : ¬f C = f a
h : a = C
⊢ f C = f C

Tactic:
]

Goal state after:
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 : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : f c = f a
hC : ¬f C = f a
h : C = b
⊢ f b = f b

Tactic:
]

Goal state after:
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 : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : ¬f c = f a
h : a = c
⊢ f c = f c

Tactic:
]

Goal state after: