mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: case mp.refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
lt : y ∈ eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (p.sum (x + y - x))

Tactic:
simp only [add_sub_cancel_left]

Goal state after:
case mp.refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
lt : y ∈ eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (p.sum y)
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
y : F
x : E
hg : AnalyticAt 𝕜 g (f x)
hf : AnalyticAt 𝕜 f x
hy : f x = y
⊢ AnalyticAt 𝕜 (g ∘ f) x

Tactic:
exact hg.comp hf

Goal state after:
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
y : F
x : E
hg : CPolynomialAt 𝕜 g (f x)
hf : CPolynomialAt 𝕜 f x
hy : f x = y
⊢ CPolynomialAt 𝕜 (g ∘ f) x

Tactic:
exact hg.comp hf

Goal state after:
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case zero.succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
IH :
  ∀ (hi : i < b.length),
    0 < b.blocksFun ⟨i, hi⟩ →
      (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum
hi : i + 1 < b.length
hj : 0 < b.blocksFun ⟨i + 1, hi⟩
A : i < b.length
B : i < (List.map sum (a.blocks.splitWrtComposition b)).length
C : 0 < b.blocksFun ⟨i, A⟩
this : take (take i b.blocks).sum a.blocks = take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)
⊢ (take (take (i + 1) b.blocks).sum a.blocks).sum =
    (take (take i b.blocks).sum a.blocks).sum + (List.map sum (a.blocks.splitWrtComposition b))[i]

Tactic:
this,

Goal state after:
case zero.succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
IH :
  ∀ (hi : i < b.length),
    0 < b.blocksFun ⟨i, hi⟩ →
      (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum
hi : i + 1 < b.length
hj : 0 < b.blocksFun ⟨i + 1, hi⟩
A : i < b.length
B : i < (List.map sum (a.blocks.splitWrtComposition b)).length
C : 0 < b.blocksFun ⟨i, A⟩
this : take (take i b.blocks).sum a.blocks = take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)
⊢ (take (take (i + 1) b.blocks).sum a.blocks).sum =
    (take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)).sum +
      (List.map sum (a.blocks.splitWrtComposition b))[i]
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  j < b.blocksFun ⟨i, hi⟩ →
    a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < (a.sigmaCompositionAux b ⟨i, ⋯⟩).length
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ a.sizeUpTo (b.sizeUpTo i + (j + 1)) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo (j + 1)

Tactic:
this,

Goal state after:
case succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  j < b.blocksFun ⟨i, hi⟩ →
    a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < (a.sigmaCompositionAux b ⟨i, ⋯⟩).length
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ a.sizeUpTo (b.sizeUpTo i + j).succ = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo (j + 1)
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: case inr
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
r : ℝ≥0∞
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
h : ∀ (i : ι), r ≤ (p i).radius
r' : ℝ≥0
hr'✝ : ↑r' < r
C : ι → ℝ
C_pos : ∀ (i : ι), C i > 0
hC : ∀ (i : ι) (n : ℕ), ‖p i n‖ * ↑r' ^ n ≤ C i
D : ℝ
D_nonneg : D ≥ 0
hD : ∀ (i : ι), C i ≤ D
n : ℕ
hr' : 0 < ↑r' ^ n
⊢ ‖ContinuousMultilinearMap.pi fun i ↦ p i n‖ * ↑r' ^ n ≤ D

Tactic:
← le_div_iff₀ hr',

Goal state after:
case inr
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
r : ℝ≥0∞
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
h : ∀ (i : ι), r ≤ (p i).radius
r' : ℝ≥0
hr'✝ : ↑r' < r
C : ι → ℝ
C_pos : ∀ (i : ι), C i > 0
hC : ∀ (i : ι) (n : ℕ), ‖p i n‖ * ↑r' ^ n ≤ C i
D : ℝ
D_nonneg : D ≥ 0
hD : ∀ (i : ι), C i ≤ D
n : ℕ
hr' : 0 < ↑r' ^ n
⊢ ‖ContinuousMultilinearMap.pi fun i ↦ p i n‖ ≤ D / ↑r' ^ n
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case e'_3
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n✝ n : ℕ
c : Composition n
d : (i : Fin c.length) → Composition (c.blocksFun i)
this : List.map sum (ofFn fun i ↦ (d i).blocks) = c.blocks
e_2✝ :
  ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.gather
        { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ }).length =
    c.length
⊢ (fun i ↦
      ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.sigmaCompositionAux
          { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ } i).blocks) ≍
    fun i ↦ (d i).blocks

Tactic:
rw [Fin.heq_fun_iff]
      · intro i
        dsimp [Composition.sigmaCompositionAux]
        rw [getElem_of_eq (splitWrtComposition_flatten _ _ _)]
        · simp only [List.getElem_ofFn]
        · simp only [map_ofFn, Function.comp_def]
      · congr

Goal state after:
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
z : Aˣ
hA : Nontrivial A
f1 : A → A := fun a ↦ a * z.inv
f2 : A → A := fun b ↦ (1 - b)⁻¹ʳ
f3 : A → A := fun c ↦ 1 - z.inv * c
this✝ : Metric.ball ↑z ‖↑z⁻¹‖⁻¹ ∈ 𝓝 ↑z
y : A
hy : ‖y - ↑z‖ < ‖↑z⁻¹‖⁻¹
this : y = ↑(z.ofNearby y hy)
⊢ (f1 ∘ f2 ∘ f3) y = y⁻¹ʳ

Tactic:
this,

Goal state after:
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
z : Aˣ
hA : Nontrivial A
f1 : A → A := fun a ↦ a * z.inv
f2 : A → A := fun b ↦ (1 - b)⁻¹ʳ
f3 : A → A := fun c ↦ 1 - z.inv * c
this✝ : Metric.ball ↑z ‖↑z⁻¹‖⁻¹ ∈ 𝓝 ↑z
y : A
hy : ‖y - ↑z‖ < ‖↑z⁻¹‖⁻¹
this : y = ↑(z.ofNearby y hy)
⊢ (f1 ∘ f2 ∘ f3) ↑(z.ofNearby y hy) = (↑(z.ofNearby y hy))⁻¹ʳ
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

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
iso : E ≃L[𝕜] F
f : G → E
x : G
⊢ DifferentiableWithinAt 𝕜 (⇑iso ∘ f) univ x ↔ DifferentiableWithinAt 𝕜 f univ x

Tactic:
iso.comp_differentiableWithinAt_iff

Goal state after:
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
iso : E ≃L[𝕜] F
f : G → E
x : G
⊢ DifferentiableWithinAt 𝕜 f univ x ↔ DifferentiableWithinAt 𝕜 f univ x
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

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
iso : E ≃L[𝕜] F
f : G → E
s : Set G
x : G
f' : G →L[𝕜] F
⊢ HasFDerivWithinAt (⇑iso ∘ f) f' s x ↔ HasFDerivWithinAt (⇑iso ∘ f) (ContinuousLinearMap.id 𝕜 F ∘SL f') s x

Tactic:
ContinuousLinearMap.id_comp

Goal state after:
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
iso : E ≃L[𝕜] F
f : G → E
s : Set G
x : G
f' : G →L[𝕜] F
⊢ HasFDerivWithinAt (⇑iso ∘ f) f' s x ↔ HasFDerivWithinAt (⇑iso ∘ f) f' s x
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
h : HasFDerivWithinAt f f' s x
hf' : ∃ C, AntilipschitzWith C ⇑f'
t : Set F
ht : ¬AccPt (f x) (𝓟 t)
⊢ ∀ᶠ (z : E) in 𝓝[s \ {x}] x, f z ∉ t

Tactic:
accPt_iff_frequently_nhdsNE,

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
h : HasFDerivWithinAt f f' s x
hf' : ∃ C, AntilipschitzWith C ⇑f'
t : Set F
ht : ¬∃ᶠ (y : F) in 𝓝[≠] f x, y ∈ t
⊢ ∀ᶠ (z : E) in 𝓝[s \ {x}] x, f z ∉ t
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] F)

Tactic:
wlog hF : T2Space F generalizing F

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] F)

𝕜 : Type u_1
E : Type u_2
F✝ : Type u_3
ι : Type u_4
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜 F✝
inst✝⁸ : UniformSpace F✝
inst✝⁷ : IsUniformAddGroup F✝
inst✝⁶ : ContinuousSMul 𝕜 E
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
F : Type u_3
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : UniformSpace F
inst✝² : IsUniformAddGroup F
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
hF : T2Space F
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] F)
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F✝ : Type u_3
ι : Type u_4
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜 F✝
inst✝⁸ : UniformSpace F✝
inst✝⁷ : IsUniformAddGroup F✝
inst✝⁶ : ContinuousSMul 𝕜 E
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
F : Type u_3
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : UniformSpace F
inst✝² : IsUniformAddGroup F
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
hF : T2Space F
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] F)

Tactic:
have := ContinuousMultilinearMap.completeSpace (F := F) h

Goal state after:
𝕜 : Type u_1
E : Type u_2
F✝ : Type u_3
ι : Type u_4
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜 F✝
inst✝⁸ : UniformSpace F✝
inst✝⁷ : IsUniformAddGroup F✝
inst✝⁶ : ContinuousSMul 𝕜 E
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
F : Type u_3
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : UniformSpace F
inst✝² : IsUniformAddGroup F
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
hF : T2Space F
this : CompleteSpace (ContinuousMultilinearMap 𝕜 (fun i ↦ E) F)
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] F)
Module: Mathlib.Analysis.Normed.Group.Completion

Goal state before:
pp: case hp
E : Type u_1
inst✝ : SeminormedAddCommGroup E
⊢ IsClosed {x | dist x.1 x.2 = ‖-x.1 + x.2‖}

Tactic:
refine isClosed_eq (Completion.uniformContinuous_extension₂ _).continuous ?_

Goal state after:
case hp
E : Type u_1
inst✝ : SeminormedAddCommGroup E
⊢ Continuous fun x ↦ ‖-x.1 + x.2‖
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hu : UniqueDiffOn 𝕜 (insert x s)
z : E
hz : z ∈ insert x s ∩ Metric.eball x r
⊢ fderivWithin 𝕜 f (insert x s) z = (fun z ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin (z - x) 1)) z

Tactic:
dsimp only
    rw [← h.fderivWithin_eq _ _ hu, add_sub_cancel]
    · simpa only [edist_eq_enorm_sub, Metric.mem_eball] using! hz.2
    · simpa using! hz.1

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
n : ℕ
c a : 𝕜
this : (fun z ↦ f (c - z)) = fun z ↦ (fun w ↦ f (c + w)) (-z)
⊢ ((iteratedFDerivWithin 𝕜 n (fun z ↦ f (c - z)) s a) fun x ↦ 1) =
    ((-1) ^ n • iteratedFDerivWithin 𝕜 n f (c +ᵥ -s) (c - a)) fun x ↦ 1

Tactic:
this,

Goal state after:
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
n : ℕ
c a : 𝕜
this : (fun z ↦ f (c - z)) = fun z ↦ (fun w ↦ f (c + w)) (-z)
⊢ ((iteratedFDerivWithin 𝕜 n (fun z ↦ (fun w ↦ f (c + w)) (-z)) s a) fun x ↦ 1) =
    ((-1) ^ n • iteratedFDerivWithin 𝕜 n f (c +ᵥ -s) (c - a)) fun x ↦ 1
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: case pos
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
f : F
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun y ↦ c y • f) s x = derivWithin c s x • f

Tactic:
exact (hc.hasDerivWithinAt.smul_const f).derivWithin hsx

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : AnalyticOnNhd 𝕜 f s
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ AnalyticOnNhd 𝕜 (fderiv 𝕜 f) s

Tactic:
rw [← hs.analyticOn_iff_analyticOnNhd] at h ⊢
  exact (h.fderivWithin hs.uniqueDiffOn).congr (fun x hx ↦ (fderivWithin_of_isOpen hs hx).symm)

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Polynomial

Goal state before:
pp: case add
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
p✝ : 𝕜[X]
x : 𝕜
p q : 𝕜[X]
hp : HasStrictDerivAt (fun x ↦ eval x p) (eval x (derivative p)) x
hq : HasStrictDerivAt (fun x ↦ eval x q) (eval x (derivative q)) x
⊢ HasStrictDerivAt (fun x ↦ eval x (p + q)) (eval x (derivative (p + q))) x

Tactic:
| add p q hp hq =>

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fa : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)
⊢ HasStrictFDerivAt (fun fx ↦ fx.1 fx.2)
    (apply 𝕜 E F fa.2 ∘SL ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i) +
      fa.1.linearDeriv fa.2 ∘SL ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
    fa

Tactic:
let f := ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F)
    |>.continuousMultilinearMapOption
  have Hf := (f.hasStrictFDerivAt (fun _ ↦ fa)).comp (f := fun fx _ ↦ fx) fa
    (hasStrictFDerivAt_pi.2 fun _ ↦ hasStrictFDerivAt_id _)
  convert! Hf using 1
  ext g
  · suffices ∑ i, fa.1 (Function.update fa.2 i 0) =
        ∑ i, fa.1 fun j ↦ (Function.update (fun _ ↦ fa) (some i) (g, 0) (some j)).2 j by
      simpa [f, ContinuousLinearMap.continuousMultilinearMapOption]
    congr with i
    congr with j
    rcases eq_or_ne j i with rfl | hij <;> simp [*]
  · suffices ∑ i, fa.1 (Function.update fa.2 i (g i)) =
        ∑ x, fa.1 fun i ↦ (Function.update (fun x ↦ fa) (some x) (0, g) (some i)).2 i by
      simpa [f, ContinuousLinearMap.continuousMultilinearMapOption]
    congr with i
    congr with j
    rcases eq_or_ne j i with rfl | hij <;> simp [*]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ
x v : (i : ι) → E i
m : Fin n → (i : ι) → E i
e : Fin n.succ ↪ ι
k : ι
hke : k ∈ range ⇑e
⊢ (if h : k ∈ range ⇑e then Fin.cons v m (e.toEquivRange.symm ⟨k, ⋯⟩) k else x k) =
    if h : k ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst then
      m (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨k, h⟩) k
    else
      Function.update ((Pi.compRightL 𝕜 E Subtype.val) x) ((Equiv.embeddingFinSucc n ι) e).snd
        (v ↑((Equiv.embeddingFinSucc n ι) e).snd) ⟨k, h⟩

pp: case neg
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ
x v : (i : ι) → E i
m : Fin n → (i : ι) → E i
e : Fin n.succ ↪ ι
k : ι
hke : k ∉ range ⇑e
⊢ (if h : k ∈ range ⇑e then Fin.cons v m (e.toEquivRange.symm ⟨k, ⋯⟩) k else x k) =
    if h : k ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst then
      m (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨k, h⟩) k
    else
      Function.update ((Pi.compRightL 𝕜 E Subtype.val) x) ((Equiv.embeddingFinSucc n ι) e).snd
        (v ↑((Equiv.embeddingFinSucc n ι) e).snd) ⟨k, h⟩

Tactic:
· simp only [hke, ↓reduceDIte]
      split_ifs with hkf
      · simp only [← Equiv.succ_embeddingFinSucc_fst_symm_apply e hkf hke, Fin.cons_succ]
      · obtain rfl : k = e 0 := by
          rcases hke with ⟨j, rfl⟩
          simpa using hkf
        simp only [Function.Embedding.toEquivRange_symm_apply_self, Fin.cons_zero, Function.update,
          Pi.compRightL_apply]
        split_ifs with h
        · congr!
        · exfalso
          apply h
          simp_rw [← Equiv.embeddingFinSucc_snd e]

Goal state after:
case neg
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ
x v : (i : ι) → E i
m : Fin n → (i : ι) → E i
e : Fin n.succ ↪ ι
k : ι
hke : k ∉ range ⇑e
⊢ (if h : k ∈ range ⇑e then Fin.cons v m (e.toEquivRange.symm ⟨k, ⋯⟩) k else x k) =
    if h : k ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst then
      m (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨k, h⟩) k
    else
      Function.update ((Pi.compRightL 𝕜 E Subtype.val) x) ((Equiv.embeddingFinSucc n ι) e).snd
        (v ↑((Equiv.embeddingFinSucc n ι) e).snd) ⟨k, h⟩
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
L : Filter 𝕜
⊢ Tendsto (fun y ↦ slope f x y - f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) ↔ Tendsto (slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 f')

Tactic:
by
      rw [← nhds_translation_sub f', tendsto_comap_iff]; rfl

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : ι → α → ℂ
p : Filter ι
g : α → ℂ
hf : TendstoUniformly f g p
⊢ TendstoUniformly (fun n x ↦ (f n x).re) (fun y ↦ (g y).re) p

Tactic:
by
  apply UniformContinuous.comp_tendstoUniformly uniformContinuous_re hf

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: c : ℂ
R θ : ℝ
⊢ circleMap c R θ - c = circleMap 0 R θ

Tactic:
by
  simp [circleMap]

Goal state after:
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : E
n : ℕ
z : 𝕜
a : ℕ → E
hs : HasSum (fun m ↦ z ^ m • a m) s
ha : ∀ k < n, a k = 0
hn : n > 0
h : ¬z = 0
h1 : ∑ i ∈ Finset.range n, z ^ i • a i = 0
⊢ HasSum (fun m ↦ z ^ (m + n) • a (m + n)) s

Tactic:
simpa [h1] using (hasSum_nat_add_iff' n).mpr hs

Goal state after:
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
⊢ (swap dslope z₀)^[p.order] f z₀ ≠ 0

Tactic:
rw [← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
⊢ (fslope^[p.order] p 0) 1 ≠ 0