mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Comp

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
f : E → F
f' : E →L[𝕜] F
L : Filter (E × E)
g : F → G
g' : F →L[𝕜] G
L' : Filter (F × F)
hg : HasFDerivAtFilter g g' L'
hf : HasFDerivAtFilter f f' L
hL : Tendsto (Prod.map f f) L L'
x✝ : E × E
⊢ (g ∘ f) x✝.1 - (g ∘ f) x✝.2 - (g' ∘SL f') (x✝.1 - x✝.2) =
    g (f x✝.1) - g (f x✝.2) - g' (f x✝.1 - f x✝.2) + g' (f x✝.1 - f x✝.2 - f' (x✝.1 - x✝.2))

Tactic:
simp

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

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
f : E → F
g : F → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hgf : HasFPowerSeriesAt (g ∘ f) (q.comp p) x
hf : HasFPowerSeriesAt f p x
hq : 0 < q.radius
⊢ x + 0 = x

Tactic:
simp

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

Goal state before:
pp: case e'_6
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
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
u : E →L[𝕜] F
f : F → G
pf : FormalMultilinearSeries 𝕜 F G
s : Set F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f pf s (u x) r
y✝ : E
hy1 : x + y✝ ∈ insert x (⇑u ⁻¹' s)
hy2 : y✝ ∈ Metric.eball 0 (r / ‖u‖ₑ)
⊢ (f ∘ ⇑u) (x + y✝) = f (u x + ↑u y✝)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
𝕜' : Type u_9
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
f : E → F
p : FormalMultilinearSeries 𝕜' E F
x : E
s : Set E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
n : ℕ
⊢ ‖FormalMultilinearSeries.restrictScalars 𝕜 p n‖ ≤ ‖p n‖

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
𝕜' : Type u_9
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
f : E → F
p : FormalMultilinearSeries 𝕜' E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
n : ℕ
⊢ ‖FormalMultilinearSeries.restrictScalars 𝕜 p n‖ ≤ ‖p n‖

Tactic:
simp

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
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
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v

Tactic:
let g : (Σ c : Composition n, ∀ i : Fin c.length, Composition (c.blocksFun i)) → H := fun c =>
    r c.1.length fun i : Fin c.1.length =>
      q (c.2 i).length (applyComposition p (c.2 i) (v ∘ c.1.embedding i))

Goal state after:
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
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
A : Type u_7
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ Tendsto (fun n ↦ ‖1‖ / ‖1‖) atTop (𝓝 ↑1)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
A : Type u_7
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ 1 = ↑1⁻¹

Tactic:
simp

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✝ : NormOneClass A
⊢ Tendsto (fun n ↦ ‖1‖ / ‖1‖) atTop (𝓝 ↑1)

Tactic:
simp

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

Goal state before:
pp: case hasSum
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
y : A
hy : ‖y‖ < 1
⊢ HasSum (fun n ↦ (formalMultilinearSeries_geometric 𝕜 A n) fun x ↦ y) (1 - (0 + y))⁻¹ʳ

Tactic:
simp only [zero_add, NormedRing.inverse_one_sub _ hy, Units.oneSub, Units.inv_mk,
      formalMultilinearSeries_geometric, ContinuousMultilinearMap.mkPiAlgebraFin_apply,
      List.ofFn_const, List.prod_replicate]

Goal state after:
case hasSum
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
y : A
hy : ‖y‖ < 1
⊢ HasSum (fun n ↦ y ^ n) (∑' (n : ℕ), y ^ n)
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✝ : NormOneClass A
⊢ Tendsto (fun n ↦ ‖(-1) ^ n‖ / ‖(-1) ^ n.succ‖) atTop (𝓝 ↑1)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : HasSummableGeomSeries A
inst✝ : Nontrivial A
x✝ : A
⊢ (1 + x✝)⁻¹ʳ = ((fun x ↦ (1 - x)⁻¹ʳ) ∘ ⇑(-ContinuousLinearMap.id 𝕜 A)) x✝

Tactic:
simp

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 := ⋯
f2 : A → A := ⋯
f3 : A → A := ⋯
⊢ 0 < ‖↑z⁻¹‖⁻¹

Tactic:
simp

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
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
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H := ⋯
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H := ⋯
a : Composition n
b : Composition a.length
a✝ : ⟨a, b⟩ ∈ Finset.univ
i : ℕ
hi1 : i < b.length
hi2 : i < (a.gather b).length
⊢ ∀ (i_1 : ℕ) (him : i_1 < b.blocksFun ⟨i, hi1⟩) (hin : i_1 < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length),
    (p.applyComposition ⟨a, b⟩.fst v ∘ ⇑(⟨a, b⟩.snd.embedding ⟨i, hi1⟩)) ⟨i_1, him⟩ =
      p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩)
        (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ⟨i_1, hin⟩

Tactic:
intro j

Goal state after:
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
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
a : Composition n
b : Composition a.length
a✝ : ⟨a, b⟩ ∈ Finset.univ
i : ℕ
hi1 : i < b.length
hi2 : i < (a.gather b).length
j : ℕ
⊢ ∀ (him : j < b.blocksFun ⟨i, hi1⟩) (hin : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length),
    (p.applyComposition ⟨a, b⟩.fst v ∘ ⇑(⟨a, b⟩.snd.embedding ⟨i, hi1⟩)) ⟨j, him⟩ =
      p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩)
        (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ⟨j, hin⟩
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)
⊢ ↑(Units.oneSub (1 - ↑z⁻¹ * y) ⋯) = ↑z⁻¹ * y

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
A : Type u_9
inst✝¹ : NormedCommRing A
inst✝ : NormedAlgebra 𝕜 A
f : α → E → A
c : E
s : Set E
N : Finset α
h : ∀ n ∈ N, AnalyticWithinAt 𝕜 (f n) s c
x✝ : E
⊢ (∏ n ∈ N, f n) x✝ = ∏ n ∈ N, f n x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
α : Type u_9
A : Type u_10
inst✝¹ : NormedCommRing A
inst✝ : NormedAlgebra 𝕜 A
f : α → E → A
c : E
N : Finset α
h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c
x✝ : E
⊢ (∏ n ∈ N, f n) x✝ = ∏ n ∈ N, f n x✝

Tactic:
simp

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

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
b : E × F → G
h : IsBoundedBilinearMap 𝕜 b
p : E × F
T : Type (max u_3 u_2) := (E × F) × E × F
⊢ ‖(0.2.1, 0.1.2)‖ = 0

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
R : Type u_4
inst✝⁴ : Monoid R
inst✝³ : DistribMulAction R F
inst✝² : SMulCommClass 𝕜 R F
inst✝¹ : ContinuousConstSMul R F
c : R
inst✝ : Invertible c
h : DifferentiableWithinAt 𝕜 (c • f) s x
⊢ f x = (⅟c • c • f) x

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : E → F
x✝ : E
s : Set E
R : Type u_4
inst✝⁴ : Monoid R
inst✝³ : DistribMulAction R F
inst✝² : SMulCommClass 𝕜 R F
inst✝¹ : ContinuousConstSMul R F
c : R
inst✝ : Invertible c
h : DifferentiableWithinAt 𝕜 (c • f) s x✝
x : E
⊢ f x = (⅟c • c • f) x

Tactic:
simp

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

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
R : Type u_4
inst✝³ : DivisionSemiring R
inst✝² : Module R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
hs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 (0 • f) s x = 0 • fderivWithin 𝕜 f s x

Tactic:
simp

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

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
x : E
ι : Type u_4
u : Finset ι
A : ι → E → F
A' : ι → E →L[𝕜] F
h : ∀ i ∈ u, HasStrictFDerivAt (A i) (A' i) x
x✝ : E
⊢ (∑ i ∈ u, A i) x✝ = ∑ i ∈ u, A i x✝

Tactic:
simp

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

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
L : Filter (E × E)
ι : Type u_4
u : Finset ι
A : ι → E → F
A' : ι → E →L[𝕜] F
h : ∀ i ∈ u, (fun p ↦ A i p.1 - A i p.2 - (A' i) (p.1 - p.2)) =o[L] fun p ↦ p.1 - p.2
x✝ : E × E
⊢ ∑ i ∈ u, A i x✝.1 - ∑ i ∈ u, A i x✝.2 - (∑ i ∈ u, A' i) (x✝.1 - x✝.2) =
    (∑ i ∈ u, fun p ↦ A i p.1 - A i p.2 - (A' i) (p.1 - p.2)) x✝

Tactic:
simp

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

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
L : Filter (E × E)
ι : Type u_4
u : Finset ι
A : ι → E → F
A' : ι → E →L[𝕜] F
h : ∀ i ∈ u, HasFDerivAtFilter (A i) (A' i) L
x✝ : E
⊢ (∑ i ∈ u, A i) x✝ = ∑ i ∈ u, A i x✝

Tactic:
simp

Goal state after:
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 : F → G
s : Set F
x : E
f' : F →L[𝕜] G
H : HasFDerivWithinAt (f ∘ ⇑iso) (f' ∘SL ↑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
A : f = (f ∘ ⇑iso) ∘ ⇑iso.symm
B : f' = (f' ∘SL ↑iso) ∘SL ↑iso.symm
⊢ HasFDerivWithinAt ((f ∘ ⇑iso) ∘ ⇑iso.symm) ((f' ∘SL ↑iso) ∘SL ↑iso.symm) s (iso x)

Tactic:
apply H.comp (iso x) iso.symm.hasFDerivWithinAt

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 : F → G
s : Set F
x : E
f' : F →L[𝕜] G
H : HasFDerivWithinAt (f ∘ ⇑iso) (f' ∘SL ↑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
A : f = (f ∘ ⇑iso) ∘ ⇑iso.symm
B : f' = (f' ∘SL ↑iso) ∘SL ↑iso.symm
⊢ MapsTo (⇑iso.symm) s (⇑iso ⁻¹' s)
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring A
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid N
inst✝⁷ : TopologicalSpace M
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ContinuousAdd N
inst✝⁴ : Module A M
inst✝³ : Module A N
inst✝² : Module R N
inst✝¹ : ContinuousConstSMul R N
inst✝ : SMulCommClass A R N
⊢ ∀ (x y : M [⋀^ι]→L[A] N), (x + y).toAlternatingMap = x.toAlternatingMap + y.toAlternatingMap

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring A
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid N
inst✝⁷ : TopologicalSpace M
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ContinuousAdd N
inst✝⁴ : Module A M
inst✝³ : Module A N
inst✝² : Module R N
inst✝¹ : ContinuousConstSMul R N
inst✝ : SMulCommClass A R N
⊢ ∀ (m : R) (x : M [⋀^ι]→L[A] N), (m • x).toAlternatingMap = (RingHom.id R) m • x.toAlternatingMap

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
g g' : M' [⋀^ι]→L[R] N
x✝ : ι → M
⊢ ((g + g').compContinuousLinearMap f) x✝ = (g.compContinuousLinearMap f + g'.compContinuousLinearMap f) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
c : R
g : M' [⋀^ι]→L[R] N
x✝ : ι → M
⊢ ((c • g).compContinuousLinearMap f) x✝ = ((RingHom.id R) c • g.compContinuousLinearMap f) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
c : R
f : N →L[R] N'
g : M [⋀^ι]→L[R] N
x✝ : ι → M
⊢ (f.compContinuousAlternatingMap (c • g)) x✝ = (c • f.compContinuousAlternatingMap g) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : IsTopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : ContinuousMultilinearMap R (fun x ↦ M) N
x✝ : ι → M
⊢ (let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ 0;
      { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ })
      x✝ =
    0 x✝

Tactic:
simp

Goal state after:
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
c : F
h : HasFDerivWithinAt f f' s x
hf' : ∃ C, AntilipschitzWith C ⇑f'
⊢ ∀ᶠ (z : E) in 𝓝[s \ {x}] x, f z ≠ c

Tactic:
rw [← eventually_map (m := f) (P := fun z ↦ z ≠ c)]

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
c : F
h : HasFDerivWithinAt f f' s x
hf' : ∃ C, AntilipschitzWith C ⇑f'
⊢ ∀ᶠ (b : F) in map f (𝓝[s \ {x}] x), b ≠ c
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
s : Set E
f' : E →L[𝕜] F
x : E
h : HasFDerivWithinAt f f' s x
y : E
hy : y ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
n : ι
hn : x + d n ∈ s
⊢ f (x + d n) = f x + (f (x + d n) - f x)

Tactic:
simp

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

Goal state before:
pp: case hds
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
s : Set E
f' : E →L[𝕜] F
x : E
h : HasFDerivWithinAt f f' s x
y : E
hy : y ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
⊢ ∀ᶠ (n : ι) in l, f x + (f (x + d n) - f x) ∈ f '' s

Tactic:
exact hds.mono fun n hn ↦ ⟨x + d n, hn, by simp⟩

Goal state after:
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
s : Set E
f' : E →L[𝕜] F
x : E
c : 𝕜ˣ
hc : ↑c ≠ 0
x✝ : E
⊢ (f' ∘SL ↑(ContinuousLinearEquiv.smulLeft c)) x✝ = (c • f') x✝

Tactic:
simp

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

Goal state before:
pp: case inl
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
s : Set E
x : E
⊢ fderivWithin 𝕜 (fun x ↦ f (0 • x)) s x = fderivWithin 𝕜 (0 • f) (0 • s) (0 • x)

Tactic:
simp

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

Goal state before:
pp: case inl
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
s : Set E
x : E
hs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 (fun x ↦ f (0 • x)) s x = 0 • fderivWithin 𝕜 f (0 • s) (0 • x)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
⊢ fderivWithin 𝕜 (fun a ↦ f (-a)) s x = -fderivWithin 𝕜 f (-s) (-x)

Tactic:
have t1 := fderivWithin_comp_smul_eq_fderivWithin_smul (-1 : 𝕜) (f := f) (s := s) (x := x)

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
t1 : fderivWithin 𝕜 (fun x ↦ f (-1 • x)) s x = fderivWithin 𝕜 (-1 • f) (-1 • s) (-1 • x)
⊢ fderivWithin 𝕜 (fun a ↦ f (-a)) s x = -fderivWithin 𝕜 f (-s) (-x)