mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Homology.Nondegenerate

Goal state before:
pp: case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasCoproducts C
inst✝ : Preadditive C
X : SSet
R : C
x : X _⦋0⦌
hx : x ∈ X.degenerate 0
⊢ X.ιChainComplex x ≫ PInfty.f 0 = 0

Tactic:
simp at hx

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplicesColimit

Goal state before:
pp: X : SSet
⊢ ∀ (n : ℕ), X.nonDegenerate n ⊆ (⨆ i, i.subcomplex).obj (op ⦋n⦌)

Tactic:
intro n x hx

Goal state after:
X : SSet
n : ℕ
x : X _⦋n⦌
hx : x ∈ X.nonDegenerate n
⊢ x ∈ (⨆ i, i.subcomplex).obj (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: X : TopCat
f : SSet.toTop.obj Δ[0] ⟶ X
this : sSetTopAdj.unit.app Δ[0] = SSet.const (TopCat.toSSetObj₀Equiv.symm default)
⊢ sSetTopAdj.unit.app Δ[0] ≫ TopCat.toSSet.map f =
    SSet.const
      ((X.toSSetObjEquiv (op ⦋0⦌)).symm { toFun := fun x ↦ (ConcreteCategory.hom f) default, continuous_toFun := ⋯ })

Tactic:
this

Goal state after:
X : TopCat
f : SSet.toTop.obj Δ[0] ⟶ X
this : sSetTopAdj.unit.app Δ[0] = SSet.const (TopCat.toSSetObj₀Equiv.symm default)
⊢ SSet.const (TopCat.toSSetObj₀Equiv.symm default) ≫ TopCat.toSSet.map f =
    SSet.const
      ((X.toSSetObjEquiv (op ⦋0⦌)).symm { toFun := fun x ↦ (ConcreteCategory.hom f) default, continuous_toFun := ⋯ })
Module: Mathlib.AlgebraicTopology.SingularHomology.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasCoproducts C
inst✝¹ : Preadditive C
n : ℕ
inst✝ : HasPullbacks C
X : C
X✝ Y✝ : TopCat
f : X✝ ⟶ Y✝
x✝ : Mono f
⊢ ((SimplicialObject.whiskering (Type w) C).obj (sigmaConst.obj X)).PreservesMonomorphisms

Tactic:
dsimp [SSet, SimplicialObject.whiskering, SimplicialObject]

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasCoproducts C
inst✝¹ : Preadditive C
n : ℕ
inst✝ : HasPullbacks C
X : C
X✝ Y✝ : TopCat
f : X✝ ⟶ Y✝
x✝ : Mono f
⊢ ((Functor.whiskeringRight SimplexCategoryᵒᵖ (Type w) C).obj (sigmaConst.obj X)).PreservesMonomorphisms
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: case e_6
⊢ stdSimplex.map (⇑(ConcreteCategory.hom (SimplexCategory.δ 1))) (⦋0⦌.toTopHomeo default) = ⟨Pi.single 0 1, ⋯⟩

Tactic:
Subsingleton.elim (⦋0⦌.toTopHomeo default) (stdSimplex.vertex 0),

Goal state after:
case e_6
⊢ stdSimplex.map (⇑(ConcreteCategory.hom (SimplexCategory.δ 1))) (stdSimplex.vertex 0) = ⟨Pi.single 0 1, ⋯⟩
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: case e_6
⊢ stdSimplex.map (⇑(ConcreteCategory.hom (SimplexCategory.δ 0))) (⦋0⦌.toTopHomeo default) = ⟨Pi.single 1 1, ⋯⟩

Tactic:
Subsingleton.elim (⦋0⦌.toTopHomeo default) (stdSimplex.vertex 0),

Goal state after:
case e_6
⊢ stdSimplex.map (⇑(ConcreteCategory.hom (SimplexCategory.δ 0))) (stdSimplex.vertex 0) = ⟨Pi.single 1 1, ⋯⟩
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: ⊢ (ConcreteCategory.hom (toSSetObjI.app (op ⦋0⦌))) (const 1 1 (op ⦋0⦌)) = TopCat.toSSetObj₀Equiv.symm 1

Tactic:
by
  apply yonedaEquiv.symm.injective
  trans stdSimplex.δ 0 ≫ toSSetObjI
  · simp [← yonedaEquiv_symm_comp, stdSimplex.δ_zero_eq_const]
  · simp

Goal state after:
Module: Mathlib.Analysis.Normed.Field.WithAbs

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : Semiring S
inst✝³ : PartialOrder S
inst✝² : CommRing R
T : Type u_3
inst✝¹ : Field T
inst✝ : Algebra R T
w : AbsoluteValue T ℝ
r : R
⊢ UniformContinuous fun x ↦ r • x

Tactic:
[

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : Semiring S
inst✝³ : PartialOrder S
inst✝² : CommRing R
T : Type u_3
inst✝¹ : Field T
inst✝ : Algebra R T
w : AbsoluteValue T ℝ
r : R
⊢ UniformContinuous fun x ↦ r • x
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Zero E
inst✝² : MulActionWithZero 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : 1 < ‖c‖
cn_ne : ∀ (n : ℕ), c ^ n ≠ 0
x : E
⊢ Tendsto (fun n ↦ (c ^ n)⁻¹) atTop (𝓝 0)

Tactic:
[

Goal state after:
E : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Zero E
inst✝² : MulActionWithZero 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : 1 < ‖c‖
cn_ne : ∀ (n : ℕ), c ^ n ≠ 0
x : E
⊢ Tendsto (fun n ↦ (c ^ n)⁻¹) atTop (𝓝 0)
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddGroup E
inst✝³ : MulActionWithZero 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : IsOpen[inst✝²] s
x : E
hx : x ∈ s
⊢ #↑s = #E

Tactic:
exact cardinal_eq_of_mem_nhds 𝕜 (hs.mem_nhds hx)

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι : Type u_3
inst✝ : Finite ι
v : ι → AbsoluteValue R S
w : AbsoluteValue R S
a b : R
i : ι
ha : 1 < (v i) a
haj : ∀ (j : ι), j ≠ i → (v j) a < 1
haw : w a = 1
hb : 1 < (v i) b
hbw : w b < 1
c : ℕ → R := fun n ↦ a ^ n * b
hcᵢ : Tendsto (fun n ↦ (v i) (c n)) atTop atTop
hcⱼ : ∀ (j : ι), j ≠ i → Tendsto (fun n ↦ (v j) (c n)) atTop (𝓝 0)
⊢ ∃ k, 1 < (v i) k ∧ (∀ (j : ι), j ≠ i → (v j) k < 1) ∧ w k < 1

Tactic:
[

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι : Type u_3
inst✝ : Finite ι
v : ι → AbsoluteValue R S
w : AbsoluteValue R S
a b : R
i : ι
ha : 1 < (v i) a
haj : ∀ (j : ι), j ≠ i → (v j) a < 1
haw : w a = 1
hb : 1 < (v i) b
hbw : w b < 1
c : ℕ → R := fun n ↦ a ^ n * b
hcᵢ : Tendsto (fun n ↦ (v i) (c n)) atTop atTop
hcⱼ : ∀ (j : ι), j ≠ i → Tendsto (fun n ↦ (v j) (c n)) atTop (𝓝 0)
⊢ ∃ k, 1 < (v i) k ∧ (∀ (j : ι), j ≠ i → (v j) k < 1) ∧ w k < 1
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι : Type u_3
inst✝ : Finite ι
v : ι → AbsoluteValue R S
w : AbsoluteValue R S
a b : R
i : ι
ha : 1 < (v i) a
haj : ∀ (j : ι), j ≠ i → (v j) a < 1
haw : 1 < w a
hb : 1 < (v i) b
hbw : w b < 1
c : ℕ → R := fun n ↦ 1 / (1 + a⁻¹ ^ n) * b
hcᵢ : Tendsto (fun n ↦ (v i) (c n)) atTop (𝓝 ((v i) b))
hcⱼ : ∀ (j : ι), j ≠ i → Tendsto (fun n ↦ (v j) (c n)) atTop (𝓝 0)
hcₙ : Tendsto (fun n ↦ w (c n)) atTop (𝓝 (w b))
⊢ ∃ k, 1 < (v i) k ∧ (∀ (j : ι), j ≠ i → (v j) k < 1) ∧ w k < 1

Tactic:
[

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι : Type u_3
inst✝ : Finite ι
v : ι → AbsoluteValue R S
w : AbsoluteValue R S
a b : R
i : ι
ha : 1 < (v i) a
haj : ∀ (j : ι), j ≠ i → (v j) a < 1
haw : 1 < w a
hb : 1 < (v i) b
hbw : w b < 1
c : ℕ → R := fun n ↦ 1 / (1 + a⁻¹ ^ n) * b
hcᵢ : Tendsto (fun n ↦ (v i) (c n)) atTop (𝓝 ((v i) b))
hcⱼ : ∀ (j : ι), j ≠ i → Tendsto (fun n ↦ (v j) (c n)) atTop (𝓝 0)
hcₙ : Tendsto (fun n ↦ w (c n)) atTop (𝓝 (w b))
⊢ ∃ k, 1 < (v i) k ∧ (∀ (j : ι), j ≠ i → (v j) k < 1) ∧ w k < 1
Module: Mathlib.Analysis.Normed.Module.Connected

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : 1 < Module.rank ℝ E
x : E
r : ℝ
hr : 0 ≤ r
rpos : 0 < r
f : E → E := fun y ↦ x + (r * ‖y‖⁻¹) • y
A : ContinuousOn f {0}ᶜ
B : IsPathConnected {0}ᶜ
C : IsPathConnected (f '' {0}ᶜ)
this : f '' {0}ᶜ = sphere x r
⊢ IsPathConnected (sphere x r)

Tactic:
this

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : 1 < Module.rank ℝ E
x : E
r : ℝ
hr : 0 ≤ r
rpos : 0 < r
f : E → E := fun y ↦ x + (r * ‖y‖⁻¹) • y
A : ContinuousOn f {0}ᶜ
B : IsPathConnected {0}ᶜ
C : IsPathConnected (sphere x r)
this : f '' {0}ᶜ = sphere x r
⊢ IsPathConnected (sphere x r)
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : (↑f'symm.nnnorm)⁻¹ ≤ ↑c
this : y = f b
⊢ f b = y

Tactic:
this

Goal state after:
case inl
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : (↑f'symm.nnnorm)⁻¹ ≤ ↑c
this : y = f b
⊢ f b = f b
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
n : ℕ
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
h : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
⊢ ⊤ = ↑n ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z

Tactic:
simp only [ENat.top_ne_natCast, false_iff]
    contrapose h
    rw [← hf.exists_eventuallyEq_pow_smul_nonzero_iff]
    exact ⟨n, h⟩

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u : ι
⊢ Tendsto (fun i ↦ (WithAbs.equiv (v u)).symm (y i)) atTop (𝓝 (z u))

Tactic:
[

Goal state after:
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u : ι
⊢ Tendsto (fun i ↦ (WithAbs.equiv (v u)).symm (y i)) atTop (𝓝 (z u))
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: case neg
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this✝ : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u w : ι
x✝ : w ∈ Finset.univ
hw : ¬u = w
hgt : 1 < (v u) (a w)⁻¹
this : Tendsto (fun n ↦ (v u) (1 / (1 + (a w)⁻¹ ^ n))) atTop (𝓝 0)
⊢ Tendsto
    (fun i ↦
      (WithAbs.equiv (v u)).symm (1 / (1 + (a w)⁻¹ ^ i)) * (WithAbs.equiv (v u)).symm ((WithAbs.equiv (v w)) (z w)))
    atTop (𝓝 0)

Tactic:
[

Goal state after:
case neg
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this✝ : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u w : ι
x✝ : w ∈ Finset.univ
hw : ¬u = w
hgt : 1 < (v u) (a w)⁻¹
this : Tendsto (fun n ↦ (v u) (1 / (1 + (a w)⁻¹ ^ n))) atTop (𝓝 0)
⊢ Tendsto
    (fun i ↦
      (WithAbs.equiv (v u)).symm (1 / (1 + (a w)⁻¹ ^ i)) * (WithAbs.equiv (v u)).symm ((WithAbs.equiv (v w)) (z w)))
    atTop (𝓝 0)
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
g : 𝕜 → E
z₀ : 𝕜
f : 𝕜 → 𝕜
hg : analyticOrderAt g z₀ = ⊤
⊢ analyticOrderAt (f • g) z₀ = ⊤

Tactic:
rw [analyticOrderAt_eq_top, eventually_nhds_iff] at *
  obtain ⟨t, h₁t, h₂t, h₃t⟩ := hg
  exact ⟨t, fun y hy ↦ by simp [h₁t y hy], h₂t, h₃t⟩

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
inst✝¹ : CompleteSpace E
inst✝ : CharZero 𝕜
s : ℕ
hrne : s + 1 ≠ 0
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = (z - x) ^ (s + 1) • F z + f x
U : Set 𝕜
hUf : ∀ y ∈ U, f y = (y - x) ^ (s + 1) • F y + f x ∧ AnalyticAt 𝕜 F y
hUo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hUx : x ∈ U
this : ∀ y ∈ U, deriv f y = (y - x) ^ (s + 1) • deriv F y + (s + 1) • (y - x) ^ s • F y
⊢ AnalyticAt 𝕜 (fun y ↦ (s + 1) • (y - x) ^ s • F y) x

Tactic:
[

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
inst✝¹ : CompleteSpace E
inst✝ : CharZero 𝕜
s : ℕ
hrne : s + 1 ≠ 0
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = (z - x) ^ (s + 1) • F z + f x
U : Set 𝕜
hUf : ∀ y ∈ U, f y = (y - x) ^ (s + 1) • F y + f x ∧ AnalyticAt 𝕜 F y
hUo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hUx : x ∈ U
this : ∀ y ∈ U, deriv f y = (y - x) ^ (s + 1) • deriv F y + (s + 1) • (y - x) ^ s • F y
⊢ AnalyticAt 𝕜 (fun y ↦ (s + 1) • (y - x) ^ s • F y) x
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case top
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
hf' : deriv f x ≠ 0
h : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ⊤
⊢ ⊤ = 1

Tactic:
[

Goal state after:
case top
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
hf' : deriv f x ≠ 0
h : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ⊤
⊢ ⊤ = 1
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case a
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
r : ℕ
h : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ↑r
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z - f x = (z - x) ^ r • F z
hf' : 1 < r
⊢ deriv f x = 0

Tactic:
[

Goal state after:
case a
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
r : ℕ
h : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ↑r
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z - f x = (z - x) ^ r • F z
hf' : 1 < r
⊢ deriv f x = 0
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

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
s : Set E
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
A : PartialEquiv E F := hf.toPartialEquiv hc
Af : ∀ (z : E), ↑A z = f z
x' : E
x's : x' ∈ s
hx : f x' ∈ f '' s
y' : E
y's : y' ∈ s
hy : f y' ∈ f '' s
⊢ ↑‖↑f'.symm‖₊ * ‖f' (x' - y' - f'.symm (↑A x' - ↑A y'))‖ = ↑‖↑f'.symm‖₊ * ‖↑A y' - ↑A x' - f' (y' - x')‖

Tactic:
by
      congr 2
      simp only [ContinuousLinearEquiv.apply_symm_apply, map_sub]
      abel

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

Goal state before:
pp: x : ℝ
hx : 0 < x
this : Real.log = fun x ↦ (log ↑x).re
⊢ AnalyticAt ℝ Real.log x

Tactic:
this

Goal state after:
x : ℝ
hx : 0 < x
this : Real.log = fun x ↦ (log ↑x).re
⊢ AnalyticAt ℝ (fun x ↦ (log ↑x).re) x
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hzero : f z₀ = 0
n : ℕ
h : ∀ i < n, iteratedDeriv (i + 1) f z₀ = 0
k : ℕ
hk : k < n + 1
⊢ iteratedDeriv k f z₀ = 0

Tactic:
cases k with

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hzero : f z₀ = 0
⊢ analyticOrderAt (deriv f) z₀ = ⊤ ↔ analyticOrderAt f z₀ = ⊤

Tactic:
[

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hzero : f z₀ = 0
⊢ analyticOrderAt (deriv f) z₀ = ⊤ ↔ analyticOrderAt f z₀ = ⊤
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case succ
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
n : ℕ
f : 𝕜 → E
IH : analyticOrderAt (deriv f) z₀ = ↑n ↔ (∀ k < n, iteratedDeriv k (deriv f) z₀ = 0) ∧ iteratedDeriv n (deriv f) z₀ ≠ 0
hf : AnalyticAt 𝕜 f z₀
⊢ analyticOrderAt f z₀ = ↑(n + 1) ↔ (∀ k < n + 1, iteratedDeriv k f z₀ = 0) ∧ iteratedDeriv (n + 1) f z₀ ≠ 0

Tactic:
[

Goal state after:
case succ
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
n : ℕ
f : 𝕜 → E
IH : analyticOrderAt (deriv f) z₀ = ↑n ↔ (∀ k < n, iteratedDeriv k (deriv f) z₀ = 0) ∧ iteratedDeriv n (deriv f) z₀ ≠ 0
hf : AnalyticAt 𝕜 f z₀
⊢ analyticOrderAt f z₀ = ↑(n + 1) ↔ (∀ k < n + 1, iteratedDeriv k f z₀ = 0) ∧ iteratedDeriv (n + 1) f z₀ ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
habc : ∀ (kn : ℕ), ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
this : Tendsto (fun k ↦ (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k)) atTop (𝓝 1)
⊢ Tendsto (fun n ↦ ‖ordinaryHypergeometricCoefficient a b c n‖ / ‖ordinaryHypergeometricCoefficient a b c n.succ‖) atTop
    (𝓝 ↑1)

Tactic:
[

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
habc : ∀ (kn : ℕ), ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
this : Tendsto (fun k ↦ (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k)) atTop (𝓝 1)
⊢ Tendsto (fun n ↦ ‖ordinaryHypergeometricCoefficient a b c n‖ / ‖ordinaryHypergeometricCoefficient a b c n.succ‖) atTop
    (𝓝 ↑1)
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: case convert_5
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
habc : ∀ (kn : ℕ), ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
this : ∀ (k : ℕ), (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k) = (c + ↑k) / (a + ↑k) * ((1 + ↑k) / (b + ↑k))
⊢ Tendsto (fun k ↦ (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k)) atTop (𝓝 1)

Tactic:
[

Goal state after:
case convert_5
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
habc : ∀ (kn : ℕ), ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
this : ∀ (k : ℕ), (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k) = (c + ↑k) / (a + ↑k) * ((1 + ↑k) / (b + ↑k))
⊢ Tendsto (fun k ↦ (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k)) atTop (𝓝 1)
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: case convert_5
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
habc : ∀ (kn : ℕ), ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
this : ∀ (k : ℕ), (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k) = (c + ↑k) / (a + ↑k) * ((1 + ↑k) / (b + ↑k))
⊢ Tendsto (fun k ↦ (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k)) atTop (𝓝 1)

Tactic:
this

Goal state after:
case convert_5
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
habc : ∀ (kn : ℕ), ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
this : ∀ (k : ℕ), (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k) = (c + ↑k) / (a + ↑k) * ((1 + ↑k) / (b + ↑k))
⊢ Tendsto (fun k ↦ (c + ↑k) / (a + ↑k) * ((1 + ↑k) / (b + ↑k))) atTop (𝓝 1)
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
n : ℕ
hg : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • g z) x
⊢ (fun z ↦ (z - x) ^ max m n • (f + g) z) = fun z ↦
    (z - x) ^ (max m n - m) • (z - x) ^ m • f z + (z - x) ^ (max m n - n) • (z - x) ^ n • g z

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
n : ℕ
hg : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • g z) x
⊢ (fun z ↦ (z - x) ^ max m n • (f + g) z) = fun z ↦
    (z - x) ^ (max m n - m) • (z - x) ^ m • f z + (z - x) ^ (max m n - n) • (z - x) ^ n • g z
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
n : ℕ
hg : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • g z) x
this :
  (fun z ↦ (z - x) ^ max m n • (f + g) z) = fun z ↦
    (z - x) ^ (max m n - m) • (z - x) ^ m • f z + (z - x) ^ (max m n - n) • (z - x) ^ n • g z
⊢ AnalyticAt 𝕜 (fun z ↦ (z - x) ^ max m n • (f + g) z) x

Tactic:
this

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
n : ℕ
hg : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • g z) x
this :
  (fun z ↦ (z - x) ^ max m n • (f + g) z) = fun z ↦
    (z - x) ^ (max m n - m) • (z - x) ^ m • f z + (z - x) ^ (max m n - n) • (z - x) ^ n • g z
⊢ AnalyticAt 𝕜 (fun z ↦ (z - x) ^ (max m n - m) • (z - x) ^ m • f z + (z - x) ^ (max m n - n) • (z - x) ^ n • g z) x
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: case empty
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_5
s : Finset ι
F : ι → 𝕜 → 𝕜'
x : 𝕜
hf : ∀ σ ∈ ∅, MeromorphicAt (F σ) x
⊢ MeromorphicAt (∏ i ∈ ∅, F i) x

Tactic:
rw [Finset.prod_empty]

Goal state after:
case empty
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_5
s : Finset ι
F : ι → 𝕜 → 𝕜'
x : 𝕜
hf : ∀ σ ∈ ∅, MeromorphicAt (F σ) x
⊢ MeromorphicAt 1 x
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
z : 𝕜
⊢ (f - g) z = (f + -g) z

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
z : 𝕜
⊢ (f - g) z = (f + -g) z