mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: case a.refine_1
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝² : LinearOrder ι
f : P.RankFunction ι
inst✝¹ : OrderBot ι
inst✝ : SuccOrder ι
i : ι
hi : Order.IsSuccLimit i
⊢ A ≤ ⨆ (_ : ⊥ < i), f.filtration ⊥

Tactic:
exact le_trans (by simp) (le_iSup _ hi.bot_lt)

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.Limits

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u'
inst✝³ : Category.{v', u'} K
F : K ⥤ C ⥤ Type w'
c : Cone F
hc : (Y : C) → IsLimit (((evaluation C (Type w')).obj Y).mapCone c)
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
hK : HasCardinalLT (Arrow K) κ
J : Type w
inst✝¹ : SmallCategory J
inst✝ : IsCardinalFiltered J κ
X : J ⥤ C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
this : IsFiltered J
y : ↑(F ⋙ (evaluation C (Type w')).obj cX.pt).sections
hy : (Types.isLimitEquivSections (hc cX.pt)).symm y = x
j₀ : J
z : (k : K) → (F.obj k).obj (X.obj j₀)
hz : ∀ (k : K), ↑y k = (ConcreteCategory.hom ((F.obj k).map (cX.ι.app j₀))) (z k)
H :
  ∀ {k k' : K} (φ : k ⟶ k'),
    ∃ j f,
      (ConcreteCategory.hom ((X ⋙ F.obj k').map f)) ((ConcreteCategory.hom ((F.map φ).app (X.obj j₀))) (z k)) =
        (ConcreteCategory.hom ((X ⋙ F.obj k').map f)) (z k')
j : {k k' : K} → (k ⟶ k') → J := fun {k k'} φ ↦ ⋯.choose
g : {k k' : K} → (φ : k ⟶ k') → j₀ ⟶ j φ := fun {k k'} φ ↦ ⋯.choose
hg :
  ∀ {k k' : K} (φ : k ⟶ k'),
    (ConcreteCategory.hom ((F.obj k').map (X.map (g φ)))) ((ConcreteCategory.hom ((F.map φ).app (X.obj j₀))) (z k)) =
      (ConcreteCategory.hom ((F.obj k').map (X.map (g φ)))) (z k')
j₁ : J
α : j₀ ⟶ j₁
β : ⦃k k' : K⦄ → (φ : k ⟶ k') → j φ ⟶ j₁
hα : ∀ ⦃k k' : K⦄ (φ : k ⟶ k'), α = g φ ≫ β φ
k k' : K
φ : k ⟶ k'
⊢ (ConcreteCategory.hom ((F.obj k').map (X.map α))) ((ConcreteCategory.hom ((F.map φ).app (X.obj j₀))) (z k)) =
    (ConcreteCategory.hom ((F.obj k').map (X.map α))) (z k')

Tactic:
simp [hα φ, hg]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
α : Fin (n + 1) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • X.δ x.2 ≫ H.h x.1
β : Fin (n + 3) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • H.h x.2 ≫ Y.δ x.1
h₂ : ∀ (x : Fin (n + 1) × Fin (n + 2)), x.1.castSucc < x.2 → α x = -β (x.2.succ, x.1.castSucc)
S : Finset (Fin (n + 1) × Fin (n + 2)) := {x | x.1.castSucc < x.2}
γ₁ : Fin (n + 1) × Fin (n + 2) → Fin (n + 2 + 1) × Fin (n + 1 + 1) := fun x ↦ (x.2.castSucc, x.1.succ)
γ₂ : Fin (n + 1) × Fin (n + 2) → Fin (n + 2 + 1) × Fin (n + 1 + 1) := fun x ↦ (x.2.succ, x.1.castSucc)
γ₃ : Fin (n + 1) → Fin (n + 1 + 1 + 1) × Fin (n + 1 + 1) := fun i ↦ (i.castSucc.succ, i.succ)
γ₄ : Fin (n + 1) → Fin (n + 1 + 1 + 1) × Fin (n + 1 + 1) := fun i ↦ (i.castSucc.succ, i.castSucc)
hγ₁ : Function.Injective γ₁
hγ₂ : Function.Injective γ₂
hγ₃ : Function.Injective γ₃
hγ₄ : Function.Injective γ₄
eq₁ : H.h 0 ≫ Y.δ 0 = β (0, 0)
eq₂ : H.h (Fin.last (n + 1)) ≫ Y.δ (Fin.last (n + 1 + 1)) = -β (Fin.last (n + 2), Fin.last (n + 1))
eq₃ : ∑ x ∈ Sᶜ, α x = -∑ y ∈ Finset.image γ₁ Sᶜ, β y
eq₄ : ∑ x ∈ S, α x = -∑ y ∈ Finset.image γ₂ S, β y
eq₅ : ∑ x, β (γ₄ x) = -∑ x, β (γ₃ x)
h₁ : Disjoint (Finset.image γ₁ Sᶜ) (Finset.image γ₂ S)
⊢ H.h (Fin.last (n + 1)) ≫ Y.δ (Fin.last (n + 1 + 1)) = -∑ x, α x + -∑ x, β x + H.h 0 ≫ Y.δ 0

Tactic:
have h₂ : Disjoint (Finset.image γ₃ .univ) (Finset.image γ₄ .univ) := by
    rw [Finset.disjoint_iff_ne]
    grind

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
α : Fin (n + 1) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • X.δ x.2 ≫ H.h x.1
β : Fin (n + 3) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • H.h x.2 ≫ Y.δ x.1
h₂✝ : ∀ (x : Fin (n + 1) × Fin (n + 2)), x.1.castSucc < x.2 → α x = -β (x.2.succ, x.1.castSucc)
S : Finset (Fin (n + 1) × Fin (n + 2)) := {x | x.1.castSucc < x.2}
γ₁ : Fin (n + 1) × Fin (n + 2) → Fin (n + 2 + 1) × Fin (n + 1 + 1) := fun x ↦ (x.2.castSucc, x.1.succ)
γ₂ : Fin (n + 1) × Fin (n + 2) → Fin (n + 2 + 1) × Fin (n + 1 + 1) := fun x ↦ (x.2.succ, x.1.castSucc)
γ₃ : Fin (n + 1) → Fin (n + 1 + 1 + 1) × Fin (n + 1 + 1) := fun i ↦ (i.castSucc.succ, i.succ)
γ₄ : Fin (n + 1) → Fin (n + 1 + 1 + 1) × Fin (n + 1 + 1) := fun i ↦ (i.castSucc.succ, i.castSucc)
hγ₁ : Function.Injective γ₁
hγ₂ : Function.Injective γ₂
hγ₃ : Function.Injective γ₃
hγ₄ : Function.Injective γ₄
eq₁ : H.h 0 ≫ Y.δ 0 = β (0, 0)
eq₂ : H.h (Fin.last (n + 1)) ≫ Y.δ (Fin.last (n + 1 + 1)) = -β (Fin.last (n + 2), Fin.last (n + 1))
eq₃ : ∑ x ∈ Sᶜ, α x = -∑ y ∈ Finset.image γ₁ Sᶜ, β y
eq₄ : ∑ x ∈ S, α x = -∑ y ∈ Finset.image γ₂ S, β y
eq₅ : ∑ x, β (γ₄ x) = -∑ x, β (γ₃ x)
h₁ : Disjoint (Finset.image γ₁ Sᶜ) (Finset.image γ₂ S)
h₂ : Disjoint (Finset.image γ₃ Finset.univ) (Finset.image γ₄ Finset.univ)
⊢ H.h (Fin.last (n + 1)) ≫ Y.δ (Fin.last (n + 1 + 1)) = -∑ x, α x + -∑ x, β x + H.h 0 ≫ Y.δ 0
Module: Mathlib.CategoryTheory.Generator.StrongGenerator

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
hP : ∀ (X : C), ∃ J x, P.colimitsOfShape J X
X Y : C
i : X ⟶ Y
x✝ : Mono i
hi : ∀ (G : C), P G → Function.Surjective fun f ↦ f ≫ i
J : Type w
w✝ : Category.{w', w} J
p : P.ColimitOfShape J Y
φ : (j : J) → p.diag.obj j ⟶ X
hφ : ∀ (j : J), φ j ≫ i = p.ι.app j
c : Cocone p.diag := { pt := X, ι := { app := φ, naturality := ⋯ } }
j : J
⊢ c.ι.app j ≫ i = p.ι.app j ≫ 𝟙 Y

Tactic:
hφ,

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
hP : ∀ (X : C), ∃ J x, P.colimitsOfShape J X
X Y : C
i : X ⟶ Y
x✝ : Mono i
hi : ∀ (G : C), P G → Function.Surjective fun f ↦ f ≫ i
J : Type w
w✝ : Category.{w', w} J
p : P.ColimitOfShape J Y
φ : (j : J) → p.diag.obj j ⟶ X
hφ : ∀ (j : J), φ j ≫ i = p.ι.app j
c : Cocone p.diag := { pt := X, ι := { app := φ, naturality := ⋯ } }
j : J
⊢ p.ι.app j = p.ι.app j ≫ 𝟙 Y
Module: Mathlib.CategoryTheory.SmallRepresentatives

Goal state before:
pp: Ω : Type w
C : Type u
inst✝ : Category.{v, u} C
h₁ : Cardinal.lift.{w, u} (Cardinal.mk C) ≤ Cardinal.lift.{u, w} (Cardinal.mk Ω)
h₂ : ∀ (X Y : C), Cardinal.lift.{w, v} (Cardinal.mk (X ⟶ Y)) ≤ Cardinal.lift.{v, w} (Cardinal.mk Ω)
⊢ ∃ h, Nonempty (categoryFamily Ω h ≌ C)

Tactic:
let f₁ := (Cardinal.lift_mk_le'.1 h₁).some

Goal state after:
Ω : Type w
C : Type u
inst✝ : Category.{v, u} C
h₁ : Cardinal.lift.{w, u} (Cardinal.mk C) ≤ Cardinal.lift.{u, w} (Cardinal.mk Ω)
h₂ : ∀ (X Y : C), Cardinal.lift.{w, v} (Cardinal.mk (X ⟶ Y)) ≤ Cardinal.lift.{v, w} (Cardinal.mk Ω)
f₁ : C ↪ Ω := ⋯.some
⊢ ∃ h, Nonempty (categoryFamily Ω h ≌ C)
Module: Mathlib.CategoryTheory.SmallRepresentatives

Goal state before:
pp: κ : Cardinal.{w}
C : Type u
inst✝ : Category.{v, u} C
hC : HasCardinalLT (Arrow C) κ
Ω : Type w := κ.ord.ToType
ι : Arrow C ↪ Ω
⊢ ∃ S, Nonempty (categoryFamily κ S ≌ C)

Tactic:
have h₁ : Cardinal.lift.{w} (Cardinal.mk C) ≤
      Cardinal.lift.{u} (Cardinal.mk Ω) := by
    rw [Cardinal.lift_mk_le']
    refine ⟨Function.Embedding.trans { toFun X := Arrow.mk (𝟙 X), inj' := ?_ } ι⟩
    intro X Y h
    exact congr_arg Arrow.leftFunc.obj h

Goal state after:
κ : Cardinal.{w}
C : Type u
inst✝ : Category.{v, u} C
hC : HasCardinalLT (Arrow C) κ
Ω : Type w := κ.ord.ToType
ι : Arrow C ↪ Ω
h₁ : Cardinal.lift.{w, u} (Cardinal.mk C) ≤ Cardinal.lift.{u, w} (Cardinal.mk Ω)
⊢ ∃ S, Nonempty (categoryFamily κ S ≌ C)
Module: Mathlib.CategoryTheory.SmallRepresentatives

Goal state before:
pp: κ : Cardinal.{w}
C : Type u
inst✝ : Category.{v, u} C
hC : HasCardinalLT (Arrow C) κ
Ω : Type w := κ.ord.ToType
ι : Arrow C ↪ Ω
h₁ : Cardinal.lift.{w, u} (Cardinal.mk C) ≤ Cardinal.lift.{u, w} (Cardinal.mk Ω)
h₂ : ∀ (X Y : C), Cardinal.lift.{w, v} (Cardinal.mk (X ⟶ Y)) ≤ Cardinal.lift.{v, w} (Cardinal.mk Ω)
⊢ ∃ S, Nonempty (categoryFamily κ S ≌ C)

Tactic:
let f₁ := (Cardinal.lift_mk_le'.1 h₁).some

Goal state after:
κ : Cardinal.{w}
C : Type u
inst✝ : Category.{v, u} C
hC : HasCardinalLT (Arrow C) κ
Ω : Type w := κ.ord.ToType
ι : Arrow C ↪ Ω
h₁ : Cardinal.lift.{w, u} (Cardinal.mk C) ≤ Cardinal.lift.{u, w} (Cardinal.mk Ω)
h₂ : ∀ (X Y : C), Cardinal.lift.{w, v} (Cardinal.mk (X ⟶ Y)) ≤ Cardinal.lift.{v, w} (Cardinal.mk Ω)
f₁ : C ↪ Ω := ⋯.some
⊢ ∃ S, Nonempty (categoryFamily κ S ≌ C)
Module: Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : HasPushouts C
A B X Y Z W : C
f : A ⟶ B
g : X ⟶ Y
h : Z ⊗ W ⟶ X
⊢ (α_ A Z W).hom ≫ A ◁ h ≫ f ▷ X ≫ pushout.inl (f ▷ X) (A ◁ g) =
    (α_ A Z W).hom ≫ A ◁ (h ≫ g) ≫ pushout.inr (f ▷ X) (A ◁ g)

Tactic:
pushout.condition,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : HasPushouts C
A B X Y Z W : C
f : A ⟶ B
g : X ⟶ Y
h : Z ⊗ W ⟶ X
⊢ (α_ A Z W).hom ≫ A ◁ h ≫ A ◁ g ≫ pushout.inr (f ▷ X) (A ◁ g) =
    (α_ A Z W).hom ≫ A ◁ (h ≫ g) ≫ pushout.inr (f ▷ X) (A ◁ g)
Module: Mathlib.CategoryTheory.LiftingProperties.ParametrizedAdjunction

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
inst✝ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
X₃ Y₃ : C₃
f₃ : X₃ ⟶ Y₃
sq₁₂ : F.PushoutObjObj f₁ f₂
sq₁₃ : G.PullbackObjObj f₁ f₃
β : Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π
⊢ (fun α ↦
        Arrow.homMk (adj₂.homEquiv (sq₁₂.inl ≫ Arrow.Hom.left α))
          (⋯.lift (adj₂.homEquiv (sq₁₂.inr ≫ Arrow.Hom.left α)) (adj₂.homEquiv (Arrow.Hom.right α)) ⋯) ⋯)
      ((fun β ↦
          Arrow.homMk
            (⋯.desc (adj₂.homEquiv.symm (Arrow.Hom.left β)) (adj₂.homEquiv.symm (Arrow.Hom.right β ≫ sq₁₃.fst)) ⋯)
            (adj₂.homEquiv.symm (Arrow.Hom.right β ≫ sq₁₃.snd)) ⋯)
        β) =
    β

Tactic:
ext
    · simp
    · apply sq₁₃.isPullback.hom_ext <;> simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.PushoutProduct

Goal state before:
pp: X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono j
hi : innerAnodyneExtensions i
⊢ innerAnodyneExtensions sq₁₂.ι

Tactic:
refine (innerAnodyneExtensions.arrow_mk_iso_iff ?_).1
    (innerAnodyneExtensions_pushoutObjObjι sq₁₂.flipTensor hi)
  exact Arrow.isoMk (Iso.refl _) (β_ _ _)

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 → ∀ s ∈ {s | ∃ a, s = Set.Ioi a ∨ s = Set.Iio a}, 0 ∈ s → ∃ a, ∀ (b : ℕ), a ≤ b → (v j) (c b) ∈ s
r₁ : ℕ
hr₁ : ∀ (a : ℕ), r₁ ≤ a → 2 ≤ (v i) (c a)
rₙ : (j : ι) → j ≠ i → ℕ
hrₙ : ∀ (j : ι) (hj : j ≠ i) (b : ℕ), rₙ j hj ≤ b → (v j) (c b) ∈ Set.Iio 1
this : Fintype ι
r : ℕ := Finset.univ.sup fun j ↦ if h : j = i then r₁ else rₙ j h
⊢ w (c r) < 1

Tactic:
simpa [c, haw]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Connected

Goal state before:
pp: 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
⊢ IsPathConnected (sphere x r)

Tactic:
by
  /- when `r > 0`, we write the sphere as the image of `{0}ᶜ` under the map
  `y ↦ x + (r * ‖y‖⁻¹) • y`. Since the image under a continuous map of a path connected set
  is path connected, this concludes the proof. -/
  rcases hr.eq_or_lt with rfl | rpos
  · simpa using isPathConnected_singleton x
  let f : E → E := fun y ↦ x + (r * ‖y‖⁻¹) • y
  have A : ContinuousOn f {0}ᶜ := by
    intro y hy
    apply (continuousAt_const.add _).continuousWithinAt
    apply (continuousAt_const.mul (ContinuousAt.inv₀ continuousAt_id.norm ?_)).smul continuousAt_id
    simpa using hy
  have B : IsPathConnected ({0}ᶜ : Set E) := isPathConnected_compl_singleton_of_one_lt_rank h 0
  have C : IsPathConnected (f '' {0}ᶜ) := B.image' A
  have : f '' {0}ᶜ = sphere x r := by
    apply Subset.antisymm
    · rintro - ⟨y, hy, rfl⟩
      have : ‖y‖ ≠ 0 := by simpa using hy
      simp [f, norm_smul, abs_of_nonneg hr, mul_assoc, inv_mul_cancel₀ this]
    · intro y hy
      refine ⟨y - x, ?_, ?_⟩
      · intro H
        simp only [mem_singleton_iff, sub_eq_zero] at H
        simp only [H, mem_sphere_iff_norm, sub_self, norm_zero] at hy
        exact rpos.ne hy
      · simp [f, mem_sphere_iff_norm.1 hy, mul_inv_cancel₀ rpos.ne']
  rwa [this] at C

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

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
a : E
hf : HasStrictFDerivAt f (↑f') a
⊢ ∃ s,
    a ∈ s ∧
      IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s ∧ ApproximatesLinearOn f (↑f') s (‖↑f'.symm‖₊⁻¹ / 2)

Tactic:
simp only [← and_assoc]
  refine ((nhds_basis_opens a).exists_iff fun s t => ApproximatesLinearOn.mono_set).1 ?_
  exact
    hf.approximates_deriv_on_nhds <|
      f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' => half_pos <| inv_pos.2 hf'

Goal state after:
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
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
⊢ ↑n ≤ ⊤ ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h : ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
⊢ ↑n ≤ ↑⋯.choose ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z

Tactic:
· simpa using ⟨0, analyticAt_const .., by simpa⟩

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h : ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
⊢ ↑n ≤ ↑⋯.choose ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: 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 : ↑c < (↑f'symm.nnnorm)⁻¹
If' : 0 < ↑f'symm.nnnorm
Icf' : ↑c * ↑f'symm.nnnorm < 1
Jcf' : 1 - ↑c * ↑f'symm.nnnorm ≠ 0
g : E → E := fun x ↦ x + f'symm.toFun (y - f x)
hg : g = fun x ↦ x + f'symm.toFun (y - f x)
u : ℕ → E := fun n ↦ g^[n] b
hu : u = fun n ↦ g^[n] b
usucc : ∀ (n : ℕ), u (n + 1) = g (u n)
A : ∀ (z : E), dist (g z) z ≤ ↑f'symm.nnnorm * dist (f z) y
B : ∀ z ∈ closedBall b ε, g z ∈ closedBall b ε → dist (f (g z)) y ≤ ↑c * ↑f'symm.nnnorm * dist (f z) y
C :
  ∀ (n : ℕ) (w : E),
    dist w b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y →
      w ∈ closedBall b ε
n : ℕ
IH :
  dist (f (u n)) y ≤ (↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y ∧
    dist (u n) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
Ign :
  dist (g (u n)) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n.succ) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ ↑c * ↑f'symm.nnnorm * ((↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y) = (↑c * ↑f'symm.nnnorm) ^ n.succ * dist (f b) y

Tactic:
by simp only [pow_succ']; ring

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

Goal state before:
pp: case neg
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
f : 𝕜 → E
hf : AnalyticAt 𝕜 f 0
n : ℕ
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F 0
hF : ∀ᶠ (z : 𝕜) in 𝓝 0, f z = ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0 + z ^ n • F z
U : Set 𝕜
hU0 : U ∈ 𝓝 0
hU' : ∀ y ∈ U, f y = ∑ i ∈ Finset.range n, (y ^ i / ↑i.factorial) • iteratedDeriv i f 0 + y ^ n • F y
z : 𝕜
hz : z ∉ U
⊢ f z =
    ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0 +
      z ^ n •
        (fun z ↦
            if z ∈ U then F z
            else (z ^ n)⁻¹ • (f z - ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0))
          z

Tactic:
· simp only [if_neg hz]
      rw [smul_inv_smul₀]
      · module
      · contrapose hz
        exact (pow_eq_zero_iff'.mp hz).1 ▸ mem_of_mem_nhds hU0

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
f : 𝕜 → E
hf : AnalyticAt 𝕜 f 0
n : ℕ
⊢ ∃ F,
    AnalyticAt 𝕜 F 0 ∧ ∀ (z : 𝕜), f z = ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0 + z ^ n • F z

Tactic:
classical
  obtain ⟨F, hFa, hF⟩ := hf.exists_eventuallyEq_sum_add_pow_mul n
  obtain ⟨U, hU0, hU'⟩ := by rwa [eventually_iff_exists_mem] at hF
  refine ⟨fun z ↦ if z ∈ U then F z else (z ^ n)⁻¹ • (f z
      - (∑ i ∈ .range n, (z ^ i / i.factorial) • iteratedDeriv i f 0)), ?_, fun z ↦ ?_⟩
  · exact hFa.congr (by filter_upwards [hU0] using by simp +contextual)
  · by_cases hz : z ∈ U
    · simpa [hz] using hU' z hz
    · simp only [if_neg hz]
      rw [smul_inv_smul₀]
      · module
      · contrapose hz
        exact (pow_eq_zero_iff'.mp hz).1 ▸ mem_of_mem_nhds hU0

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
f : 𝕜 → 𝕜
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
⊢ analyticOrderAt (f ^ 0) z₀ = 0 • analyticOrderAt f z₀

Tactic:
by simp [analyticOrderAt_eq_zero]

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

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
z₀ : 𝕜
hg : AnalyticAt 𝕜 g z₀
hg' : deriv g z₀ ≠ 0
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
hf : AnalyticAt 𝕜 f (g z₀)
⊢ analyticOrderAt (f ∘ g) z₀ = analyticOrderAt f (g z₀)

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
z₀ : 𝕜
hg : AnalyticAt 𝕜 g z₀
hg' : deriv g z₀ ≠ 0
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
hf : ¬AnalyticAt 𝕜 f (g z₀)
⊢ analyticOrderAt (f ∘ g) z₀ = analyticOrderAt f (g z₀)

Tactic:
· simp [hf.analyticOrderAt_comp hg, hg.analyticOrderAt_sub_eq_one_of_deriv_ne_zero hg']

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
z₀ : 𝕜
hg : AnalyticAt 𝕜 g z₀
hg' : deriv g z₀ ≠ 0
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
hf : ¬AnalyticAt 𝕜 f (g z₀)
⊢ analyticOrderAt (f ∘ g) z₀ = analyticOrderAt f (g z₀)
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
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
z : ↑U
hz : z ∈ {u | analyticOrderAt f ↑u = ⊤}ᶜ
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0
t' : Set 𝕜
h₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y ≠ 0
h₂t' : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t'
h₃t' : ↑z ∈ t'
w : ↑U
hw : w ∈ Subtype.val ⁻¹' t'
h₁w : w = z
⊢ ¬analyticOrderAt f ↑w = ⊤

Tactic:
rwa [h₁w]

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
h_eq : ¬(fun z ↦ (z - x) ^ m • f z) =ᶠ[𝓝 x] 0
n : ℕ
g : 𝕜 → 𝕜'
hg_an : AnalyticAt 𝕜 g x
hg_ne : g x ≠ 0
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ m • f z = (z - x) ^ n • g z
this : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ (m + 1)) x
z : 𝕜
hg_ne' : g z ≠ 0
hz_ne : z ≠ x
hfg : ((z - x) ^ m)⁻¹ • (f z)⁻¹ = ((z - x) ^ n)⁻¹ • (g z)⁻¹
⊢ (z - x) ^ (m + 1) • g⁻¹ z = (z - x) ^ (n + 1) • f⁻¹ z

Tactic:
rw [inv_smul_eq_iff₀ (pow_ne_zero m (sub_ne_zero.mpr hz_ne)), smul_comm,
        eq_inv_smul_iff₀ (pow_ne_zero n (sub_ne_zero.mpr hz_ne))] at hfg

Goal state after:
case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
h_eq : ¬(fun z ↦ (z - x) ^ m • f z) =ᶠ[𝓝 x] 0
n : ℕ
g : 𝕜 → 𝕜'
hg_an : AnalyticAt 𝕜 g x
hg_ne : g x ≠ 0
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ m • f z = (z - x) ^ n • g z
this : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ (m + 1)) x
z : 𝕜
hg_ne' : g z ≠ 0
hz_ne : z ≠ x
hfg : (z - x) ^ n • (f z)⁻¹ = (z - x) ^ m • (g z)⁻¹
⊢ (z - x) ^ (m + 1) • g⁻¹ z = (z - x) ^ (n + 1) • f⁻¹ z
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv

Goal state before:
pp: case succ
n : ℕ
ih : iteratedDeriv (n + 1) sin = iteratedDeriv n cos
⊢ iteratedDeriv (n + 1 + 1) sin = iteratedDeriv (n + 1) cos

Tactic:
| succ n ih =>

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv

Goal state before:
pp: case succ
n : ℕ
ih : iteratedDeriv (n + 1) cos = -iteratedDeriv n sin
⊢ iteratedDeriv (n + 1 + 1) cos = -iteratedDeriv (n + 1) sin

Tactic:
| succ n ih =>

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv

Goal state before:
pp: case succ
n : ℕ
ih : iteratedDeriv (2 * n) sin = (-1) ^ n * sin
⊢ iteratedDeriv (2 * (n + 1)) sin = (-1) ^ (n + 1) * sin

Tactic:
| succ n ih =>

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv

Goal state before:
pp: case succ
n : ℕ
ih : iteratedDeriv (2 * n) cos = (-1) ^ n * cos
⊢ iteratedDeriv (2 * (n + 1)) cos = (-1) ^ (n + 1) * cos

Tactic:
| succ n ih =>

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case succ
R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
n : ℕ
ih : multichoose 1 n = 1
⊢ multichoose 1 (n + 1) = 1

Tactic:
| succ n ih =>

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case succ
R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
n : ℕ
ih : multichoose 2 n = ↑n + 1
⊢ multichoose 2 (n + 1) = ↑(n + 1) + 1

Tactic:
| succ n ih =>

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case succ
R : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
r : R
n : ℕ
ih : (descPochhammer ℤ n).smeval r = (ascPochhammer ℕ n).smeval (r - ↑n + 1)
⊢ (descPochhammer ℤ (n + 1)).smeval r = (ascPochhammer ℕ (n + 1)).smeval (r - ↑(n + 1) + 1)

Tactic:
| succ n ih =>

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case inr
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : x < 0
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y ↦ (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1))) ((r + 1) * (-↑x) ^ r * cexp (↑π * I * r)) x

Tactic:
mul_add ((π : ℂ) * _),

Goal state after:
case inr
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : x < 0
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y ↦ (-↑y) ^ (r + 1) * cexp (↑π * I * r + ↑π * I * 1)) ((r + 1) * (-↑x) ^ r * cexp (↑π * I * r)) x