mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialNerve

Goal state before:
pp: J : Type u
inst✝ : LinearOrder J
⊢ functor OrderHom.id = EnrichedFunctor.id SSet (SimplicialThickening J)

Tactic:
refine EnrichedFunctor.ext _ (fun _ ↦ rfl) fun i j ↦ ?_

Goal state after:
J : Type u
inst✝ : LinearOrder J
i j : SimplicialThickening J
⊢ (functor OrderHom.id).map i j ≫ eqToHom ⋯ = (EnrichedFunctor.id SSet (SimplicialThickening J)).map i j
Module: Mathlib.AlgebraicTopology.SimplicialNerve

Goal state before:
pp: J K L : Type u
inst✝² : LinearOrder J
inst✝¹ : LinearOrder K
inst✝ : LinearOrder L
f : J →o K
g : K →o L
⊢ functor (g.comp f) = EnrichedFunctor.comp SSet (functor f) (functor g)

Tactic:
refine EnrichedFunctor.ext _ (fun _ ↦ rfl) fun i j ↦ ?_

Goal state after:
J K L : Type u
inst✝² : LinearOrder J
inst✝¹ : LinearOrder K
inst✝ : LinearOrder L
f : J →o K
g : K →o L
i j : SimplicialThickening J
⊢ (functor (g.comp f)).map i j ≫ eqToHom ⋯ = (EnrichedFunctor.comp SSet (functor f) (functor g)).map i j
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: x✝ y✝ z✝ : AugmentedSimplexCategory
x y z : SimplexCategory
i : Fin (z.len + 1)
e₁ :
  (SimplexCategory.Hom.toOrderHom (inr' (tensorObjOf x y) z)) i = Fin.cast ⋯ (Fin.natAdd (tensorObjOf x y).len.succ i)
e₂ : (SimplexCategory.Hom.toOrderHom (inr' y z)) i = Fin.cast ⋯ (Fin.natAdd y.len.succ i)
⊢ y.len + 1 + (z.len + 1) = (tensorObjOf y z).len + 1

Tactic:
simp +arith

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: x✝ y✝ z✝ : AugmentedSimplexCategory
x y z : SimplexCategory
i : Fin (z.len + 1)
e₁ : (SimplexCategory.Hom.toOrderHom (inr' (tensorObjOf x y) z)) i = Fin.cast ⋯ (Fin.natAdd (x.len + y.len + 1).succ i)
e₂ : (SimplexCategory.Hom.toOrderHom (inr' y z)) i = Fin.cast ⋯ (Fin.natAdd y.len.succ i)
e₃ :
  (SimplexCategory.Hom.toOrderHom (inr' x (tensorObjOf y z))) (Fin.cast ⋯ (Fin.natAdd (y.len + 1) i)) =
    Fin.cast ⋯ (Fin.natAdd x.len.succ (Fin.cast ⋯ (Fin.natAdd (y.len + 1) i)))
⊢ ↑(Fin.cast ⋯ (Fin.cast ⋯ (Fin.natAdd (x.len + y.len + 1).succ i))) =
    ↑(Fin.cast ⋯ (Fin.natAdd x.len.succ (Fin.cast ⋯ (Fin.natAdd (y.len + 1) i))))

Tactic:
simp +arith

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: x✝ y✝ z✝ : AugmentedSimplexCategory
x y z : SimplexCategory
i : Fin (x.len + 1)
e₁ : (SimplexCategory.Hom.toOrderHom (inl' x y)) i = Fin.cast ⋯ (Fin.castAdd (y.len + 1) i)
e₂ :
  (SimplexCategory.Hom.toOrderHom (inl' x (tensorObjOf y z))) i = Fin.cast ⋯ (Fin.castAdd ((tensorObjOf y z).len + 1) i)
⊢ x.len + 1 + (y.len + 1) = (tensorObjOf x y).len + 1

Tactic:
simp +arith

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: x✝ y✝ z✝ : AugmentedSimplexCategory
x y z : SimplexCategory
i : Fin (x.len + 1)
e₁ : (SimplexCategory.Hom.toOrderHom (inl' x y)) i = Fin.cast ⋯ (Fin.castAdd (y.len + 1) i)
e₂ : (SimplexCategory.Hom.toOrderHom (inl' x (tensorObjOf y z))) i = Fin.cast ⋯ (Fin.castAdd (y.len + z.len + 1 + 1) i)
e₃ :
  (SimplexCategory.Hom.toOrderHom (inl' (tensorObjOf x y) z)) (Fin.cast ⋯ (Fin.castAdd (y.len + 1) i)) =
    Fin.cast ⋯ (Fin.castAdd (z.len + 1) (Fin.cast ⋯ (Fin.castAdd (y.len + 1) i)))
⊢ ↑((SimplexCategory.Hom.toOrderHom (WithInitial.down (eqToHom ⋯)))
        (Fin.cast ⋯ (Fin.castAdd (z.len + 1) (Fin.cast ⋯ (Fin.castAdd (y.len + 1) i))))) =
    ↑(Fin.cast ⋯ (Fin.castAdd (y.len + z.len + 1 + 1) i))

Tactic:
simp +arith

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: x✝ y✝ z✝ : AugmentedSimplexCategory
x y z : SimplexCategory
i : Fin (y.len + 1)
e₁ : (SimplexCategory.Hom.toOrderHom (inl' y z)) i = Fin.cast ⋯ (Fin.castAdd (z.len + 1) i)
e₂ : (SimplexCategory.Hom.toOrderHom (inr' x y)) i = Fin.cast ⋯ (Fin.natAdd x.len.succ i)
⊢ x.len + 1 + (y.len + 1) = (tensorObjOf x y).len + 1

Tactic:
simp +arith

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: x✝ y✝ z✝ : AugmentedSimplexCategory
x y z : SimplexCategory
i : Fin (y.len + 1)
e₁ : (SimplexCategory.Hom.toOrderHom (inl' y z)) i = Fin.cast ⋯ (Fin.castAdd (z.len + 1) i)
e₂ : (SimplexCategory.Hom.toOrderHom (inr' x y)) i = Fin.cast ⋯ (Fin.natAdd x.len.succ i)
e₃ :
  (SimplexCategory.Hom.toOrderHom (inl' (tensorObjOf x y) z)) (Fin.cast ⋯ (Fin.natAdd (x.len + 1) i)) =
    Fin.cast ⋯ (Fin.castAdd (z.len + 1) (Fin.cast ⋯ (Fin.natAdd (x.len + 1) i)))
⊢ y.len + 1 + (z.len + 1) = (tensorObjOf y z).len + 1

Tactic:
simp +arith

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: x✝ y✝ z✝ : AugmentedSimplexCategory
x y z : SimplexCategory
i : Fin (y.len + 1)
e₁ : (SimplexCategory.Hom.toOrderHom (inl' y z)) i = Fin.cast ⋯ (Fin.castAdd (z.len + 1) i)
e₂ : (SimplexCategory.Hom.toOrderHom (inr' x y)) i = Fin.cast ⋯ (Fin.natAdd x.len.succ i)
e₃ :
  (SimplexCategory.Hom.toOrderHom (inl' (tensorObjOf x y) z)) (Fin.cast ⋯ (Fin.natAdd (x.len + 1) i)) =
    Fin.cast ⋯ (Fin.castAdd (z.len + 1) (Fin.cast ⋯ (Fin.natAdd (x.len + 1) i)))
e₄ :
  (SimplexCategory.Hom.toOrderHom (inr' x (tensorObjOf y z))) (Fin.cast ⋯ (Fin.castAdd (z.len + 1) i)) =
    Fin.cast ⋯ (Fin.natAdd x.len.succ (Fin.cast ⋯ (Fin.castAdd (z.len + 1) i)))
⊢ ↑((SimplexCategory.Hom.toOrderHom (WithInitial.down (eqToHom ⋯)))
        (Fin.cast ⋯ (Fin.castAdd (z.len + 1) (Fin.cast ⋯ (Fin.natAdd (x.len + 1) i))))) =
    ↑(Fin.cast ⋯ (Fin.natAdd x.len.succ (Fin.cast ⋯ (Fin.castAdd (z.len + 1) i))))

Tactic:
simp +arith

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: case h₁
x y : AugmentedSimplexCategory
⊢ x.inl y ≫ (𝟙 x ⊗ₘ 𝟙 y) = x.inl y ≫ 𝟙 (x ⊗ y)

pp: case h₂
x y : AugmentedSimplexCategory
⊢ x.inr y ≫ (𝟙 x ⊗ₘ 𝟙 y) = x.inr y ≫ 𝟙 (x ⊗ y)

Tactic:
· simpa [inl, MonoidalCategoryStruct.whiskerLeft, MonoidalCategoryStruct.whiskerRight] using
      (tensorHom_comp_tensorHom (𝟙 x) (WithInitial.starInitial.to y) (𝟙 x) (𝟙 y))

Goal state after:
case h₂
x y : AugmentedSimplexCategory
⊢ x.inr y ≫ (𝟙 x ⊗ₘ 𝟙 y) = x.inr y ≫ 𝟙 (x ⊗ y)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PairingCore

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
inst✝ : P.IsProper
s : ↑P.II
⊢ (ConcreteCategory.hom (SimplicialObject.δ X (⋯.index ⋯))) ((↑(P.p s)).cast ⋯).simplex ∈ X.nonDegenerate (↑s).dim

Tactic:
(P.isUniquelyCodimOneFace s).δ_index rfl

Goal state after:
X : SSet
A : X.Subcomplex
P : A.Pairing
inst✝ : P.IsProper
s : ↑P.II
⊢ ((↑s).cast ⋯).simplex ∈ X.nonDegenerate (↑s).dim
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PairingCore

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
inst✝ : P.IsProper
s : ↑P.II
⊢ (ConcreteCategory.hom (SimplicialObject.δ X (⋯.index ⋯))) ((↑(P.p s)).cast ⋯).simplex ∉ A.obj (Opposite.op ⦋(↑s).dim⦌)

Tactic:
(P.isUniquelyCodimOneFace s).δ_index rfl

Goal state after:
X : SSet
A : X.Subcomplex
P : A.Pairing
inst✝ : P.IsProper
s : ↑P.II
⊢ ((↑s).cast ⋯).simplex ∉ A.obj (Opposite.op ⦋(↑s).dim⦌)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝¹ : LinearOrder ι
f : P.RankFunction ι
inst✝ : P.IsProper
j : ι
c : f.Cell j
⊢ c.ιSigmaHorn ≫ f.m j = c.horn.ι ≫ c.ιSigmaStdSimplex

Tactic:
by
  simp [m]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case pos.snd
m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ (k.castSucc, (s.δ.cast ⋯).simplex.2 (min s.δ ⋯)).2 = ((objEquiv (s.x.cast ⋯).simplex) s.index.castSucc).2

Tactic:
· change (s.x.cast s.hd).simplex.2
        (s.index.castSucc.succAbove (min s.δ rfl)) = _
      rw [s.isIndex.min_δ, Fin.succAbove_castSucc_self]
      exact s.isIndex.simplex_snd_succ

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.PushoutProduct

Goal state before:
pp: case refine_1.refine_1.refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasPushouts C
inst✝² : CartesianMonoidalCategory C
inst✝¹ : MonoidalClosed C
inst✝ : HasInitial C
X : Arrow C
⊢ ((curriedTensor C).map X.hom).app (Arrow.mk (initial.to (𝟙_ C))).left ≫
      (initialIsInitial.ofIso (zeroMul initialIsInitial).symm).to X.left =
    ((curriedTensor C).obj X.left).map (Arrow.mk (initial.to (𝟙_ C))).hom ≫ (ρ_ X.left).hom

Tactic:
· apply (initialIsInitial.ofIso (zeroMul initialIsInitial).symm).hom_ext

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Homology.Nondegenerate

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasCoproducts C
inst✝ : Preadditive C
X : SSet
R : C
n : ℕ
x : X _⦋n⦌
⊢ X.ιNormalizedChainComplex x ≫ (X.fromNormalizedChainComplex R).f n = X.ιChainComplex x ≫ PInfty.f n

Tactic:
by
  dsimp [ιNormalizedChainComplex]
  rw [Category.assoc, toNormalizedChainComplex_f_fromNormalizedChainComplex_f]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n ↦ x + c ^ n) atTop (𝓝 (x + 0))
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Tactic:
rw [add_zero] at A

Goal state after:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n ↦ x + c ^ n) atTop (𝓝 x)
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x
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
⊢ 0 < ↑f'symm.nnnorm

Tactic:
← inv_pos

Goal state after:
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
⊢ 0 < (↑f'symm.nnnorm)⁻¹
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
x✝ : IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F)) ∧ Function.Surjective ⇑(WithAbs.congr v w (RingEquiv.refl F))
hi : IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F))
right✝ : Function.Surjective ⇑(WithAbs.congr v w (RingEquiv.refl F))
x : F
| v x < 1

Tactic:
rw [← WithAbs.ofAbs_toAbs v x]

Goal state after:
F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
x✝ : IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F)) ∧ Function.Surjective ⇑(WithAbs.congr v w (RingEquiv.refl F))
hi : IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F))
right✝ : Function.Surjective ⇑(WithAbs.congr v w (RingEquiv.refl F))
x : F
| v (WithAbs.toAbs v x).ofAbs < 1
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)
⊢ Tendsto (fun n i ↦ (WithAbs.equiv (v i)).symm (y n)) atTop (𝓝 z)

Tactic:
refine tendsto_pi_nhds.mpr fun u ↦ ?_

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.Analytic.Order

Goal state before:
pp: case top
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
h : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ⊤
⊢ analyticOrderAt (deriv f) x + 1 = ⊤

Tactic:
suffices analyticOrderAt (deriv f) x = ⊤ by simp_all

Goal state after:
case top
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
h : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ⊤
⊢ analyticOrderAt (deriv f) x = ⊤
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
ε : ℝ
f : E → F
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
inst✝ : CompleteSpace E
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ Subsingleton F ∨ c < f'.toNonlinearRightInverse.nnnorm⁻¹

Tactic:
f'.toEquiv.subsingleton_congr

Goal state after:
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
ε : ℝ
f : E → F
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
inst✝ : CompleteSpace E
hf : ApproximatesLinearOn f (↑f') s c
hc✝ : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
hc : Subsingleton F ∨ c < ‖↑f'.symm‖₊⁻¹
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
__PartialEquiv✝ : PartialEquiv E F := hf.toPartialEquiv hc✝
⊢ Subsingleton F ∨ c < f'.toNonlinearRightInverse.nnnorm⁻¹
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
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
z : ↑U
hz : z ∈ {u | analyticOrderAt f ↑u = ⊤}
| ∃ t ⊆ {u | analyticOrderAt f ↑u = ⊤}, IsOpen[instTopologicalSpaceSubtype] t ∧ z ∈ t

Tactic:
arg 1

Goal state after:
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 = ⊤}
| fun t ↦ t ⊆ {u | analyticOrderAt f ↑u = ⊤} ∧ IsOpen[instTopologicalSpaceSubtype] t ∧ z ∈ t
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
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
z : ↑U
hz : z ∈ {u | analyticOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| {u | analyticOrderAt f ↑u = ⊤}

Tactic:
arg 1

Goal state after:
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 = ⊤}
x✝ : Set ↑U
| fun u ↦ analyticOrderAt f ↑u = ⊤
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case inr
x y : ℂ
h : x ≠ 0 ∨ y ≠ 0
hx : ¬x = 0
⊢ HasStrictDerivAt (fun y ↦ x ^ y) (x ^ y * log x) y

Tactic:
· simpa only [cpow_def_of_ne_zero hx, mul_one] using!
      ((hasStrictDerivAt_id y).const_mul (log x)).cexp

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: n k : ℕ
h : n < k
⊢ multichoose (-↑n) k = 0

Tactic:
rw [← nsmul_right_inj (Nat.factorial_ne_zero k),
    factorial_nsmul_multichoose_eq_ascPochhammer, smeval_ascPochhammer_neg_of_lt h, smul_zero]

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

Goal state before:
pp: case neg
f : ℂ → ℂ
s : Set ℂ
x : ℂ
hf : DifferentiableWithinAt ℂ f s x
c : ℂ
h : ¬UniqueDiffWithinAt ℂ s x
⊢ derivWithin (fun x ↦ c ^ f x) s x = log c * derivWithin f s x * c ^ f x

pp: case pos
f : ℂ → ℂ
s : Set ℂ
x : ℂ
hf : DifferentiableWithinAt ℂ f s x
c : ℂ
h : UniqueDiffWithinAt ℂ s x
⊢ derivWithin (fun x ↦ c ^ f x) s x = log c * derivWithin f s x * c ^ f x

Tactic:
· rw [derivWithin_zero_of_not_uniqueDiffWithinAt h,
      derivWithin_zero_of_not_uniqueDiffWithinAt h, mul_zero, zero_mul]

Goal state after:
case pos
f : ℂ → ℂ
s : Set ℂ
x : ℂ
hf : DifferentiableWithinAt ℂ f s x
c : ℂ
h : UniqueDiffWithinAt ℂ s x
⊢ derivWithin (fun x ↦ c ^ f x) s x = log c * derivWithin f s x * c ^ f x
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
inst✝¹ : SecondCountableTopology 𝕜
inst✝ : CompleteSpace E
h : MeromorphicOn f U
⊢ ({z | AnalyticAt 𝕜 f z}ᶜ ∩ U).Countable

Tactic:
by
  apply (HereditarilyLindelofSpace.isLindelof _).countable_of_isDiscrete
    (isDiscrete_of_codiscreteWithin _)
  simpa using! eventually_codiscreteWithin_analyticAt f h

Goal state after:
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
f : 𝕜 → E
c : 𝕜
⊢ Meromorphic (f ∘ fun x ↦ x - c) ↔ Meromorphic f

Tactic:
nth_rw 2 [← meromorphic_comp_add_const_iff_meromorphic (c := -c)]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
⊢ Meromorphic (f ∘ fun x ↦ x - c) ↔ Meromorphic (f ∘ fun x ↦ x + -c)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: 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)
this : HasDerivAt (fun y ↦ (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1))) ((r + 1) * (-↑x) ^ r * cexp (↑π * I * r)) x
⊢ HasDerivAt (fun x ↦ (-↑x) ^ (r + 1) * cexp (↑π * I * (r + 1)) / (r + 1)) ((-↑x) ^ r * cexp (↑π * I * r)) x

Tactic:
convert! this.div_const (r + 1) using 1

Goal state after:
case e'_9
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)
this : HasDerivAt (fun y ↦ (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1))) ((r + 1) * (-↑x) ^ r * cexp (↑π * I * r)) x
⊢ (-↑x) ^ r * cexp (↑π * I * r) = (r + 1) * (-↑x) ^ r * cexp (↑π * I * r) / (r + 1)
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
⊢ (iteratedFDerivWithin 𝕜 n f s x) v = ∑ σ, (p n) fun i ↦ v (σ i)

Tactic:
have fcont : ContDiffOn 𝕜 (↑n) f s := by
    apply AnalyticOn.contDiffOn _ hs
    simpa [I] using h'

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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
⊢ (iteratedFDerivWithin 𝕜 n f s x) v = ∑ σ, (p n) fun i ↦ v (σ i)
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case h₀
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
y : E
hy : y ∈ Metric.eball 0 r
⊢ ∀ b ∈ Finset.range (n + 1), b ≠ n → ((q b) fun x ↦ y) = 0

Tactic:
intro i hi h'i

Goal state after:
case h₀
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
y : E
hy : y ∈ Metric.eball 0 r
i : ℕ
hi : i ∈ Finset.range (n + 1)
h'i : i ≠ n
⊢ ((q i) fun x ↦ y) = 0
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case refine_2
w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall (fun z ↦ 1 / w ^ 2) (FormalMultilinearSeries.ofScalars ℂ fun i ↦ Nat.casesOn i (w ^ (-2)) 0) x
    ‖w - x‖ₑ

Tactic:
· convert! hasFPowerSeriesOnBall_const.mono _ le_top
    · ext (_ | _) <;> simp [zpow_ofNat]
    · simpa [sub_eq_zero]

Goal state after:
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: case packingRadius
X : Type u_1
Y : Type u_2
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
Z : Type u_3
inst✝ : MetricSpace Z
D : DeloneSet X
f : X ≃ Y
g : Y ≃ Z
K₁f K₂f K₁g K₂g : ℝ≥0
hf₁_pos : 0 < K₁f
hf₂_pos : 0 < K₂f
hg₁_pos : 0 < K₁g
hg₂_pos : 0 < K₂g
hf_anti : AntilipschitzWith K₁f ⇑f
hf_lip : LipschitzWith K₂f ⇑f
hg_anti : AntilipschitzWith K₁g ⇑g
hg_lip : LipschitzWith K₂g ⇑g
⊢ ↑(mapBilipschitz g K₁g K₂g hg₁_pos hg₂_pos hg_anti hg_lip
          (mapBilipschitz f K₁f K₂f hf₁_pos hf₂_pos hf_anti hf_lip D)).packingRadius =
    ↑(mapBilipschitz (f.trans g) (K₁f * K₁g) (K₂g * K₂f) ⋯ ⋯ ⋯ ⋯ D).packingRadius

Tactic:
simp only [mapBilipschitz_packingRadius, NNReal.coe_div, div_div]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: ⊢ expGrowthSup ⊤ = ⊤

Tactic:
by
  apply top_le_iff.1
  rw [← expGrowthInf_top]
  exact expGrowthInf_le_expGrowthSup

Goal state after:
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
v_0 : 0 < linearGrowthInf fun n ↦ ↑(v n)
u_0 : ¬u = ⊥
a : EReal
a_0 : 0 < a
a_v : a < linearGrowthInf fun n ↦ ↑(v n)
b : EReal
b_0 : 0 < b
b_u : b < linearGrowthSup u
a' : EReal
a_a' : a < a'
a_v' : a' < linearGrowthInf fun n ↦ ↑(v n)
⊢ ∃ᶠ (n : ℕ) in atTop, a * b * ↑n ≤ (u ∘ v) n

Tactic:
have a_top' := ne_top_of_lt a_v'

Goal state after:
case neg
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
v_0 : 0 < linearGrowthInf fun n ↦ ↑(v n)
u_0 : ¬u = ⊥
a : EReal
a_0 : 0 < a
a_v : a < linearGrowthInf fun n ↦ ↑(v n)
b : EReal
b_0 : 0 < b
b_u : b < linearGrowthSup u
a' : EReal
a_a' : a < a'
a_v' : a' < linearGrowthInf fun n ↦ ↑(v n)
a_top' : a' ≠ ⊤
⊢ ∃ᶠ (n : ℕ) in atTop, a * b * ↑n ≤ (u ∘ v) n