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

Goal state before:
pp: X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
x : X _⦋n + 1⦌
⊢ (ConcreteCategory.hom
          ((↾fun x ↦
              (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1 + 1⦌)))
                ↑(prodStdSimplex.nonDegenerateEquiv₁ j.succ)) ≫
            δ Y i.castSucc)).toFun
      x =
    (ConcreteCategory.hom
          (δ X i ≫
            ↾fun x ↦
              (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
                ↑(prodStdSimplex.nonDegenerateEquiv₁ j))).toFun
      x

Tactic:
simp only [TypeCat.Fun.toFun_apply, types_comp_apply, TypeCat.hom_ofHom, TypeCat.Fun.coe_mk,
      ← SSet.δ_naturality_apply]

Goal state after:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
x : X _⦋n + 1⦌
⊢ (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
      ((ConcreteCategory.hom (δ (Δ[n + 1] ⊗ Δ[1]) i.castSucc)) ↑(prodStdSimplex.nonDegenerateEquiv₁ j.succ)) =
    (ConcreteCategory.hom ((yonedaEquiv.symm ((ConcreteCategory.hom (δ X i)) x) ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
      ↑(prodStdSimplex.nonDegenerateEquiv₁ j)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Homotopy

Goal state before:
pp: X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 1)
x : X _⦋n + 1⦌
⊢ (ConcreteCategory.hom
          ((↾fun x ↦
              (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1 + 1⦌)))
                ↑(prodStdSimplex.nonDegenerateEquiv₁ i.succ)) ≫
            δ Y i.castSucc.succ)).toFun
      x =
    (ConcreteCategory.hom
          ((↾fun x ↦
              (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1 + 1⦌)))
                ↑(prodStdSimplex.nonDegenerateEquiv₁ i.castSucc)) ≫
            δ Y i.castSucc.succ)).toFun
      x

Tactic:
simp only [TypeCat.Fun.toFun_apply, types_comp_apply, TypeCat.hom_ofHom, TypeCat.Fun.coe_mk,
      ← SSet.δ_naturality_apply]

Goal state after:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 1)
x : X _⦋n + 1⦌
⊢ (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
      ((ConcreteCategory.hom (δ (Δ[n + 1] ⊗ Δ[1]) i.castSucc.succ)) ↑(prodStdSimplex.nonDegenerateEquiv₁ i.succ)) =
    (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
      ((ConcreteCategory.hom (δ (Δ[n + 1] ⊗ Δ[1]) i.castSucc.succ)) ↑(prodStdSimplex.nonDegenerateEquiv₁ i.castSucc))
Module: Mathlib.AlgebraicTopology.SimplicialSet.Homotopy

Goal state before:
pp: X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
x : X _⦋n + 1⦌
⊢ (ConcreteCategory.hom
          ((↾fun x ↦
              (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1 + 1⦌)))
                ↑(prodStdSimplex.nonDegenerateEquiv₁ j.castSucc)) ≫
            δ Y i.succ)).toFun
      x =
    (ConcreteCategory.hom
          (δ X i ≫
            ↾fun x ↦
              (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
                ↑(prodStdSimplex.nonDegenerateEquiv₁ j))).toFun
      x

Tactic:
simp only [TypeCat.Fun.toFun_apply, types_comp_apply, TypeCat.hom_ofHom, TypeCat.Fun.coe_mk,
      ← SSet.δ_naturality_apply]

Goal state after:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
x : X _⦋n + 1⦌
⊢ (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
      ((ConcreteCategory.hom (δ (Δ[n + 1] ⊗ Δ[1]) i.succ)) ↑(prodStdSimplex.nonDegenerateEquiv₁ j.castSucc)) =
    (ConcreteCategory.hom ((yonedaEquiv.symm ((ConcreteCategory.hom (δ X i)) x) ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
      ↑(prodStdSimplex.nonDegenerateEquiv₁ j)
Module: Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋0⦌
f g fg : X.PtSimplex n x
i : Fin n
h : f.MulStruct g fg i
j : Fin n
hij : i.rev = j
⊢ yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.rev.rev.succ.succ ≫ h.map))) =
    yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.succ.succ ≫ h.map)))

Tactic:
Fin.rev_rev

Goal state after:
X : SSet
n : ℕ
x : X _⦋0⦌
f g fg : X.PtSimplex n x
i : Fin n
h : f.MulStruct g fg i
j : Fin n
hij : i.rev = j
⊢ yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.succ.succ ≫ h.map))) =
    yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.succ.succ ≫ h.map)))
Module: Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋0⦌
f g fg : X.PtSimplex n x
i : Fin n
h : f.MulStruct g fg i
j : Fin n
hij : i.rev = j
⊢ yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.rev.rev.castSucc.succ ≫ h.map))) =
    yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.castSucc.succ ≫ h.map)))

Tactic:
Fin.rev_rev

Goal state after:
X : SSet
n : ℕ
x : X _⦋0⦌
f g fg : X.PtSimplex n x
i : Fin n
h : f.MulStruct g fg i
j : Fin n
hij : i.rev = j
⊢ yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.castSucc.succ ≫ h.map))) =
    yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.castSucc.succ ≫ h.map)))
Module: Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋0⦌
f g fg : X.PtSimplex n x
i : Fin n
h : f.MulStruct g fg i
j : Fin n
hij : i.rev = j
⊢ yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.rev.rev.castSucc.castSucc ≫ h.map))) =
    yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.castSucc.castSucc ≫ h.map)))

Tactic:
Fin.rev_rev

Goal state after:
X : SSet
n : ℕ
x : X _⦋0⦌
f g fg : X.PtSimplex n x
i : Fin n
h : f.MulStruct g fg i
j : Fin n
hij : i.rev = j
⊢ yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.castSucc.castSucc ≫ h.map))) =
    yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv (stdSimplex.δ i.castSucc.castSucc ≫ h.map)))
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

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

Tactic:
stdSimplex.map_vertex

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

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

Tactic:
stdSimplex.map_vertex

Goal state after:
case e_6
⊢ stdSimplex.vertex ((ConcreteCategory.hom (SimplexCategory.δ 0)) 0) = ⟨Pi.single 1 1, ⋯⟩
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: E : Type u
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Nontrivial E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s.Countable
H : ¬interior s = ∅
⊢ ℵ₀ < ℵ₀

Tactic:
calc
    (ℵ₀ : Cardinal.{u}) < 𝔠 := aleph0_lt_continuum
    _ ≤ #(interior s) :=
      continuum_le_cardinal_of_isOpen 𝕜 isOpen_interior (notMem_singleton_empty.1 H)
    _ ≤ #s := mk_le_mk_of_subset interior_subset
    _ ≤ ℵ₀ := le_aleph0 hs

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

Goal state before:
pp: case «0»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ ((hY.spineEquiv 2 f₂._proof_4)
          (f₂ f₀ f₁ hδ₁ hδ₀ hY ((ConcreteCategory.hom (X.map (σ₂ 1 _proof_15 _proof_12).op)) x))).arrow
      ((fun i ↦ i) ⟨0, ⋯⟩) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 1 _proof_15 _proof_12).op)) (f₁ x))).arrow
      ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
dsimp
    rw [spineEquiv_f₂_arrow_zero]
    simp [StrictSegal.spineEquiv, SimplexCategory.mkOfSucc_zero_eq_δ,
      ← Functor.map_comp_apply, ← op_comp]

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

Goal state before:
pp: case refine_3
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))
hcₙ : Tendsto (fun n ↦ w (c n)) atTop (𝓝 (w b))
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 : ∀ (b : ℕ), r₁ ≤ b → 1 < (v i) (c b)
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
rN : ℕ
hrN : ∀ (b : ℕ), rN ≤ b → w (c b) < 1
this : Fintype ι
r : ℕ := max (Finset.univ.sup fun j ↦ if h : j = i then r₁ else rₙ j h) rN
⊢ w (c r) < 1

Tactic:
· exact hrN _ <| le_max_iff.2 (.inr le_rfl)

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

Goal state before:
pp: case «1»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op))
      (f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 δ₂_zero_comp_σ₂_one._proof_4 δ₂_zero_comp_σ₂_one._proof_3).op)) x)) =
    (ConcreteCategory.hom (Y.map (Hom.tr (SimplexCategory.mkOfSucc 1) ⋯ f₂._proof_4).op))
      ((ConcreteCategory.hom (Y.map (σ₂ 1 _proof_15 _proof_12).op)) (f₁ x))

Tactic:
SimplexCategory.mkOfSucc_one_eq_δ,

Goal state after:
case «1»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op))
      (f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 δ₂_zero_comp_σ₂_one._proof_4 δ₂_zero_comp_σ₂_one._proof_3).op)) x)) =
    (ConcreteCategory.hom (Y.map (Hom.tr (SimplexCategory.δ 0) ⋯ f₂._proof_4).op))
      ((ConcreteCategory.hom (Y.map (σ₂ 1 _proof_15 _proof_12).op)) (f₁ x))
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
a : F
ha₀ : a ≠ 0
ha₁ : v a ≠ 1
b : F
hb₀ : b ≠ 0
hb₁ : v b ≠ 1
h_ne : log (v b) / log (w b) ≠ log (v a) / log (w a)
ha : 1 < v a
hb : 1 < v b
h_lt : log (v b) / log (v a) < log (w b) / log (w a)
hwa : 1 < w a
hwb : 1 < w b
q : ℚ
hq₁ : log (v b ^ q.den) < log (v a ^ q.num)
hq₂ : ↑q.num / ↑q.den < log (w b) / log (w a)
⊢ False

Tactic:
log_lt_log_iff (pow_pos (by linarith) _) (zpow_pos (by linarith) _),

Goal state after:
F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
a : F
ha₀ : a ≠ 0
ha₁ : v a ≠ 1
b : F
hb₀ : b ≠ 0
hb₁ : v b ≠ 1
h_ne : log (v b) / log (w b) ≠ log (v a) / log (w a)
ha : 1 < v a
hb : 1 < v b
h_lt : log (v b) / log (v a) < log (w b) / log (w a)
hwa : 1 < w a
hwb : 1 < w b
q : ℚ
hq₁ : v b ^ q.den < v a ^ q.num
hq₂ : ↑q.num / ↑q.den < log (w b) / log (w a)
⊢ False
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: case pos.inr.inl
F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
hw : w.IsNontrivial
a : F
ha₀ : a ≠ 0
ha₁ : w a ≠ 1
b : F
hb₀ : b ≠ 0
hb₁ : w b = 1
⊢ v b ^ (log (w a) / log (v a)) = w b

Tactic:
simp [hb₁, h.eq_one_iff.2 hb₁]

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:
← Fintype.sum_pi_single u z,

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 (∑ j, Pi.single j (z j) 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
⊢ 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 (Pi.single w (z w) u))

Tactic:
rw [Pi.single_eq_of_ne (M := fun i ↦ WithAbs (v i)) hw (z w)]

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
⊢ 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: case coe
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 m a✝ : ℕ
⊢ ↑a✝ = ↑m ↔ ↑m ≤ ↑a✝ ∧ ¬↑m + 1 ≤ ↑a✝

Tactic:
norm_cast; lia

Goal state after:
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:
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
⊢ MeromorphicAt (f + g) x ↔ MeromorphicAt g x

Tactic:
by
  exact ⟨fun h ↦ by simpa using h.sub hf, fun _ ↦ by fun_prop⟩

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
x : 𝕜
f g : 𝕜 → E
hf : MeromorphicAt f x
⊢ MeromorphicAt (f - g) x ↔ MeromorphicAt g x

Tactic:
by
  exact ⟨fun h ↦ by simpa using h.sub hf, fun _ ↦ by fun_prop⟩

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝⁵ : Semiring R
S : Type u_2
inst✝⁴ : NonAssocSemiring S
inst✝³ : Pow S ℕ
inst✝² : Module R S
inst✝¹ : IsScalarTower R S S
inst✝ : NatPowAssoc S
x : S
n : ℕ
⊢ (ascPochhammer R n).smeval x = (ascPochhammer ℕ n).smeval x

Tactic:
induction n with
  | zero => simp only [ascPochhammer_zero, smeval_one]
  | succ n hn =>
    simp only [ascPochhammer_succ_right, mul_add, smeval_add, smeval_mul_X, ← Nat.cast_comm]
    simp only [← C_eq_natCast, smeval_C_mul, hn, Nat.cast_smul_eq_nsmul R n]
    simp only [nsmul_eq_mul, Nat.cast_id]

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

Goal state before:
pp: case neg
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : 𝕜
f : 𝕜' → F
g : 𝕜 → 𝕜'
hg : AnalyticAt 𝕜 g x
r : ℕ
hr : AnalyticAt 𝕜' (fun z ↦ (z - g x) ^ r • f z) (g x)
hg' : ¬analyticOrderAt (fun x_1 ↦ g x_1 - g x) x = ⊤
⊢ MeromorphicAt (f ∘ g) x

Tactic:
obtain ⟨n, hn⟩ := WithTop.ne_top_iff_exists.mp hg'
    obtain ⟨h, han, hne, heq⟩ := (hg.fun_sub analyticAt_const).analyticOrderAt_eq_natCast.mp hn.symm
    set j := fun z ↦ (z - g x) ^ r • f z
    have : AnalyticAt 𝕜 (fun i ↦ (h i)⁻¹ ^ r • j (g i)) x :=
      ((han.inv hne).pow r).smul (hr.restrictScalars.comp hg)
    refine ⟨n * r, this.congr ?_⟩
    filter_upwards [heq, han.continuousAt.tendsto.eventually_ne hne] with z hz hzne
    simp only [j, inv_pow, Function.comp_apply, inv_smul_eq_iff₀ (pow_ne_zero r hzne)]
    rw [hz, smul_comm, ← smul_assoc, pow_mul, smul_pow]

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n : ℕ
⊢ (ascPochhammer ℤ n).smeval ((X - ↑n + 1).smeval r) = (ascPochhammer ℕ n).smeval (r - ↑n + 1)

Tactic:
add_comm_sub,

Goal state after:
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n : ℕ
⊢ (ascPochhammer ℤ n).smeval ((X + (1 - ↑n)).smeval r) = (ascPochhammer ℕ n).smeval (r - ↑n + 1)
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case succ
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
p : FormalMultilinearSeries 𝕜 E F
k : ℕ
ih : ∀ {n : ℕ}, p (n + k) = 0 → p.iteratedFDerivSeries k n = 0
n : ℕ
h : p (n + (k + 1)) = 0
x✝¹ : Fin n → E
x✝ : Fin (k + 1) → E
⊢ ((p.iteratedFDerivSeries (k + 1) n) x✝¹) x✝ = (0 x✝¹) x✝

Tactic:
simp only [iteratedFDerivSeries, Nat.succ_eq_add_one,
      ContinuousLinearMap.compFormalMultilinearSeries_apply,
      ContinuousLinearMap.compContinuousMultilinearMap_coe, ContinuousLinearEquiv.coe_coe,
      LinearIsometryEquiv.coe_toContinuousLinearEquiv, Function.comp_apply,
      continuousMultilinearCurryLeftEquiv_symm_apply, _root_.zero_apply,
      derivSeries_eq_zero _ (ih (p.congr_zero (Nat.succ_add_eq_add_succ _ _).symm h))]

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

Goal state before:
pp: r : ℝ
hr : r < 1
hr' : r ≠ 0
⊢ ¬DifferentiableAt ℝ (fun x ↦ x ^ r) 0

Tactic:
by_contra h

Goal state after:
r : ℝ
hr : r < 1
hr' : r ≠ 0
h : DifferentiableAt ℝ (fun x ↦ x ^ r) 0
⊢ False
Module: Mathlib.Topology.MetricSpace.Cover

Goal state before:
pp: X : Type u_1
inst✝ : EMetricSpace X
s N : Set X
⊢ IsCover 0 s N ↔ s ⊆ N

Tactic:
simp [isCover_iff_subset_iUnion_closedEBall]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: x : ℝ≥0∞
⊢ x.log.exp = x

Tactic:
by_cases hx_top : x = ∞
  · simp [hx_top]
  by_cases hx_zero : x = 0
  · simp [hx_zero]
  have hx_pos : 0 < x.toReal := ENNReal.toReal_pos hx_zero hx_top
  rw [← ENNReal.ofReal_toReal hx_top, log_ofReal_of_pos hx_pos, exp_coe, Real.exp_log hx_pos]

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

Goal state before:
pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthInf fun x ↦ v x + b) ≤ linearGrowthInf v

Tactic:
apply (linearGrowthInf_add_le' _ _).trans_eq <;> rw [linearGrowthSup_const b_bot.ne' hb]

Goal state after:
case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v + 0 = linearGrowthInf v

u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊥ ∨ 0 ≠ ⊤

u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊤ ∨ 0 ≠ ⊥
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k (k * f)
x : α
hx : k x ≠ 0
⊢ (k⁻¹ * (k * f)) x = f x

Tactic:
by simp [← mul_assoc, inv_mul_cancel₀ hx]

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

Goal state before:
pp: case pos
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : u = ⊥
⊢ linearGrowthInf ⊥ ≤ (linearGrowthSup fun n ↦ ↑(v n)) * linearGrowthInf ⊥

Tactic:
linearGrowthInf_bot

Goal state after:
case pos
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : u = ⊥
⊢ ⊥ ≤ (linearGrowthSup fun n ↦ ↑(v n)) * ⊥