mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sheaves.Abelian

Goal state before:
pp: case mpr
C : Type v
inst✝⁶ : Category.{u, v} C
inst✝⁵ : HasColimits C
inst✝⁴ : HasLimits C
FC : C → C → Type u_1
CC : C → Type u
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝² : PreservesFilteredColimits (CategoryTheory.forget C)
inst✝¹ : PreservesLimits (CategoryTheory.forget C)
inst✝ : Abelian C
X : TopCat
S : ShortComplex (Sheaf C X)
h : ∀ (x : ↑X), IsZero (S.map (forget C X ⋙ stalkFunctor C x)).homology
⊢ S.Exact

Tactic:
ShortComplex.exact_iff_isZero_homology,

Goal state after:
case mpr
C : Type v
inst✝⁶ : Category.{u, v} C
inst✝⁵ : HasColimits C
inst✝⁴ : HasLimits C
FC : C → C → Type u_1
CC : C → Type u
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝² : PreservesFilteredColimits (CategoryTheory.forget C)
inst✝¹ : PreservesLimits (CategoryTheory.forget C)
inst✝ : Abelian C
X : TopCat
S : ShortComplex (Sheaf C X)
h : ∀ (x : ↑X), IsZero (S.map (forget C X ⋙ stalkFunctor C x)).homology
⊢ IsZero S.homology
Module: Mathlib.Topology.Sheaves.Alexandrov

Goal state before:
pp: case e_a
X✝ : Type v
inst✝⁶ : TopologicalSpace X✝
inst✝⁵ : Preorder X✝
inst✝⁴ : Topology.IsUpperSet X✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasLimits C
F✝ : X✝ ⥤ C
X : TopCat
inst✝¹ : Preorder ↑X
inst✝ : Topology.IsUpperSet ↑X
F : ↑X ⥤ C
α : Type v
Us : α → Opens ↑X
S : Cone ((ObjectProperty.ι fun V ↦ ∃ i, V ≤ Us i).op ⋙ principalsKanExtension F)
m : S.pt ⟶ ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op).pt
hm✝ :
  ∀ (j : (ObjectProperty.FullSubcategory fun V ↦ ∃ i, V ≤ Us i)ᵒᵖ),
    m ≫ ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op).π.app j = S.π.app j
left✝ : Discrete PUnit.{1}
x : ↑X
f : (fromPUnit (Opposite.op (iSup Us))).obj left✝ ⟶ (principals ↑X).obj x
hm :
  m ≫
      ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op).π.app
        (Opposite.op { obj := principalOpen x, property := ⋯ }) =
    S.π.app (Opposite.op { obj := principalOpen x, property := ⋯ })
⊢ ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op).π.app
        (Opposite.op { obj := principalOpen x, property := ⋯ }) ≫
      limit.π (generator (principalOpen (StructuredArrow.right { left := left✝, right := x, hom := f })) ⋙ F)
        { left := { as := PUnit.unit }, right := StructuredArrow.right { left := left✝, right := x, hom := f },
          hom := 𝟙 (Opposite.op (principalOpen (StructuredArrow.right { left := left✝, right := x, hom := f }))) } =
    limit.π (generator (iSup Us) ⋙ F) { left := left✝, right := x, hom := f }

Tactic:
apply limit.lift_π

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
⊢ ∃ β x F, ∀ (x y : ↑t), F x = F y → dist ↑x ↑y < δ

Tactic:
let ε := 1 / 5 * δ

Goal state after:
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
⊢ ∃ β x F, ∀ (x y : ↑t), F x = F y → dist ↑x ↑y < δ