mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case refine_2
δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set p.Rep
hs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)
E : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) := fun p ↦ ⟨N p, fun a b ↦ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x : q.Rep
this : x ∈ ⋃ y ∈ s q, ball y ε
y : q.Rep
ys : y ∈ s q
hy : x ∈ ball y ε
i : ℕ := ↑((E q) ⟨y, ys⟩)
⊢ ∃ z, dist x (Φ z) ≤ ε

Tactic:
let hi := ((E q) ⟨y, ys⟩).is_lt

Goal state after:
case refine_2
δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set p.Rep
hs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)
E : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) := fun p ↦ ⟨N p, fun a b ↦ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x : q.Rep
this : x ∈ ⋃ y ∈ s q, ball y ε
y : q.Rep
ys : y ∈ s q
hy : x ∈ ball y ε
i : ℕ := ↑((E q) ⟨y, ys⟩)
hi : ↑((E q) ⟨y, ys⟩) < Nat.card ↑(s q) := Fin.is_lt ((E q) ⟨y, ys⟩)
⊢ ∃ z, dist x (Φ z) ≤ ε
Module: Mathlib.Topology.Sheaves.EtaleSpace

Goal state before:
pp: X : TopCat
C : Type u
inst✝⁴ : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
inst✝¹ : Limits.HasColimits C
F : Presheaf C X
inst✝ : Limits.PreservesFilteredColimits (forget C)
U : Opens ↑X
hF_bij : ∀ (x : ↑X) (hx : x ∈ U), Bijective ⇑(ConcreteCategory.hom (F.germ U x hx))
x : ↑X
hx : x ∈ U
⊢ LeftInverse
    (fun x_1 ↦
      match x_1 with
      | (⟨y, hy⟩, WithTopology.toTopology .(⊥) g) =>
        ⟨{ base := y, germ := (ConcreteCategory.hom (F.germ U y hy)) (surjInv ⋯ g) }, hy⟩)
    fun s ↦ (⟨(↑s).base, ⋯⟩, WithTopology.toTopology ⊥ ((ConcreteCategory.hom (F.germ U x hx)) (surjInv ⋯ (↑s).germ)))

Tactic:
by
    rintro ⟨⟨base, s⟩, hs⟩
    simp only
    congr 2
    rw [leftInverse_surjInv (hF_bij _ _), surjInv_eq (hF_bij _ _).surjective]

Goal state after:
Module: Mathlib.Topology.Sheaves.Flasque

Goal state before:
pp: X : TopCat
U : Opens ↑X
F G : Sheaf AddCommGrpCat X
g : F ⟶ G
s : ↑(G.obj.obj (op U))
c : Set (Under g s)
h : IsChain (fun x y ↦ Nonempty (y ⟶ x)) c
f : ↑c → Opens ↑X := fun x ↦ unop (StructuredArrow.right ↑x).fst
t : ToType (F.obj.obj (op (iSup f)))
ht : IsGluing F.obj f (fun x ↦ (StructuredArrow.right ↑x).snd) t
right✝ :
  ∀ (y : ToType (F.obj.obj (op (iSup f)))),
    (fun s_1 ↦ IsGluing F.obj f (fun x ↦ (StructuredArrow.right ↑x).snd) s_1) y → y = t
⊢ ∃ ub, ∀ a ∈ c, Nonempty (ub ⟶ a)

Tactic:
have le₁ : iSup f ≤ U := iSup_le <| fun j => leOfHom j.1.hom.1.unop

Goal state after:
X : TopCat
U : Opens ↑X
F G : Sheaf AddCommGrpCat X
g : F ⟶ G
s : ↑(G.obj.obj (op U))
c : Set (Under g s)
h : IsChain (fun x y ↦ Nonempty (y ⟶ x)) c
f : ↑c → Opens ↑X := fun x ↦ unop (StructuredArrow.right ↑x).fst
t : ToType (F.obj.obj (op (iSup f)))
ht : IsGluing F.obj f (fun x ↦ (StructuredArrow.right ↑x).snd) t
right✝ :
  ∀ (y : ToType (F.obj.obj (op (iSup f)))),
    (fun s_1 ↦ IsGluing F.obj f (fun x ↦ (StructuredArrow.right ↑x).snd) s_1) y → y = t
le₁ : iSup f ≤ U
⊢ ∃ ub, ∀ a ∈ c, Nonempty (ub ⟶ a)
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c c' : Cone ((diagram U).op ⋙ F)
f : c ⟶ c'
⊢ f.hom ≫ (coneEquivFunctorObj F U c').π.app WalkingParallelPair.zero =
    (coneEquivFunctorObj F U c).π.app WalkingParallelPair.zero

Tactic:
· dsimp
            ext
            simp

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c c' : Cone ((diagram U).op ⋙ F)
f : c ⟶ c'
⊢ f.hom ≫ (coneEquivFunctorObj F U c').π.app WalkingParallelPair.one =
    (coneEquivFunctorObj F U c).π.app WalkingParallelPair.one

Tactic:
· dsimp
            ext
            simp

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

Goal state before:
pp: case refine_2
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ y, ⨅ x, f (inl x, inr y) ≤ ⨆ y, ⨅ x, g (inl x, inr y) + dist f g
y : Y
⊢ BddBelow (range fun x ↦ g (inl x, inr y))

Tactic:
· change BddBelow (range fun x : X => g (inl x, inr y))
      exact ⟨cg, forall_mem_range.2 fun i => Hcg _⟩

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: case zero
X : Type u_1
inst✝ : TopologicalSpace X
m : ℕ
hmn : m ≤ 0
H : HasSmallInductiveDimensionLT X m
⊢ HasSmallInductiveDimensionLT X 0

Tactic:
simp_all

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
n : ℕ
h : smallInductiveDimension X ≤ ↑n
⊢ HasSmallInductiveDimensionLE X n

Tactic:
apply hasSmallInductiveDimensionLT_of_smallInductiveDimension_lt (h.trans_lt _)
  exact_mod_cast n.lt_add_one

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

Goal state before:
pp: X : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X n)
inst✝ : ∀ (n : ℕ), Nonempty (X n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
⊢ CompleteSpace GHSpace

Tactic:
have : ∀ n : ℕ, 0 < d n := fun _ ↦ by positivity

Goal state after:
X : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X n)
inst✝ : ∀ (n : ℕ), Nonempty (X n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
⊢ CompleteSpace GHSpace