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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
x : ↑X
hx : x ∈ U
this✝¹ : Presheaf.IsLocallySurjective S.g.hom
W : Opens ↑X
Wle : W ≤ U
hW : x ∈ W
t₁ : ↑(S.X₂.obj.obj (op W))
ht₁ : (ConcreteCategory.hom (S.g.hom.app (op W))) t₁ = (s |_ W) Wle
t₂ : ToType (S.X₂.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W))) :=
  ((StructuredArrow.right t).snd |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ -
    (t₁ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
this✝ : (ConcreteCategory.hom (S.g.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₂ = 0
t₃ : ↑(S.X₁.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W)))
ht₃ : (ConcreteCategory.hom (S.f.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₃ = t₂
t₄ : ↑(S.X₁.obj.obj (op W))
ht₄ : (t₄ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = t₃
f : Fin 2 → Opens ↑X := ![unop (StructuredArrow.right t).fst, W]
sf : (i : Fin 2) → ↑(S.X₂.obj.obj (op (f i))) :=
  fun x ↦
    match x with
    | 0 => (StructuredArrow.right t).snd
    | 1 => t₁ + (ConcreteCategory.hom (S.f.hom.app (op W))) t₄
this : (sf 0 |_ (W ⊓ unop (StructuredArrow.right t).fst)) ⋯ = (sf 1 |_ (W ⊓ unop (StructuredArrow.right t).fst)) ⋯
⊢ (ConcreteCategory.hom (S.X₂.obj.map ((f 1).infLERight (f 0)).op)) (sf 0) =
    (ConcreteCategory.hom (S.X₂.obj.map ((f 1).infLELeft (f 0)).op)) (sf 1)

Tactic:
exact this

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
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
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 y : ↑(s p)
this : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
⊢ j < N q

Tactic:
rwa [Npq] at hjp

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

Goal state before:
pp: 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
⊢ ⨆ x, ⨅ y, f (inl x, inr y) ≤ HD g + dist f g

Tactic:
by grw [HD_lipschitz_aux1 f g, HD, ← le_max_left]

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
b : β
y : ↑Y
x : ↑X
⊢ x ∈ sublevelLeft X f b ↑y ↔ f ↑x ↑y ≤ b

Tactic:
simp [sublevelLeft]

Goal state after:
Module: Mathlib.Topology.Subpath

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ : ↑I
⊢ (γ.subpath t₀ t₁).symm = γ.subpath t₁ t₀

Tactic:
ext s
  simp [subpath]

Goal state after:
Module: Mathlib.Topology.Subpath

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t : ↑I
⊢ γ.subpath t t = refl (γ t)

Tactic:
ext s
  simp [subpath]

Goal state after:
Module: Mathlib.Topology.Subpath

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
⊢ γ.subpath 0 1 = γ.cast ⋯ ⋯

Tactic:
ext s
  simp [subpath]

Goal state after:
Module: Mathlib.Topology.Subpath

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ t₂ x✝¹ x✝ : ↑I
hx : x✝ ∈ {0, 1}
⊢ {
        toFun := fun x ↦
          ((γ.subpath t₀ (Icc.convexComb t₁ t₂ (x✝¹, x).1)).trans (γ.subpath (Icc.convexComb t₁ t₂ (x✝¹, x).1) t₂))
            (x✝¹, x).2,
        continuous_toFun := ⋯ }
      x✝ =
    ((γ.subpath t₀ t₁).trans (γ.subpath t₁ t₂)).toContinuousMap x✝

Tactic:
by
    rcases hx with rfl | rfl <;>
    simp

Goal state after:
Module: Mathlib.Topology.UniformSpace.OfCompactT2

Goal state before:
pp: γ : Type u_1
inst✝² : TopologicalSpace γ
inst✝¹ : CompactSpace γ
inst✝ : R1Space γ
𝓝Δ : Filter (γ × γ) := 𝓝ˢ (diagonal γ)
F : Filter (γ × γ) := 𝓝Δ.lift' fun s ↦ s ○ s
V : Set (γ × γ)
V_in : V ∈ 𝓝Δ
H : ¬F ⊓ 𝓟 Vᶜ = ⊥
⊢ False

Tactic:
have : NeBot (F ⊓ 𝓟 Vᶜ) := ⟨H⟩

Goal state after:
γ : Type u_1
inst✝² : TopologicalSpace γ
inst✝¹ : CompactSpace γ
inst✝ : R1Space γ
𝓝Δ : Filter (γ × γ) := 𝓝ˢ (diagonal γ)
F : Filter (γ × γ) := 𝓝Δ.lift' fun s ↦ s ○ s
V : Set (γ × γ)
V_in : V ∈ 𝓝Δ
H : ¬F ⊓ 𝓟 Vᶜ = ⊥
this : (F ⊓ 𝓟 Vᶜ).NeBot
⊢ False