mathlib-initiative/mathlib-tactics huggingface dataset:
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
E1 : ∀ (y : Y), (⨅ x, g (inl x, inr y)) + dist f g = ⨅ x, g (inl x, inr y) + dist f g
⊢ BddAbove (range fun y ↦ ⨅ x, g (inl x, inr y))

Tactic:
simpa using HD_bound_aux2 _ 0

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : Cb X Y
h : f ∈ candidatesB X Y
r : ℝ
hr : HD (optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
z : Y
I1 : ⨆ y, ⨅ x, (optimalGHDist X Y) (inl x, inr y) < r
⊢ BddAbove (range fun y ↦ ⨅ x, (optimalGHDist X Y) (inl x, inr y))

Tactic:
simpa using HD_bound_aux2 _ 0

Goal state after:
Module: Mathlib.Topology.UniformSpace.Ultra.Basic

Goal state before:
pp: case refine_2
X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
⊢ ∀ (i' : Set X), x ∈ i' ∧ IsClopen i' → id i' ∈ 𝓝 x

Tactic:
rintro u ⟨hx, hu⟩

Goal state after:
case refine_2
X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
u : Set X
hx : x ∈ u
hu : IsClopen u
⊢ id u ∈ 𝓝 x