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

Goal state before:
pp: case intro
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Finite ↑s
hs : s.Nontrivial
val✝ : Fintype ↑s
w : α × α
hxy : w ∈ s.offDiag.toFinset
hed : s.offDiag.toFinset.inf' ⋯ (uncurry dist) = uncurry dist w
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' ⋯ (uncurry dist) = dist x y

Tactic:
mem_toFinset

Goal state after:
case intro
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Finite ↑s
hs : s.Nontrivial
val✝ : Fintype ↑s
w : α × α
hed : s.offDiag.toFinset.inf' ⋯ (uncurry dist) = uncurry dist w
hxy : w ∈ s.offDiag
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' ⋯ (uncurry dist) = dist x y
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Finite ↑s
hs : s.Nontrivial
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y

Tactic:
by
  cases nonempty_fintype s
  simp_rw [hs.infsep_of_fintype]
  rcases Finset.exists_mem_eq_inf' (s := s.offDiag.toFinset) (by simpa) (uncurry dist) with
    ⟨w, hxy, hed⟩
  simp_rw [mem_toFinset] at hxy
  exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩

Goal state after:
Module: Mathlib.Topology.OmegaCompletePartialOrder

Goal state before:
pp: case left
α : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
⊢ ∀ (i : ℕ), c i ≤ ωSup c

pp: case right
α : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
⊢ ∀ (y : α), (∀ (i : ℕ), c i ≤ y) → ωSup c ≤ y

Tactic:
· apply le_ωSup

Goal state after:
case right
α : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
⊢ ∀ (y : α), (∀ (i : ℕ), c i ≤ y) → ωSup c ≤ y
Module: Mathlib.Topology.Sheaves.Skyscraper

Goal state before:
pp: X : TopCat
p₀ : ↑X
inst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝¹ : Category.{u, v} C
A : C
inst✝ : HasTerminal C
⊢ IsTerminal ((skyscraperPresheaf PUnit.unit A).obj (op ⊥))

Tactic:
by
          dsimp [skyscraperPresheaf]
          rw [if_neg]
          · exact terminalIsTerminal
          · #adaptation_note /-- 2024-03-24
            Previously the universe annotation was not needed here. -/
            exact Set.notMem_empty PUnit.unit.{u + 1}

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

Goal state before:
pp: case mp
C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
X : TopCat
ℱ 𝒢 : Presheaf C X
inst✝¹ : Limits.HasColimits C
inst✝ : Limits.PreservesFilteredColimits (forget C)
T : ℱ ⟶ 𝒢
hT : IsLocallySurjective T
⊢ ∀ (x : ↑X), Function.Surjective ⇑(ConcreteCategory.hom ((stalkFunctor C x).map T))

Tactic:
intro x g

Goal state after:
case mp
C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
X : TopCat
ℱ 𝒢 : Presheaf C X
inst✝¹ : Limits.HasColimits C
inst✝ : Limits.PreservesFilteredColimits (forget C)
T : ℱ ⟶ 𝒢
hT : IsLocallySurjective T
x : ↑X
g : CC ((stalkFunctor C x).obj 𝒢)
⊢ ∃ a, (ConcreteCategory.hom ((stalkFunctor C x).map T)) a = g
Module: Mathlib.Topology.Sheaves.MayerVietoris

Goal state before:
pp: case refine_1
T : Type u
inst✝ : TopologicalSpace T
sq : Square (Opens T)
h₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃
h₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃
⊢ (s : PullbackCone sq.f₂₄ sq.f₃₄) → s.pt ⟶ sq.X₁

Tactic:
intro s
        apply homOfLE
        rw [h₁, le_inf_iff]
        exact ⟨leOfHom s.fst, leOfHom s.snd⟩

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

Goal state before:
pp: 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 d : Cone (SheafConditionEqualizerProducts.diagram F U)
f : c ⟶ d
⊢ f.hom ≫ 𝟙 d.pt = 𝟙 c.pt ≫ f.hom

Tactic:
simp only [Category.comp_id, Category.id_comp]

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

Goal state before:
pp: X : TopCat
F : Presheaf (Type v) X
U U' : (Opens ↑X)ᵒᵖ
f : U ⟶ U'
⊢ (F.map f ≫ ↾fun f ↦ ⟨fun x ↦ (ConcreteCategory.hom (F.germ (unop U') ↑x ⋯)) f, ⋯⟩) =
    (↾fun f ↦ ⟨fun x ↦ (ConcreteCategory.hom (F.germ (unop U) ↑x ⋯)) f, ⋯⟩) ≫ F.sheafify.obj.map f

Tactic:
ext x
    apply Subtype.ext -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): Added `apply`
    ext ⟨u, m⟩
    exact germ_res_apply F f.unop u m x

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
this : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Tactic:
have isom : ∀ n, Isometry (coeZ ∘ Φ n ∘ (Y n).embed) := by
    intro n
    refine UniformSpace.Completion.coe_isometry.comp ?_
    exact (toInductiveLimit_isometry _ _).comp (Y n).isom

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
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
⊢ ∃ x, Tendsto u atTop (𝓝 x)
Module: Mathlib.Topology.Subpath

Goal state before:
pp: case succ
X : Type u_1
inst✝ : TopologicalSpace X
x : X
n✝ : ℕ
a✝ : (concat (fun x_1 ↦ x) fun x_1 ↦ refl x) = refl x
⊢ (concat (fun x_1 ↦ x) fun x_1 ↦ refl x) = refl x

Tactic:
rw [concat_succ]
    convert! refl_trans_refl

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

Goal state before:
pp: X : Type u_1
s : SetRel X X
t u v : Set X
inst✝ : s.IsTrans
htu : t ×ˢ u ⊆ s
huv : u ×ˢ v ⊆ s
hu : u.Nonempty
⊢ t ×ˢ v ⊆ s

Tactic:
by
  rintro ⟨a, b⟩ hab
  simp only [mem_prod] at hab
  obtain ⟨x, hx⟩ := hu
  exact s.trans (@htu ⟨a, x⟩ ⟨hab.left, hx⟩) (@huv ⟨x, b⟩ ⟨hx, hab.right⟩)

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: case a
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
cY : Convex ℝ Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
⊢ sup_inf ∈ lowerBounds {x | ∃ x_1 ∈ X, sup_y x_1 = x}

Tactic:
mem_lowerBounds

Goal state after:
case a
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
cY : Convex ℝ Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
⊢ ∀ x ∈ {x | ∃ x_1 ∈ X, sup_y x_1 = x}, sup_inf ≤ x