mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Tactic.NormNum.Irrational

Goal state before:
pp: x y : ℝ
x_num x_den y_num y_den k_den : ℕ
hy_isNNRat : IsNNRat y y_num y_den
hx_coprime : x_num.Coprime x_den
hy_coprime : y_num.Coprime y_den
hd1 : k_den ^ y_den < x_den
hd2 : x_den < (k_den + 1) ^ y_den
hx_inv : Invertible ↑x_den
hx_eq : x = ↑x_num * ⅟↑x_den
⊢ Irrational (x ^ y)⁻¹

Tactic:
← Real.inv_rpow (by simp only [hx_eq, invOf_eq_inv]; positivity)

Goal state after:
x y : ℝ
x_num x_den y_num y_den k_den : ℕ
hy_isNNRat : IsNNRat y y_num y_den
hx_coprime : x_num.Coprime x_den
hy_coprime : y_num.Coprime y_den
hd1 : k_den ^ y_den < x_den
hd2 : x_den < (k_den + 1) ^ y_den
hx_inv : Invertible ↑x_den
hx_eq : x = ↑x_num * ⅟↑x_den
⊢ Irrational (x⁻¹ ^ y)
Module: Mathlib.Tactic.NormNum.RealSqrt

Goal state before:
pp: n sn d sd : ℕ
hn : sn * sn = n
hd : sd * sd = d
inv✝ : Invertible ↑d
⊢ ↑(sd * sd) ≠ 0

Tactic:
hd

Goal state after:
n sn d sd : ℕ
hn : sn * sn = n
hd : sd * sd = d
inv✝ : Invertible ↑d
⊢ ↑d ≠ 0
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case h
α : Type u
β : Type v
inst✝² : DecidableEq α
inst✝¹ : Zero β
inst✝ : DecidableEq β
a : α
A : List ((_ : α) × β)
y : β
h : ¬(List.dlookup a A).getD 0 = 0
⊢ ⟨a, (List.dlookup a A).getD 0⟩ ∈ A.dedupKeys ∧ (decide ¬(List.dlookup a A).getD 0 = 0) = true

Tactic:
← List.dlookup_dedupKeys

Goal state after:
case h
α : Type u
β : Type v
inst✝² : DecidableEq α
inst✝¹ : Zero β
inst✝ : DecidableEq β
a : α
A : List ((_ : α) × β)
y : β
h : ¬(List.dlookup a A.dedupKeys).getD 0 = 0
⊢ ⟨a, (List.dlookup a A.dedupKeys).getD 0⟩ ∈ A.dedupKeys ∧ (decide ¬(List.dlookup a A.dedupKeys).getD 0 = 0) = true
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case neg.cons.cons
α : Type u
inst✝ : DecidableEq α
x val x' : α
xs : List α
y : α
ys : List α
xs_ih : val ∈ ys ↔ x ∈ xs
h₁ : (x' :: xs).length = (y :: ys).length
h : ¬x' = x
h₃ : dlookup x (map Prod.toSigma (xs.zip ys)) = some val
h₅ : List.Pairwise (fun x1 x2 ↦ x1 ≠ x2) xs
h₀ : ∀ a' ∈ xs, x' ≠ a'
h₄ : List.Pairwise (fun x1 x2 ↦ x1 ≠ x2) ys
h₂ : ∀ a' ∈ ys, y ≠ a'
h₆ : xs.length = ys.length
⊢ val ∈ ys

Tactic:
rw [← Option.mem_def, List.mem_dlookup_iff] at h₃

Goal state after:
case neg.cons.cons
α : Type u
inst✝ : DecidableEq α
x val x' : α
xs : List α
y : α
ys : List α
xs_ih : val ∈ ys ↔ x ∈ xs
h₁ : (x' :: xs).length = (y :: ys).length
h : ¬x' = x
h₃ : ⟨x, val⟩ ∈ map Prod.toSigma (xs.zip ys)
h₅ : List.Pairwise (fun x1 x2 ↦ x1 ≠ x2) xs
h₀ : ∀ a' ∈ xs, x' ≠ a'
h₄ : List.Pairwise (fun x1 x2 ↦ x1 ≠ x2) ys
h₂ : ∀ a' ∈ ys, y ≠ a'
h₆ : xs.length = ys.length
⊢ val ∈ ys

case neg.cons.cons
α : Type u
inst✝ : DecidableEq α
x val x' : α
xs : List α
y : α
ys : List α
xs_ih : val ∈ ys ↔ x ∈ xs
h₁ : (x' :: xs).length = (y :: ys).length
h : ¬x' = x
h₃ : val ∈ dlookup x (map Prod.toSigma (xs.zip ys))
h₅ : List.Pairwise (fun x1 x2 ↦ x1 ≠ x2) xs
h₀ : ∀ a' ∈ xs, x' ≠ a'
h₄ : List.Pairwise (fun x1 x2 ↦ x1 ≠ x2) ys
h₂ : ∀ a' ∈ ys, y ≠ a'
h₆ : xs.length = ys.length
⊢ (map Prod.toSigma (xs.zip ys)).NodupKeys
Module: Mathlib.Testing.Plausible.Functions

Goal state before:
pp: case neg.cons.cons
α : Type u
inst✝ : DecidableEq α
x val x' : α
xs : List α
y : α
ys : List α
xs_ih : val ∈ ys ↔ x ∈ xs
h₁ : (x' :: xs).length = (y :: ys).length
h : ¬x' = x
h₃ : val ∈ dlookup x (map Prod.toSigma (xs.zip ys))
h₅ : List.Pairwise (fun x1 x2 ↦ x1 ≠ x2) xs
h₀ : ∀ a' ∈ xs, x' ≠ a'
h₄ : List.Pairwise (fun x1 x2 ↦ x1 ≠ x2) ys
h₂ : ∀ a' ∈ ys, y ≠ a'
h₆ : xs.length = ys.length
⊢ (map (fun x ↦ x.1) (xs.zip ys)).Nodup

Tactic:
rwa [List.map_fst_zip (le_of_eq h₆)]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.EmbeddingOfLocal

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜₁ E
inst✝⁷ : Module 𝕜₂ F
σ : 𝕜₁ →+* 𝕜₂
f : E →ₛₗ[σ] F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜₁ E
inst✝¹ : ContinuousSMul 𝕜₂ F
inst✝ : RingHomIsometric σ
V : Set E
V_mem : V ∈ 𝓝 0
H : IsInducing (V.domRestrict ⇑f)
⊢ inst✝⁶ = TopologicalSpace.induced (⇑f) inst✝⁵

Tactic:
have := topologicalAddGroup_induced f

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜₁ E
inst✝⁷ : Module 𝕜₂ F
σ : 𝕜₁ →+* 𝕜₂
f : E →ₛₗ[σ] F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜₁ E
inst✝¹ : ContinuousSMul 𝕜₂ F
inst✝ : RingHomIsometric σ
V : Set E
V_mem : V ∈ 𝓝 0
H : IsInducing (V.domRestrict ⇑f)
this : IsTopologicalAddGroup E
⊢ inst✝⁶ = TopologicalSpace.induced (⇑f) inst✝⁵
Module: Mathlib.Topology.Algebra.PontryaginDual

Goal state before:
pp: case refine_1
A : Type u_1
B : Type u_2
C : Type u_3
G : Type u_4
H : Type u_5
inst✝¹² : Monoid A
inst✝¹¹ : Monoid B
inst✝¹⁰ : Monoid C
inst✝⁹ : CommGroup G
inst✝⁸ : Group H
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalSpace H
inst✝² : IsTopologicalGroup G
inst✝¹ : IsTopologicalGroup H
inst✝ : LocallyCompactSpace H
Vn : ℕ → Set Circle := fun n ↦ Circle.centeredArc (π / 2 ^ (n + 1))
hVn : ∀ (n : ℕ) (x : Circle), x ∈ Vn n ↔ |(↑x).arg| < π / 2 ^ (n + 1)
n : ℕ
x : Circle
h1 : |(↑x).arg| < π / 2 ^ (n + 1)
h2 : |(↑x * ↑x).arg| < π / 2 ^ (n + 1)
⊢ 2 * |(↑x).arg| < π

Tactic:
← lt_div_iff₀' two_pos

Goal state after:
case refine_1
A : Type u_1
B : Type u_2
C : Type u_3
G : Type u_4
H : Type u_5
inst✝¹² : Monoid A
inst✝¹¹ : Monoid B
inst✝¹⁰ : Monoid C
inst✝⁹ : CommGroup G
inst✝⁸ : Group H
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalSpace H
inst✝² : IsTopologicalGroup G
inst✝¹ : IsTopologicalGroup H
inst✝ : LocallyCompactSpace H
Vn : ℕ → Set Circle := fun n ↦ Circle.centeredArc (π / 2 ^ (n + 1))
hVn : ∀ (n : ℕ) (x : Circle), x ∈ Vn n ↔ |(↑x).arg| < π / 2 ^ (n + 1)
n : ℕ
x : Circle
h1 : |(↑x).arg| < π / 2 ^ (n + 1)
h2 : |(↑x * ↑x).arg| < π / 2 ^ (n + 1)
⊢ |(↑x).arg| < π / 2
Module: Mathlib.Topology.Algebra.Ring.Compact

Goal state before:
pp: case hab
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : TopologicalSpace R
inst✝³ : IsTopologicalRing R
inst✝² : CompactSpace R
inst✝¹ : T2Space R
inst✝ : IsArtinianRing R
n : ℕ
hn : ⊥.jacobson ^ n = 0
⊢ ∏ p, p.asIdeal ≤ ⊥.jacobson

Tactic:
Ideal.jacobson_bot,

Goal state after:
case hab
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : TopologicalSpace R
inst✝³ : IsTopologicalRing R
inst✝² : CompactSpace R
inst✝¹ : T2Space R
inst✝ : IsArtinianRing R
n : ℕ
hn : ⊥.jacobson ^ n = 0
⊢ ∏ p, p.asIdeal ≤ Ring.jacobson R
Module: Mathlib.Topology.Baire.BaireMeasurable

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
s : Set α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
h : MeasurableSet s
⊢ BaireMeasurableSet s

Tactic:
borelize α

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
s : Set α
inst✝ : BorelSpace α
h : MeasurableSet s
this✝ : MeasurableSpace α := borel α
⊢ BaireMeasurableSet s
Module: Mathlib.Topology.Baire.BaireMeasurable

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h : IsOpen[inst✝] s
⊢ BaireMeasurableSet s

Tactic:
borelize α

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h : IsOpen[inst✝] s
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ BaireMeasurableSet s
Module: Mathlib.Topology.Baire.BaireMeasurable

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h : BaireMeasurableSet s
⊢ ∃ u, IsOpen[inst✝] u ∧ s =ᶠ[residual α] u

Tactic:
borelize α

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h : BaireMeasurableSet s
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ ∃ u, IsOpen[inst✝] u ∧ s =ᶠ[residual α] u
Module: Mathlib.Topology.Category.TopCat.Sphere

Goal state before:
pp: n : ℕ
t : Set (ULift.{?u.17, 0} ↑(Metric.closedBall 0 1))
x✝ : IsOpen t
s : Set ↑(Metric.closedBall 0 1)
r : Set (EuclideanSpace ℝ (Fin n))
hro : IsOpen r
hrs : Subtype.val ⁻¹' r = s
hst : ULift.down ⁻¹' s = t
⊢ ∃ t_1,
    IsOpen t_1 ∧
      ULift.down ⁻¹' t_1 =
        (fun x ↦
            match x with
            | { down := ⟨p, hp⟩ } => { down := ⟨p, ⋯⟩ }) ⁻¹'
          t

Tactic:
← hst,

Goal state after:
n : ℕ
t : Set (ULift.{?u.17, 0} ↑(Metric.closedBall 0 1))
x✝ : IsOpen t
s : Set ↑(Metric.closedBall 0 1)
r : Set (EuclideanSpace ℝ (Fin n))
hro : IsOpen r
hrs : Subtype.val ⁻¹' r = s
hst : ULift.down ⁻¹' s = t
⊢ ∃ t,
    IsOpen t ∧
      ULift.down ⁻¹' t =
        (fun x ↦
            match x with
            | { down := ⟨p, hp⟩ } => { down := ⟨p, ⋯⟩ }) ⁻¹'
          ULift.down ⁻¹' s
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
A : Set X
⊢ IsClosed[inst✝¹] (derivedSet A)

Tactic:
rw [← derivedSet_closure, isClosed_iff_derivedSet_subset]
  apply derivedSet_mono
  simp [← isClosed_iff_derivedSet_subset]

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
⊢ perfectKernel ∅ = ∅

Tactic:
by
  simpa using perfectKernel_subset ∅

Goal state after:
Module: Mathlib.Topology.CWComplex.Classical.Finite

Goal state before:
pp: case inr
X : Type u_2
inst✝¹ : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
finite : _root_.Finite ((n : ℕ) × cell C n)
h✝ : Nonempty ((n : ℕ) × cell C n)
x✝ : Fintype ((n : ℕ) × cell C n)
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → IsEmpty (cell C b)

Tactic:
let A := (Finset.univ : Finset (Σ n, cell C n)).image Sigma.fst

Goal state after:
case inr
X : Type u_2
inst✝¹ : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
finite : _root_.Finite ((n : ℕ) × cell C n)
h✝ : Nonempty ((n : ℕ) × cell C n)
x✝ : Fintype ((n : ℕ) × cell C n)
A : Finset ℕ := Finset.image Sigma.fst Finset.univ
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → IsEmpty (cell C b)
Module: Mathlib.Topology.Convenient.Category

Goal state before:
pp: ι : Type t
X : ι → Type u
inst✝ : (i : ι) → TopologicalSpace (X i)
Y Z : ContinuousGeneratedByCat X
g : (toTopCat X).obj Y ⟶ (toTopCat X).obj Z
⊢ ContinuousGeneratedBy X (⇑WithGeneratedByTopology.equiv ∘ ⇑(TopCat.Hom.hom g) ∘ ⇑WithGeneratedByTopology.equiv.symm)

Tactic:
by
      rw [continuousGeneratedBy_iff]
      exact g.hom.continuous

Goal state after:
Module: Mathlib.Topology.CWComplex.Classical.Basic

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
I J : (n : ℕ) → Set (cell C n)
hIJ : D ∪ ⋃ n, ⋃ j, openCell n ↑j = D ∪ ⋃ n, ⋃ j, openCell n ↑j
n : ℕ
⊢ I n ⊆ J n

Tactic:
by
  intro i hi
  by_contra hJ
  have h : openCell n i ⊆ D ∪ ⋃ n, ⋃ (j : J n), openCell (C := C) n j :=
    hIJ.symm ▸ subset_union_of_subset_right
      (subset_iUnion_of_subset n (subset_iUnion_of_subset ⟨i, hi⟩ (subset_refl (openCell n i)))) D
  have h' : Disjoint (openCell n i) (D ∪ ⋃ n, ⋃ (j : J n), openCell (C := C) n j) := by
    simp_rw [disjoint_union_right, disjoint_iUnion_right]
    exact ⟨disjointBase n i, fun m j ↦ disjoint_openCell_of_ne (by lia)⟩
  rw [disjoint_of_subset_iff_left_eq_empty h] at h'
  exact notMem_empty _ (h' ▸ map_zero_mem_openCell n i)

Goal state after:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Span

Goal state before:
pp: I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
a : I
as : List I
ha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)
c : Products I →₀ ℤ
hc : ↑c.support ⊆ {m | ↑m ≤ as}
m : Products I
hm : c m ≠ 0
hsm :
  ∀ (c : ℤ) (x : LocallyConstant ↑(π C fun x ↦ x ∈ s) ℤ),
    e (π C fun x ↦ x ∈ s) a * c • x = c • (e (π C fun x ↦ x ∈ s) a * x)
⊢ e (π C fun x ↦ x ∈ s) a * c m • Products.eval (π C fun x ↦ x ∈ s) m ∈
    Submodule.span ℤ (Products.eval (π C fun x ↦ x ∈ s) '' {m | ↑m ≤ a :: as})

Tactic:
rw [hsm]

Goal state after:
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
a : I
as : List I
ha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)
c : Products I →₀ ℤ
hc : ↑c.support ⊆ {m | ↑m ≤ as}
m : Products I
hm : c m ≠ 0
hsm :
  ∀ (c : ℤ) (x : LocallyConstant ↑(π C fun x ↦ x ∈ s) ℤ),
    e (π C fun x ↦ x ∈ s) a * c • x = c • (e (π C fun x ↦ x ∈ s) a * x)
⊢ c m • (e (π C fun x ↦ x ∈ s) a * Products.eval (π C fun x ↦ x ∈ s) m) ∈
    Submodule.span ℤ (Products.eval (π C fun x ↦ x ∈ s) '' {m | ↑m ≤ a :: as})
Module: Mathlib.Topology.Category.Profinite.Nobeling.Basic

Goal state before:
pp: case refine_1
I : Type u
C : Set (I → Bool)
J K : I → Prop
inst✝¹ : (i : I) → Decidable (J i)
inst✝ : (i : I) → Decidable (K i)
h : ∀ (i : I), J i → K i
z : I → Bool
hz : z ∈ C
⊢ Proj J z = (Proj J ∘ Proj K) z

Tactic:
rw [proj_comp_of_subset J K h]

Goal state after:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Basic

Goal state before:
pp: case refine_2
I : Type u
C : Set (I → Bool)
J K : I → Prop
inst✝¹ : (i : I) → Decidable (J i)
inst✝ : (i : I) → Decidable (K i)
h : ∀ (i : I), J i → K i
y : I → Bool
hy : y ∈ C
⊢ (Proj J ∘ Proj K) y = Proj J y

Tactic:
rw [proj_comp_of_subset J K h]

Goal state after:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Basic

Goal state before:
pp: case neg
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
h✝ : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l✝ : Products I
L : Products I → Prop := fun m ↦ Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
l : Products I
h : ∀ y < l, L y
hl : ¬Products.isGood C l
⊢ L l

Tactic:
simp only [Products.isGood, not_not] at hl

Goal state after:
case neg
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
h✝ : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l✝ : Products I
L : Products I → Prop := fun m ↦ Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
l : Products I
h : ∀ y < l, L y
hl : Products.eval C l ∈ Submodule.span ℤ (Products.eval C '' {m | m < l})
⊢ L l
Module: Mathlib.Topology.Category.Profinite.Nobeling.Basic

Goal state before:
pp: I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : ∀ i ∈ ↑l, ord I i < o₁
f : LocallyConstant ↑(π C fun x ↦ ord I x < o₂) ℤ
m : Products I
hm : m < l
⊢ eval (π C fun x ↦ ord I x < o₂) m = f ↔ (πs' C h) (eval (π C fun x ↦ ord I x < o₁) m) = f

Tactic:
eval_πs' C h (lt_ord_of_lt hm hl)

Goal state after:
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : ∀ i ∈ ↑l, ord I i < o₁
f : LocallyConstant ↑(π C fun x ↦ ord I x < o₂) ℤ
m : Products I
hm : m < l
⊢ eval (π C fun x ↦ ord I x < o₂) m = f ↔ eval (π C fun x ↦ ord I x < o₂) m = f
Module: Mathlib.Topology.CWComplex.Classical.Basic

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
n : ℕ
j : cell C n
⊢ closedCell n j ⊆ ↑(skeletonLT C (↑n + 1))

Tactic:
by
  intro x xmem
  rw [coe_skeletonLT]
  right
  simp_rw [mem_iUnion, exists_prop]
  refine ⟨n, (by norm_cast; exact lt_add_one n), ⟨j,xmem⟩⟩

Goal state after:
Module: Mathlib.Topology.Compactness.CompactSystem

Goal state before:
pp: α : Type u_1
S : Set (Set α)
h : IsCompactSystem S
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert ∅ S
⊢ ⋂ i, s i = ∅ → ∃ n, dissipate s n = ∅

Tactic:
hd

Goal state after:
α : Type u_1
S : Set (Set α)
h : IsCompactSystem S
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert ∅ S
hd : ⋂ i, s i = ∅
⊢ ∃ n, dissipate s n = ∅
Module: Mathlib.Topology.Compactness.CompactSystem

Goal state before:
pp: case inr
α : Type u_1
S : Set (Set α)
h : ∀ (C : ℕ → Set α), (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → (⋂ i, C i).Nonempty
h✝ : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
⊢ (∀ (n : ℕ), (dissipate s n).Nonempty) → (⋂ i, s i).Nonempty

Tactic:
hd

Goal state after:
case inr
α : Type u_1
S : Set (Set α)
h : ∀ (C : ℕ → Set α), (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → (⋂ i, C i).Nonempty
h✝ : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
⊢ (⋂ i, s i).Nonempty
Module: Mathlib.Topology.Compactness.CountablyCompact

Goal state before:
pp: case neg
E : Type u_2
inst✝ : TopologicalSpace E
A : Set E
hA : IsCountablyCompact A
hl : IsLindelof A
ι : Type u_2
U : ι → Set E
hUo : ∀ (i : ι), IsOpen[inst✝] (U i)
hAU : A ⊆ ⋃ i, U i
h : IsEmpty ι
⊢ ∃ t, A ⊆ ⋃ i ∈ t, U i

Tactic:
· exact ⟨∅, by simp_all⟩

Goal state after:
Module: Mathlib.Topology.Convenient.Localization

Goal state before:
pp: ι : Type t
X : ι → Type u
inst✝ : (i : ι) → TopologicalSpace (X i)
⊢ (morphismPropertyWithGeneratedByTopologyEquiv X).IsInvertedBy (toContinuousGeneratedByCat X)

Tactic:
rintro _ _ _ ⟨Z⟩

Goal state after:
ι : Type t
X✝ : ι → Type u
inst✝ : (i : ι) → TopologicalSpace (X✝ i)
X Y Z : TopCat
⊢ IsIso ((toContinuousGeneratedByCat X✝).map (GeneratedByTopCat.adjCounit.app Z))
Module: Mathlib.Topology.ContinuousMap.SecondCountableSpace

Goal state before:
pp: case h.refine_2
X : Type u_1
Y : Type u_2
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : SecondCountableTopology X
inst✝¹ : LocallyCompactSpace X
inst✝ : SecondCountableTopology Y
this : ∀ (U : ↑(countableBasis X)), LocallyCompactSpace ↑↑U
K : (U : ↑(countableBasis X)) → CompactExhaustion ↑↑U := fun U ↦ CompactExhaustion.choice ↑↑U
f : C(X, Y)
V : Set Y
hVo : IsOpen[inst✝³] V
U : { x // x ∈ countableBasis X }
hUV : ↑U ⊆ ⇑f ⁻¹' V
x : { x // x ∈ ↑U }
hxV : ↑x ∈ ⇑f ⁻¹' V
n : ℕ
hn : (K U) n ∈ 𝓝 x
⊢ ∃ K_1 ∈ ⋃ U, range fun n ↦ Subtype.val '' (K U) n, K_1 ∈ 𝓝 ↑x ∧ MapsTo (⇑f) K_1 V

Tactic:
refine ⟨K U n, mem_iUnion.2 ⟨U, mem_range_self _⟩, ?_, ?_⟩

Goal state after:
case h.refine_2.refine_1
X : Type u_1
Y : Type u_2
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : SecondCountableTopology X
inst✝¹ : LocallyCompactSpace X
inst✝ : SecondCountableTopology Y
this : ∀ (U : ↑(countableBasis X)), LocallyCompactSpace ↑↑U
K : (U : ↑(countableBasis X)) → CompactExhaustion ↑↑U := fun U ↦ CompactExhaustion.choice ↑↑U
f : C(X, Y)
V : Set Y
hVo : IsOpen[inst✝³] V
U : { x // x ∈ countableBasis X }
hUV : ↑U ⊆ ⇑f ⁻¹' V
x : { x // x ∈ ↑U }
hxV : ↑x ∈ ⇑f ⁻¹' V
n : ℕ
hn : (K U) n ∈ 𝓝 x
⊢ Subtype.val '' (K U) n ∈ 𝓝 ↑x

case h.refine_2.refine_2
X : Type u_1
Y : Type u_2
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : SecondCountableTopology X
inst✝¹ : LocallyCompactSpace X
inst✝ : SecondCountableTopology Y
this : ∀ (U : ↑(countableBasis X)), LocallyCompactSpace ↑↑U
K : (U : ↑(countableBasis X)) → CompactExhaustion ↑↑U := fun U ↦ CompactExhaustion.choice ↑↑U
f : C(X, Y)
V : Set Y
hVo : IsOpen[inst✝³] V
U : { x // x ∈ countableBasis X }
hUV : ↑U ⊆ ⇑f ⁻¹' V
x : { x // x ∈ ↑U }
hxV : ↑x ∈ ⇑f ⁻¹' V
n : ℕ
hn : (K U) n ∈ 𝓝 x
⊢ MapsTo (⇑f) (Subtype.val '' (K U) n) V
Module: Mathlib.Topology.Convenient.Localization

Goal state before:
pp: ι : Type t
X : ι → Type u
inst✝ : (i : ι) → TopologicalSpace (X i)
⊢ (morphismPropertyWithGeneratedByTopologyEquiv X).IsInvertedBy toGeneratedByTopCat

Tactic:
rintro _ _ _ ⟨Z⟩

Goal state after:
ι : Type t
X✝ : ι → Type u
inst✝ : (i : ι) → TopologicalSpace (X✝ i)
X Y Z : TopCat
⊢ IsIso (toGeneratedByTopCat.map (GeneratedByTopCat.adjCounit.app Z))
Module: Mathlib.Topology.ContinuousMap.SecondCountableSpace

Goal state before:
pp: case h.refine_2.refine_1
X : Type u_1
Y : Type u_2
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : SecondCountableTopology X
inst✝¹ : LocallyCompactSpace X
inst✝ : SecondCountableTopology Y
this : ∀ (U : ↑(countableBasis X)), LocallyCompactSpace ↑↑U
K : (U : ↑(countableBasis X)) → CompactExhaustion ↑↑U := fun U ↦ CompactExhaustion.choice ↑↑U
f : C(X, Y)
V : Set Y
hVo : IsOpen[inst✝³] V
U : { x // x ∈ countableBasis X }
hUV : ↑U ⊆ ⇑f ⁻¹' V
x : { x // x ∈ ↑U }
hxV : ↑x ∈ ⇑f ⁻¹' V
n : ℕ
hn : (K U) n ∈ 𝓝 x
⊢ Subtype.val '' (K U) n ∈ 𝓝 ↑x

Tactic:
rw [← map_nhds_subtype_coe_eq_nhds x.2]
      exacts [image_mem_map hn, (isOpen_of_mem_countableBasis U.2).mem_nhds x.2]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.T0Sierpinski

Goal state before:
pp: case a
X : Type u_1
t : TopologicalSpace X
u : Set X
h : IsOpen[t] u
⊢ IsOpen[⨅ u, induced (fun x ↦ x ∈ u) sierpinskiSpace] u

Tactic:
rw [← generateFrom_iUnion_isOpen]

Goal state after:
case a
X : Type u_1
t : TopologicalSpace X
u : Set X
h : IsOpen[t] u
⊢ IsOpen[generateFrom (⋃ i, {s | IsOpen[induced (fun x ↦ x ∈ i) sierpinskiSpace] s})] u
Module: Mathlib.Topology.Filter

Goal state before:
pp: α : Type u_2
l₁ l₂ : Filter α
⊢ 𝓝 (l₁ ⊓ l₂) = 𝓝 l₁ ⊓ 𝓝 l₂

Tactic:
by
  simpa only [iInf_bool_eq] using! Filter.nhds_iInf fun b => cond b l₁ l₂

Goal state after:
Module: Mathlib.Topology.Filter

Goal state before:
pp: X : Type u_4
inst✝ : TopologicalSpace X
x : X
⊢ 𝓝 (𝓝 x) = ⨅ s, ⨅ (_ : IsOpen[inst✝] s), ⨅ (_ : x ∈ s), 𝓟 (Iic (𝓟 s))

Tactic:
simp only [(nhds_basis_opens x).nhds.eq_biInf, iInf_and, @iInf_comm _ (_ ∈ _)]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst✝ : DecidableEq T
hJ : J.Nonempty
ha : 1 < a
hn : 1 ≤ n
hJ_card : ↑(#J) ≤ a ^ n
i : ℕ
⊢ (logSizeBallSeq J hJ a c 0).radius ≤ n

Tactic:
simp only [radius_logSizeBallSeq_zero, logSizeRadius, ha, ↓reduceDIte]

Goal state after:
T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst✝ : DecidableEq T
hJ : J.Nonempty
ha : 1 < a
hn : 1 ≤ n
hJ_card : ↑(#J) ≤ a ^ n
i : ℕ
⊢ Nat.find ⋯ ≤ n
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: case pos
T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝ : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : (logSizeBallSeq J hJ a c i).point ∈ J
h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty
⊢ (logSizeBallSeq J hJ a c (i + 1)).point ∈ J

Tactic:
refine Finset.mem_of_subset ?_ (point_mem_finset_logSizeBallSeq hJ (i + 1) h)
      apply finset_logSizeBallSeq_subset_logSizeBallSeq_init

Goal state after:
Module: Mathlib.Topology.Instances.RatLemmas

Goal state before:
pp: s : Set ℚ
hs : IsPreconnected s
x : ℚ
hx : x ∈ s
y : ℚ
hy : y ∈ s
H : x ≠ y
hlt : x < y
z : ℝ
hz : Irrational z
hxz : ↑x < z
hzy : z < ↑y
this : (Rat.cast '' s).OrdConnected
⊢ False

Tactic:
have : z ∈ Rat.cast '' s :=
    this.out (mem_image_of_mem _ hx) (mem_image_of_mem _ hy) ⟨hxz.le, hzy.le⟩

Goal state after:
s : Set ℚ
hs : IsPreconnected s
x : ℚ
hx : x ∈ s
y : ℚ
hy : y ∈ s
H : x ≠ y
hlt : x < y
z : ℝ
hz : Irrational z
hxz : ↑x < z
hzy : z < ↑y
this✝ : (Rat.cast '' s).OrdConnected
this : z ∈ Rat.cast '' s
⊢ False
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: case inl
N : Type u_1
inst✝ : DecidableEq N
i : N
t₀ : ↑I
t : { j // j ≠ i } → ↑I
H : t₀ = 0 ∨ t₀ = 1
⊢ (insertAt i) (t₀, t) ∈ boundary N

Tactic:
use i; rwa [funSplitAt_symm_apply, dif_pos rfl]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝ : DecidableEq T
hJ : J.Nonempty
i : ℕ
ha : 1 < a
⊢ ↑(#(pairSetSeq J a c i)) ≤
    (if (logSizeBallSeq J hJ a c i).finset.Nonempty then 1 else 0) * a ^ (logSizeBallSeq J hJ a c i).radius

Tactic:
induction i with
  | zero =>
    simpa [pairSetSeq, hJ, finset_logSizeBallSeq_zero, logSizeBallStruct.ball,
      radius_logSizeBallSeq_zero] using! card_le_logSizeRadius_le_pow_logSizeRadius ha
  | succ i ih =>
    by_cases! h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty
    · simpa [pairSetSeq, logSizeBallStruct.ball, h, hJ]
        using! card_le_logSizeRadius_le_pow_logSizeRadius ha
    simp [pairSetSeq, logSizeBallStruct.ball, h, hJ]

Goal state after:
Module: Mathlib.Topology.Instances.CantorSet

Goal state before:
pp: ⊢ ⋂ n, preCantorSet n = (⋂ i, (fun x ↦ x / 3) '' preCantorSet i) ∪ ⋂ i, (fun x ↦ (2 + x) / 3) '' preCantorSet i

Tactic:
← Function.comp_def,

Goal state after:
⊢ ⋂ n, preCantorSet n =
    Set.iInter ((Set.image fun x ↦ x / 3) ∘ preCantorSet) ∪ Set.iInter ((Set.image fun x ↦ (2 + x) / 3) ∘ preCantorSet)