mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Star.LinearMap

Goal state before:
pp: R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁵ : Semiring R
inst✝¹⁴ : InvolutiveStar R
inst✝¹³ : AddCommMonoid E
inst✝¹² : Module R E
inst✝¹¹ : StarAddMonoid E
inst✝¹⁰ : StarModule R E
inst✝⁹ : AddCommMonoid F
inst✝⁸ : Module R F
inst✝⁷ : StarAddMonoid F
inst✝⁶ : StarModule R F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : ContinuousStar E
inst✝² : ContinuousStar F
inst✝¹ : SMulCommClass R R F
inst✝ : ContinuousConstSMul R F
x✝¹ : R
x✝ : WithConv (E →L[R] F)
⊢ star (x✝¹ • x✝) = star x✝¹ • star x✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithZeroMulInt

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : Valued R (WithZero (Multiplicative ℤ))
x : R
hx : v x ≤ exp (-1)
⊢ exp (-1) < exp 0

Tactic:
exp_lt_exp

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : Valued R (WithZero (Multiplicative ℤ))
x : R
hx : v x ≤ exp (-1)
⊢ -1 < 0
Module: Mathlib.Topology.Algebra.Valued.WithZeroMulInt

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : Valued R (WithZero (Multiplicative ℤ))
x : R
hx : v x ≤ exp (-1)
⊢ Tendsto (fun n ↦ x ^ n) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithZeroMulInt

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : Valued R (WithZero (Multiplicative ℤ))
x : R
hx : v x ≤ exp (-1)
γ : (WithZero (Multiplicative ℤ))ˣ
⊢ exp (-1) < exp 0

Tactic:
exp_lt_exp

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : Valued R (WithZero (Multiplicative ℤ))
x : R
hx : v x ≤ exp (-1)
γ : (WithZero (Multiplicative ℤ))ˣ
⊢ -1 < 0
Module: Mathlib.Topology.Algebra.Valued.WithZeroMulInt

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : Valued R (WithZero (Multiplicative ℤ))
x : R
hx : v x ≤ exp (-1)
γ : (WithZero (Multiplicative ℤ))ˣ
⊢ ∃ n, v x ^ n < ↑γ

Tactic:
by

Goal state after:
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
⊢ IsOpen
    ((fun x ↦
        match x with
        | { down := ⟨p, hp⟩ } => { down := ⟨p, ⋯⟩ }) ⁻¹'
      t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Sphere

Goal state before:
pp: n : ℕ
x : EuclideanSpace ℝ (Fin n)
hx : x ∈ Metric.sphere 0 1
y : EuclideanSpace ℝ (Fin n)
hy : y ∈ Metric.sphere 0 1
h :
  (ConcreteCategory.hom (diskBoundaryInclusion n)) { down := ⟨x, hx⟩ } =
    (ConcreteCategory.hom (diskBoundaryInclusion n)) { down := ⟨y, hy⟩ }
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Sphere

Goal state before:
pp: n : ℕ
⊢ Function.Injective ⇑(ConcreteCategory.hom (diskBoundaryInclusion n))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Sphere

Goal state before:
pp: n : ℕ
x : EuclideanSpace ℝ (Fin n)
hx : x ∈ Metric.ball 0 1
y : EuclideanSpace ℝ (Fin n)
hy : y ∈ Metric.ball 0 1
h :
  (ConcreteCategory.hom (ballInclusion n)) { down := ⟨x, hx⟩ } =
    (ConcreteCategory.hom (ballInclusion n)) { down := ⟨y, hy⟩ }
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Sphere

Goal state before:
pp: n : ℕ
⊢ Function.Injective ⇑(ConcreteCategory.hom (ballInclusion n))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Sphere

Goal state before:
pp: n : ℕ
⊢ CompactSpace ↑(𝔻 n)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Sphere

Goal state before:
pp: n : ℕ
⊢ CompactSpace ↑(∂𝔻 n)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
⊢ ∀ ⦃e : cell C 1⦄, e ∈ univ → Std.Symm fun x y ↦ cellFrontier 1 e = closedCell 0 x ∪ closedCell 0 y

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
x y z w : cell C 0
h1 : cellFrontier 1 e = closedCell 0 x ∪ closedCell 0 y
h2 : cellFrontier 1 e = closedCell 0 z ∪ closedCell 0 w
⊢ x = z ∨ x = w

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
⊢ e ∈ univ ↔ ∃ x y, cellFrontier 1 e = closedCell 0 x ∪ closedCell 0 y

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
C✝ D✝ : Set X✝
inst✝¹ : RelCWComplex C✝ D✝
X : Type u
inst✝ : TopologicalSpace X
C D : Set X
cell : ℕ → Type u
map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X
finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)
source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1
continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i)) (closedBall 0 1)
continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i).symm) (map n i).target
pairwiseDisjoint' : univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ni.snd) '' ball 0 1
disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (↑(map n i) '' ball 0 1) D
mapsTo :
  ∀ (n : ℕ) (i : cell n), MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j, ↑(map m j) '' closedBall 0 1)
closed' :
  ∀ A ⊆ C,
    (∀ (n : ℕ) (j : cell n), IsClosed[inst✝] (A ∩ ↑(map n j) '' closedBall 0 1)) ∧ IsClosed[inst✝] (A ∩ D) →
      IsClosed[inst✝] A
isClosedBase : IsClosed[inst✝] D
union' : D ∪ ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C
n : ℕ
i : cell n
⊢ ∃ I, MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
C✝ D✝ : Set X✝
inst✝¹ : RelCWComplex C✝ D✝
X : Type u
inst✝ : TopologicalSpace X
C D : Set X
cell : ℕ → Type u
map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X
finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)
source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1
continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i)) (closedBall 0 1)
continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i).symm) (map n i).target
pairwiseDisjoint' : univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ni.snd) '' ball 0 1
disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (↑(map n i) '' ball 0 1) D
mapsTo :
  ∀ (n : ℕ) (i : cell n), MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j, ↑(map m j) '' closedBall 0 1)
closed' :
  ∀ A ⊆ C,
    (∀ (n : ℕ) (j : cell n), IsClosed[inst✝] (A ∩ ↑(map n j) '' closedBall 0 1)) ∧ IsClosed[inst✝] (A ∩ D) →
      IsClosed[inst✝] A
isClosedBase : IsClosed[inst✝] D
union' : D ∪ ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C
n : ℕ
i : cell n
⊢ ∃ I, MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)

Tactic:
by
    use fun m ↦ finite_univ.toFinset (s := (univ : Set (cell m)))
    simp only [Finite.mem_toFinset, mem_univ, iUnion_true]
    exact mapsTo n i

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
x y : cell C 0
⊢ (OneSkeletonGraph C).IsLink e x y ↔ cellFrontier 1 e = {↑(map 0 x) ![], ↑(map 0 y) ![]}

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
⊢ (∃ x, (OneSkeletonGraph C).IsLoopAt e x) ↔ ∃ x, cellFrontier 1 e = closedCell 0 x

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
C✝ D : Set X✝
inst✝¹ : RelCWComplex C✝ D
X : Type u
inst✝ : TopologicalSpace X
C : Set X
cell : ℕ → Type u
map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X
finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)
source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1
continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i)) (closedBall 0 1)
continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i).symm) (map n i).target
pairwiseDisjoint' : univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ni.snd) '' ball 0 1
mapsTo :
  ∀ (n : ℕ) (i : cell n), MapsTo (↑(map n i)) (sphere 0 1) (⋃ m, ⋃ (_ : m < n), ⋃ j, ↑(map m j) '' closedBall 0 1)
closed' : ∀ A ⊆ C, (∀ (n : ℕ) (j : cell n), IsClosed[inst✝] (A ∩ ↑(map n j) '' closedBall 0 1)) → IsClosed[inst✝] A
union' : ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C
n : ℕ
i : cell n
⊢ ∃ I, MapsTo (↑(map n i)) (sphere 0 1) (⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
C✝ D : Set X✝
inst✝¹ : RelCWComplex C✝ D
X : Type u
inst✝ : TopologicalSpace X
C : Set X
cell : ℕ → Type u
map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X
finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)
source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1
continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i)) (closedBall 0 1)
continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i).symm) (map n i).target
pairwiseDisjoint' : univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ni.snd) '' ball 0 1
mapsTo :
  ∀ (n : ℕ) (i : cell n), MapsTo (↑(map n i)) (sphere 0 1) (⋃ m, ⋃ (_ : m < n), ⋃ j, ↑(map m j) '' closedBall 0 1)
closed' : ∀ A ⊆ C, (∀ (n : ℕ) (j : cell n), IsClosed[inst✝] (A ∩ ↑(map n j) '' closedBall 0 1)) → IsClosed[inst✝] A
union' : ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C
n : ℕ
i : cell n
⊢ ∃ I, MapsTo (↑(map n i)) (sphere 0 1) (⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)

Tactic:
by
    use fun m ↦ finite_univ.toFinset (s := (univ : Set (cell m)))
    simp only [Finite.mem_toFinset, mem_univ, iUnion_true]
    exact mapsTo n i

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
x✝ : ∃ x, cellFrontier 1 e = closedCell 0 x
x : cell C 0
hx : cellFrontier 1 e = closedCell 0 x
⊢ (cellFrontier 1 e).Subsingleton

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
C✝ D✝ : Set X✝
inst✝¹ : RelCWComplex C✝ D✝
X : Type u
inst✝ : TopologicalSpace X
C D : Set X
cell : ℕ → Type u
map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X
eventually_isEmpty_cell : ∀ᶠ (n : ℕ) in Filter.atTop, IsEmpty (cell n)
finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)
source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1
continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i)) (closedBall 0 1)
continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i).symm) (map n i).target
pairwiseDisjoint' : univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ni.snd) '' ball 0 1
disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (↑(map n i) '' ball 0 1) D
mapsTo :
  ∀ (n : ℕ) (i : cell n), MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j, ↑(map m j) '' closedBall 0 1)
isClosedBase : IsClosed[inst✝] D
union' : D ∪ ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C
n : ℕ
i : cell n
⊢ ∃ I, MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
h : (cellFrontier 1 e).Subsingleton
x y : cell C 0
hxy : cellFrontier 1 e = {↑(map 0 y) ![], ↑(map 0 x) ![]}
⊢ (fun x ↦ ↑(map 0 x) ![]) x ∈ {↑(map 0 y) ![], ↑(map 0 x) ![]}

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
C✝ D✝ : Set X✝
inst✝¹ : RelCWComplex C✝ D✝
X : Type u
inst✝ : TopologicalSpace X
C D : Set X
cell : ℕ → Type u
map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X
eventually_isEmpty_cell : ∀ᶠ (n : ℕ) in Filter.atTop, IsEmpty (cell n)
finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)
source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1
continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i)) (closedBall 0 1)
continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i).symm) (map n i).target
pairwiseDisjoint' : univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ni.snd) '' ball 0 1
disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (↑(map n i) '' ball 0 1) D
mapsTo :
  ∀ (n : ℕ) (i : cell n), MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j, ↑(map m j) '' closedBall 0 1)
isClosedBase : IsClosed[inst✝] D
union' : D ∪ ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C
n : ℕ
i : cell n
⊢ ∃ I, MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)

Tactic:
by
    use fun m ↦ finite_univ.toFinset (s := (univ : Set (cell m)))
    simp only [Finite.mem_toFinset, mem_univ, iUnion_true]
    exact mapsTo n i

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
h : (cellFrontier 1 e).Subsingleton
x y : cell C 0
hxy : cellFrontier 1 e = {↑(map 0 y) ![], ↑(map 0 x) ![]}
⊢ (fun x ↦ ↑(map 0 x) ![]) y ∈ {↑(map 0 y) ![], ↑(map 0 x) ![]}

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
h : (cellFrontier 1 e).Subsingleton
x : cell C 0
hxy : cellFrontier 1 e = {↑(map 0 x) ![], ↑(map 0 x) ![]}
⊢ cellFrontier 1 e = closedCell 0 x

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
e : cell C 1
⊢ (∃ x, (OneSkeletonGraph C).IsLoopAt e x) ↔ (cellFrontier 1 e).Subsingleton

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
β : Type u_2
inst✝ : TopologicalSpace β
F : Filter X
x : X
h : AccPt x F
f : X → β
hf1 : ContinuousAt f x
hf2 : Function.Injective f
⊢ AccPt (f x) (Filter.map f F)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
A B : Set X
⊢ derivedSet (A ∪ B) = derivedSet A ∪ derivedSet B

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
β : Type u_2
inst✝ : TopologicalSpace β
A : Set X
f : X → β
hf1 : Continuous[inst✝¹, inst✝] f
hf2 : Function.Injective f
⊢ f '' derivedSet A ⊆ derivedSet (f '' A)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
A : Set X
h : derivedSet A ⊆ A
a : X
ha : ClusterPt a (𝓟 A)
nh : a ∉ A
⊢ A = A \ {a}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
A : Set X
h : derivedSet A ⊆ A
⊢ IsClosed[inst✝] A

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
A : Set X
hA : IsClosed[inst✝] A
⊢ relDerivedSet A = derivedSet A

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
A : Set X
⊢ closure[inst✝] A = A ∪ derivedSet A

Tactic:
by

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

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
n : ℕ
i : cell C n
hi : i ∈ E.I n
⊢ closedCell n i ⊆ ↑E

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

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

Tactic:
by

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

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
⊢ D ∪ ⋃ n, ⋃ j, closedCell n ↑j = ↑E

Tactic:
by

Goal state after:
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:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
U : Set X
⊢ Preperfect U ↔ U = relDerivedSet U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
U : Set X
⊢ Perfect U ↔ U = derivedSet U

Tactic:
by

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

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
E : Subcomplex C
n : ℕ
i : cell C n
h : i ∉ E.I n
x✝¹ : ℕ
x✝ : ↑(E.I x✝¹)
⊢ ⟨n, i⟩ ≠ ⟨x✝¹, ↑x✝⟩

Tactic:
by

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

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
E : Subcomplex C
n : ℕ
i : cell C n
h : i ∉ E.I n
⊢ Disjoint (openCell n i) ↑E

Tactic:
by

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

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
⊢ univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ↑ni.snd) '' ball 0 1

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
C✝ D✝ : Set X✝
inst✝¹ : RelCWComplex C✝ D✝
X : Type u
inst✝ : TopologicalSpace X
C D : Set X
cell : ℕ → Type u
map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X
finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)
source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1
continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i)) (closedBall 0 1)
continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i).symm) (map n i).target
pairwiseDisjoint' : univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ni.snd) '' ball 0 1
disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (↑(map n i) '' ball 0 1) D
mapsTo :
  ∀ (n : ℕ) (i : cell n), MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j, ↑(map m j) '' closedBall 0 1)
isClosedBase : IsClosed[inst✝] D
union' : D ∪ ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C
A : Set X
asubc : A ⊆ C
h : (∀ (n : ℕ) (j : cell n), IsClosed[inst✝] (A ∩ ↑(map n j) '' closedBall 0 1)) ∧ IsClosed[inst✝] (A ∩ D)
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → IsEmpty (cell b)
this : IsClosed[inst✝] (A ∩ (D ∪ ⋃ n, ⋃ j, ↑(map (↑n) j) '' closedBall 0 1))
n : ℕ
i : n < N
⊢ ⋃ j, ↑(map (↑⟨n, i⟩) j) '' closedBall 0 1 ⊆ ⋃ j, ↑(map n j) '' closedBall 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DerivedSet

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
U : Set X
hs : IsPreconnected U
a b : Set X
h : U ⊆ a ∪ b
ha : (U ∩ derivedSet a).Nonempty
hb : (U ∩ derivedSet b).Nonempty
⊢ (U ∩ (derivedSet a ∩ derivedSet b)).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ sᵈ[0] = s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
a : Ordinal.{u}
⊢ sᵈ[a + 1] = relDerivedSet sᵈ[a]

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
a : Ordinal.{u}
ha : Order.IsSuccLimit a
⊢ sᵈ[a] = ⋂ b, sᵈ[↑b]

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ Preperfect s ↔ ∀ (a : Ordinal.{u}), sᵈ[a] = s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsClosed s
⊢ ∀ (a : Ordinal.{u}), IsClosed sᵈ[a]

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
a : Ordinal.{u}
ha : sᵈ[a + 1] = sᵈ[a]
⊢ sᵈ[a] ∈ fixedPoints ⇑relDerivedSet

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ ∃ a, sᵈ[a] ∈ fixedPoints ⇑relDerivedSet

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ perfectKernel s ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hst : s ⊆ t
⊢ perfectKernel s ⊆ perfectKernel t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

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

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
a : Ordinal.{u}
ha : sᵈ[a] ∈ fixedPoints ⇑relDerivedSet
⊢ perfectKernel s = sᵈ[a]

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s P : Set X
hP : Perfect P
hPs : P ⊆ s
⊢ P ⊆ perfectKernel s

Tactic:
by

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

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
⊢ ∀ (n : ℕ) (i : ↑(E.I n)),
    ∃ I, MapsTo (↑(map n ↑i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m ↑j) '' closedBall 0 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.CantorBendixson

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsClosed s
⊢ Perfect (perfectKernel s)

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
C✝ D✝ : Set X✝
inst✝¹ : RelCWComplex C✝ D✝
X : Type u
inst✝ : TopologicalSpace X
C D : Set X
cell : ℕ → Type u
map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X
finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)
source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1
continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i)) (closedBall 0 1)
continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (↑(map n i).symm) (map n i).target
pairwiseDisjoint' : univ.PairwiseDisjoint fun ni ↦ ↑(map ni.fst ni.snd) '' ball 0 1
disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (↑(map n i) '' ball 0 1) D
mapsTo :
  ∀ (n : ℕ) (i : cell n), MapsTo (↑(map n i)) (sphere 0 1) (D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j, ↑(map m j) '' closedBall 0 1)
isClosedBase : IsClosed[inst✝] D
union' : D ∪ ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C
A : Set X
asubc : A ⊆ C
h : (∀ (n : ℕ) (j : cell n), IsClosed[inst✝] (A ∩ ↑(map n j) '' closedBall 0 1)) ∧ IsClosed[inst✝] (A ∩ D)
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → IsEmpty (cell b)
this : IsClosed[inst✝] (A ∩ (D ∪ ⋃ n, ⋃ j, ↑(map (↑n) j) '' closedBall 0 1))
⊢ IsClosed[inst✝] (A ∩ C)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
A : Set X
hA : A ⊆ ↑E
h : (∀ (n : ℕ) (j : ↑(E.I n)), IsClosed[t] (A ∩ ↑(map n ↑j) '' closedBall 0 1)) ∧ IsClosed[t] (A ∩ D)
⊢ IsClosed[t] A

Tactic:
by

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

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
n : ℕ
i : ↑(E.I n)
⊢ openCell n i = openCell n ↑i

Tactic:
by

Goal state after: