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

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
⊢ Disjoint s (closure[inst✝] t) ↔ Disjoint s (closure[inst✝] t)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
⊢ (∃ U, IsOpen[inst✝] U ∧ t ⊆ U ∧ U ⊆ s) ↔ ∃ U, IsOpen[inst✝] U ∧ t ⊆ U ∧ U ⊆ s

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.ToIntervalMod

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ 0 ≤ b - a - ↑⌊(b - a) / p⌋ * p ∧ b - a - ↑⌊(b - a) / p⌋ * p < p

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ 0 ≤ b - a - ↑⌊(b - a) / p⌋ * p ∧ b - a - ↑⌊(b - a) / p⌋ * p < p
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
hU : IsOpen[inst✝] s
⊢ t ⊆ s ↔ t ⊆ s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s : Set X
⊢ (∀ x ∈ s, s ∈ 𝓝 x) ↔ ∀ x ∈ s, s ∈ 𝓝 x

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.ToIntervalMod

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ a < b + ↑⌊(a + p - b) / p⌋ * p ∧ 0 ≤ a + p - b - ↑⌊(a + p - b) / p⌋ * p

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ a < b + ↑⌊(a + p - b) / p⌋ * p ∧ 0 ≤ a + p - b - ↑⌊(a + p - b) / p⌋ * p
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
⊢ ⊥ = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.ToIntervalMod

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ a + p - b - ↑⌊(a + p - b) / p⌋ * p < p

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ a + p - b - ↑⌊(a + p - b) / p⌋ * p < p
Module: Mathlib.Algebra.Order.ToIntervalMod

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ b - ⌊(b - a) / p⌋ • p = a + ((b - a) / p - ↑⌊(b - a) / p⌋) * p

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ b - ⌊(b - a) / p⌋ • p = a + ((b - a) / p - ↑⌊(b - a) / p⌋) * p
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s₁ s₂ t₁ t₂ : Set X
h₁ : s₁ ∈ 𝓝ˢ t₁
h₂ : s₂ ∈ 𝓝ˢ t₂
⊢ s₁ ∪ s₂ ∈ 𝓝ˢ t₁ ⊔ 𝓝ˢ t₂

Tactic:
]

Goal state after:
X : Type u_2
inst✝ : TopologicalSpace X
s₁ s₂ t₁ t₂ : Set X
h₁ : s₁ ∈ 𝓝ˢ t₁
h₂ : s₂ ∈ 𝓝ˢ t₂
⊢ s₁ ∪ s₂ ∈ 𝓝ˢ t₁ ⊔ 𝓝ˢ t₂
Module: Mathlib.Algebra.Order.ToIntervalMod

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ b - -⌊(a + p - b) / p⌋ • p = a + p - ((a + p - b) / p - ↑⌊(a + p - b) / p⌋) * p

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
p : α
hp : 0 < p
a b : α
⊢ b - -⌊(a + p - b) / p⌋ • p = a + p - ((a + p - b) / p - ↑⌊(a + p - b) / p⌋) * p
Module: Mathlib.Algebra.Order.ToIntervalMod

Goal state before:
pp: α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
hl : a < b - toIocDiv hp a b • p
hr : b - toIocDiv hp a b • p ≤ a + p
⊢ b ≤ a + toIocDiv hp a b • p + p

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
hl : a < b - toIocDiv hp a b • p
hr : b - toIocDiv hp a b • p ≤ a + p
⊢ b ≤ a + toIocDiv hp a b • p + p
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
x : X
s : Set X
⊢ 𝓝 x ⊔ 𝓝ˢ s = 𝓝 x ⊔ 𝓝ˢ s

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.ToIntervalMod

Goal state before:
pp: α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
hl : a ≤ b - toIcoDiv hp a b • p
hr : b - toIcoDiv hp a b • p < a + p
⊢ b < a + toIcoDiv hp a b • p + p

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
hl : a ≤ b - toIcoDiv hp a b • p
hr : b - toIcoDiv hp a b • p < a + p
⊢ b < a + toIcoDiv hp a b • p + p
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
h : IsClosed[inst✝] t
⊢ 𝓝ˢ s = 𝓝ˢ s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
h : IsClosed[inst✝] t
⊢ 𝓝ˢ (s ∩ t) ⊔ 𝓝ˢ (s ∩ tᶜ) = 𝓝ˢ (s ∩ t) ⊔ 𝓝ˢ (s ∩ tᶜ)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
h : IsClosed[inst✝] t
⊢ 𝓝ˢ (s ∩ t) ⊔ 𝓝ˢ tᶜ ≤ 𝓝ˢ (s ∩ t) ⊔ 𝓝ˢ tᶜ

Tactic:
]

Goal state after:
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
h : IsClosed[inst✝] t
⊢ 𝓝ˢ (s ∩ t) ⊔ 𝓟 tᶜ = 𝓝ˢ (s ∩ t) ⊔ 𝓟 tᶜ

Tactic:
]

Goal state after:
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
h : IsClosed[inst✝] t
⊢ 𝓝ˢ s ≤ 𝓝ˢ (s ∩ t) ⊔ 𝓟 tᶜ

Tactic:
]

Goal state after:
X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
h : IsClosed[inst✝] t
⊢ 𝓝ˢ s ≤ 𝓝ˢ (s ∩ t) ⊔ 𝓟 tᶜ
Module: Mathlib.Topology.Maps.OpenQuotient

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
ind : IsInducing f
surj : Surjective f
V : Set Y
hV : IsOpen[inst✝] V
U_open : IsOpen[inst✝¹] (f ⁻¹' V)
⊢ IsOpen[inst✝] V

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
ind : IsInducing f
surj : Surjective f
V : Set Y
hV : IsOpen[inst✝] V
U_open : IsOpen[inst✝¹] (f ⁻¹' V)
⊢ IsOpen[inst✝] V
Module: Mathlib.Topology.Maps.OpenQuotient

Goal state before:
pp: case mp.mp
A : Type u_4
B : Type u_5
C : Type u_6
D : Type u_7
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace D
f : A → B
g : C → D
p : A → C
q : B → D
h : g ∘ p = q ∘ f
hf : IsInducing f
hp : Surjective p
hq : IsOpenQuotientMap q
hg : Injective g
H : q ⁻¹' q '' range f ⊆ range f
U : Set C
V : Set B
hV : IsOpen[inst✝¹] V
e : f ⁻¹' V = p ⁻¹' U
x y : A
hy : f y ∈ V
e' : q (f y) = g (p x)
⊢ p y ∈ U

Tactic:
]

Goal state after:
case mp.mp
A : Type u_4
B : Type u_5
C : Type u_6
D : Type u_7
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace D
f : A → B
g : C → D
p : A → C
q : B → D
h : g ∘ p = q ∘ f
hf : IsInducing f
hp : Surjective p
hq : IsOpenQuotientMap q
hg : Injective g
H : q ⁻¹' q '' range f ⊆ range f
U : Set C
V : Set B
hV : IsOpen[inst✝¹] V
e : f ⁻¹' V = p ⁻¹' U
x y : A
hy : f y ∈ V
e' : q (f y) = g (p x)
⊢ p y ∈ U
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u
g : Set (Set α)
a : α
this : TopologicalSpace α := generateFrom g
⊢ ⨅ s ∈ {s | a ∈ s ∧ IsOpen[generateFrom g] s}, 𝓟 s = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s

Tactic:
]

Goal state after:
α : Type u
g : Set (Set α)
a : α
this : TopologicalSpace α := generateFrom g
⊢ ⨅ s ∈ {s | a ∈ s ∧ IsOpen[generateFrom g] s}, 𝓟 s = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s
Module: Mathlib.Topology.NhdsSet

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
p : X → Prop
⊢ ((∀ᶠ (x : X) in 𝓝ˢ s, p x) ∧ ∀ᶠ (x : X) in 𝓝ˢ t, p x) ↔ (∀ᶠ (x : X) in 𝓝ˢ s, p x) ∧ ∀ᶠ (x : X) in 𝓝ˢ t, p x

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Separation.SeparatedNhds

Goal state before:
pp: case mp
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
u : ℕ → Set X
u_cov : s ⊆ ⋃ n, u n
u_props : ∀ (n : ℕ), IsOpen[inst✝] (u n) ∧ Disjoint (closure[inst✝] (u n)) t
v : ℕ → Set X
v_cov : t ⊆ ⋃ n, v n
v_props : ∀ (n : ℕ), IsOpen[inst✝] (v n) ∧ Disjoint (closure[inst✝] (v n)) s
open_lemma : ∀ (u₀ a : ℕ → Set X), (∀ (n : ℕ), IsOpen[inst✝] (u₀ n)) → IsOpen[inst✝] (⋃ n, u₀ n \ closure[inst✝] (a n))
cover_lemma :
  ∀ (h₀ : Set X) (u₀ v₀ : ℕ → Set X),
    h₀ ⊆ ⋃ n, u₀ n →
      (∀ (n : ℕ), Disjoint (closure[inst✝] (v₀ n)) h₀) → h₀ ⊆ ⋃ n, u₀ n \ closure[inst✝] (⋃ m, ⋃ (_ : m ≤ n), v₀ m)
⊢ ∀ ⦃a : X⦄,
    a ∈ ⋃ n, u n \ closure[inst✝] (⋃ m, ⋃ (_ : m ≤ n), v m) → a ∉ ⋃ n, v n \ closure[inst✝] (⋃ m, ⋃ (_ : m ≤ n), u m)

Tactic:
]

Goal state after:
case mp
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
u : ℕ → Set X
u_cov : s ⊆ ⋃ n, u n
u_props : ∀ (n : ℕ), IsOpen[inst✝] (u n) ∧ Disjoint (closure[inst✝] (u n)) t
v : ℕ → Set X
v_cov : t ⊆ ⋃ n, v n
v_props : ∀ (n : ℕ), IsOpen[inst✝] (v n) ∧ Disjoint (closure[inst✝] (v n)) s
open_lemma : ∀ (u₀ a : ℕ → Set X), (∀ (n : ℕ), IsOpen[inst✝] (u₀ n)) → IsOpen[inst✝] (⋃ n, u₀ n \ closure[inst✝] (a n))
cover_lemma :
  ∀ (h₀ : Set X) (u₀ v₀ : ℕ → Set X),
    h₀ ⊆ ⋃ n, u₀ n →
      (∀ (n : ℕ), Disjoint (closure[inst✝] (v₀ n)) h₀) → h₀ ⊆ ⋃ n, u₀ n \ closure[inst✝] (⋃ m, ⋃ (_ : m ≤ n), v₀ m)
⊢ ∀ ⦃a : X⦄,
    a ∈ ⋃ n, u n \ closure[inst✝] (⋃ m, ⋃ (_ : m ≤ n), v m) → a ∉ ⋃ n, v n \ closure[inst✝] (⋃ m, ⋃ (_ : m ≤ n), u m)
Module: Mathlib.Topology.Order

Goal state before:
pp: case inl
α : Type u
inst✝ : DecidableEq α
l : Filter α
b✝ : α
s : Set α
b : α
hb : b ∈ s
h : pure b ≤ l
hs : s ∈ update pure b l b
⊢ s ∈ update pure b l b

Tactic:
exact hs

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: case inr
α : Type u
inst✝ : DecidableEq α
l : Filter α
b✝ a : α
s : Set α
h : pure a ≤ l
hs : s ∈ update pure a l a
b : α
hb✝ : b ∈ s
hb : b ≠ a
⊢ s ∈ pure b

Tactic:
]

Goal state after:
case inr
α : Type u
inst✝ : DecidableEq α
l : Filter α
b✝ a : α
s : Set α
h : pure a ≤ l
hs : s ∈ update pure a l a
b : α
hb✝ : b ∈ s
hb : b ≠ a
⊢ s ∈ pure b
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Z
hg : IsInducing g
hf : IsInducing f
⊢ induced (g ∘ f) inst✝ = induced (g ∘ f) inst✝

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Z
hg : IsInducing g
h : IsInducing (g ∘ f)
⊢ induced (g ∘ f) inst✝ = induced (g ∘ f) inst✝

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Z
hf : Continuous[inst✝¹, inst✝²] f
hg : Continuous[inst✝², inst✝] g
hgf : IsInducing (g ∘ f)
⊢ induced f inst✝² ≤ induced f inst✝²

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
ι : Type u_3
f : ι → α
⊢ Surjective f ↔ Surjective f

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: β : Type u_2
α : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
f : α → β
h : ∀ (b : β), IsOpen[inst✝²] (f ⁻¹' {b})
s : Set β
x✝ : IsOpen[inst✝¹] s
⊢ IsOpen[inst✝²] (⋃ i ∈ s, f ⁻¹' {i})

Tactic:
]

Goal state after:
β : Type u_2
α : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
f : α → β
h : ∀ (b : β), IsOpen[inst✝²] (f ⁻¹' {b})
s : Set β
x✝ : IsOpen[inst✝¹] s
⊢ IsOpen[inst✝²] (⋃ i ∈ s, f ⁻¹' {i})
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
x : α
s : Set α
⊢ x ∈ s ↔ x ∈ s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_1
t₁ t₂ : TopologicalSpace α
h : ∀ (x : α), 𝓝 x ≤ 𝓝 x
s : Set α
⊢ (∀ x ∈ s, s ∈ 𝓝 x) → ∀ x ∈ s, s ∈ 𝓝 x

Tactic:
]

Goal state after:
α : Type u_1
t₁ t₂ : TopologicalSpace α
h : ∀ (x : α), 𝓝 x ≤ 𝓝 x
s : Set α
⊢ (∀ x ∈ s, s ∈ 𝓝 x) → ∀ x ∈ s, s ∈ 𝓝 x
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace X
hf : IsInducing f
x : X
l : Filter X
⊢ (𝓝 x ⊓ l).NeBot ↔ (𝓝 x ⊓ l).NeBot

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: Y : Type u_2
Z : Type u_3
ι : Type u_4
g : Y → Z
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : ι → Y
l : Filter ι
y : Y
hg : IsInducing g
⊢ Tendsto (g ∘ f) l (𝓝 (g y)) ↔ Tendsto (g ∘ f) l (𝓝 (g y))

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_1
β : Type u_2
g : β → α
s : Set (TopologicalSpace α)
⊢ ⨅ i, TopologicalSpace.induced g ↑i = ⨅ a, TopologicalSpace.induced g ↑a

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
s : Set (TopologicalSpace α)
⊢ ⨆ i, TopologicalSpace.coinduced f ↑i = ⨆ a, TopologicalSpace.coinduced f ↑a

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_4
β : Type u_5
e : α ≃ β
t : TopologicalSpace α
U : Set β
⊢ (∃ t_1, IsOpen[t] t_1 ∧ ⇑e.symm ⁻¹' t_1 = U) ↔ IsOpen[t] (⇑e ⁻¹' U)

Tactic:
]

Goal state after:
α : Type u_4
β : Type u_5
e : α ≃ β
t : TopologicalSpace α
U : Set β
⊢ (∃ t_1, IsOpen[t] t_1 ∧ ⇑e.symm ⁻¹' t_1 = U) ↔ IsOpen[t] (⇑e ⁻¹' U)
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_4
β : Type u_5
e : α ≃ β
⊢ TopologicalSpace.induced ⇑e.symm = TopologicalSpace.coinduced ⇑e

Tactic:
ext t U
  rw [isOpen_induced_iff, isOpen_coinduced]
  simp only [e.symm.preimage_eq_iff_eq_image, exists_eq_right, Equiv.image_symm_eq_preimage]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u
β : Type v
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Separation.SeparatedNhds

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hst' : IsOpen[inst✝] (s ∪ t)
u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hsu : s ⊆ u
htv : t ⊆ v
huv : Disjoint u v
⊢ s = s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Separation.SeparatedNhds

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hst' : IsOpen[inst✝] (s ∪ t)ᶜ
u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hsu : s ⊆ u
htv : t ⊆ v
huv : Disjoint u v
⊢ IsOpen[inst✝] sᶜ

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hst' : IsOpen[inst✝] (s ∪ t)ᶜ
u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hsu : s ⊆ u
htv : t ⊆ v
huv : Disjoint u v
⊢ IsOpen[inst✝] sᶜ
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_1
β : Type u_2
t : TopologicalSpace α
b : Set (Set β)
f : α → β
h : ∀ a ∈ b, IsOpen[t] (f ⁻¹' a)
⊢ t ≤ generateFrom (preimage f '' b)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
t : TopologicalSpace α
b : Set (Set β)
f : α → β
h : ∀ a ∈ b, IsOpen[t] (f ⁻¹' a)
⊢ t ≤ generateFrom (preimage f '' b)
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u_1
s : Set (Set α)
⊢ generateFrom (insert (⋃₀ ∅) s) = generateFrom s

Tactic:
]

Goal state after:
α : Type u_1
s : Set (Set α)
⊢ generateFrom (insert (⋃₀ ∅) s) = generateFrom s
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace X
hf : IsInducing f
s : Set X
x : X
⊢ x ∈ closure[induced f inst✝¹] s ↔ x ∈ closure[induced f inst✝¹] s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u
a : α
f : Filter α
t : TopologicalSpace α
⊢ nhdsAdjoint a f ≤ t ↔ ∀ (s : Set α), a ∈ s → IsOpen[t] s → s ∈ f

Tactic:
]

Goal state after:
α : Type u
a : α
f : Filter α
t : TopologicalSpace α
⊢ nhdsAdjoint a f ≤ t ↔ ∀ (s : Set α), a ∈ s → IsOpen[t] s → s ∈ f
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace X
hf : IsInducing f
s : Set X
⊢ IsClosed[induced f inst✝¹] s ↔ ∃ t, IsClosed[inst✝¹] t ∧ f ⁻¹' t = s

Tactic:
isClosed_induced_iff

Goal state after:
X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace X
hf : IsInducing f
s : Set X
⊢ (∃ t, IsClosed[inst✝¹] t ∧ f ⁻¹' t = s) ↔ ∃ t, IsClosed[inst✝¹] t ∧ f ⁻¹' t = s
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace X
hf : IsInducing f
s : Set X
⊢ (∃ t, IsClosed[inst✝¹] t ∧ f ⁻¹' t = s) ↔ ∃ t, IsClosed[inst✝¹] t ∧ f ⁻¹' t = s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace X
hf : IsInducing f
s : Set X
⊢ (∀ (a : X), f a ∈ closure[inst✝¹] (f '' s) → a ∈ s) ↔ ∀ (x : X), f x ∈ closure[inst✝¹] (f '' s) → x ∈ s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace X
hf : IsInducing f
s : Set X
⊢ (∃ t, IsOpen[inst✝¹] t ∧ f ⁻¹' t = s) ↔ ∃ t, IsOpen[inst✝¹] t ∧ f ⁻¹' t = s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Homeomorph.Defs

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h : X ≃ₜ Y
s : Set X
⊢ IsOpen[inst✝¹] s ↔ IsOpen[inst✝¹] s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Separation.SeparatedNhds

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hst' : IsOpen[inst✝] (s ∪ t)ᶜ
u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hsu : s ⊆ u
htv : t ⊆ v
huv : Disjoint u v
⊢ v ⊆ sᶜ

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hst' : IsOpen[inst✝] (s ∪ t)ᶜ
u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hsu : s ⊆ u
htv : t ⊆ v
huv : Disjoint u v
⊢ v ⊆ sᶜ
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: case refl
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
f : X → Y
inst✝ : IndiscreteTopology Y
hf : IsInducing f
this : TopologicalSpace Y := ⊤
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Homeomorph.Defs

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h : X ≃ₜ Y
s : Set X
⊢ IsClosed[inst✝¹] s ↔ IsClosed[inst✝¹] s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u
t : TopologicalSpace α
⊢ t = ⊤ ↔ t = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Homeomorph.Defs

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h : X ≃ₜ Y
s : Set X
⊢ closure[inst✝] (⇑h.symm ⁻¹' s) = closure[inst✝] (⇑h.symm ⁻¹' s)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u
t : TopologicalSpace α
⊢ t ≠ ⊤ ↔ t ≠ ⊤

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
hf : IsCoinducing f
s : Set Y
hs : IsOpen[inst✝] s
⊢ IsOpen[inst✝] s

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
hf : IsCoinducing f
s : Set Y
hs : IsOpen[inst✝] s
⊢ IsOpen[inst✝] s
Module: Mathlib.Topology.Homeomorph.Defs

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h : X ≃ₜ Y
s : Set X
⊢ interior (⇑h.symm ⁻¹' s) = interior (⇑h.symm ⁻¹' s)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hg : IsCoinducing g
hf : IsCoinducing f
⊢ coinduced (g ∘ f) inst✝² = coinduced (g ∘ f) inst✝²

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u
β : Type v
f : α → β
t : TopologicalSpace α
b : Set (Set β)
⊢ b ⊆ {s | IsOpen[coinduced f t] s} ↔ ∀ s ∈ b, IsOpen[t] (f ⁻¹' s)

Tactic:
]

Goal state after:
α : Type u
β : Type v
f : α → β
t : TopologicalSpace α
b : Set (Set β)
⊢ b ⊆ {s | IsOpen[coinduced f t] s} ↔ ∀ s ∈ b, IsOpen[t] (f ⁻¹' s)
Module: Mathlib.Topology.Homeomorph.Defs

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h : X ≃ₜ Y
s : Set X
⊢ frontier (⇑h.symm ⁻¹' s) = frontier (⇑h.symm ⁻¹' s)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hf : IsCoinducing f
hgf : IsCoinducing (g ∘ f)
s : Set Z
⊢ IsOpen[inst✝¹] (g ⁻¹' s) ↔ IsOpen[inst✝¹] (g ⁻¹' s)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u
β : Type v
f : α → β
t₁ t₂ : TopologicalSpace α
t₃ : TopologicalSpace β
h₁ : t₂ ≤ t₁
h₂ : t₁ ≤ induced f t₃
⊢ t₂ ≤ induced f t₃

Tactic:
]

Goal state after:
α : Type u
β : Type v
f : α → β
t₁ t₂ : TopologicalSpace α
t₃ : TopologicalSpace β
h₁ : t₂ ≤ t₁
h₂ : t₁ ≤ induced f t₃
⊢ t₂ ≤ induced f t₃
Module: Mathlib.Topology.Order

Goal state before:
pp: α : Type u
β : Type v
f : α → β
t₁ : TopologicalSpace α
t₂ t₃ : TopologicalSpace β
h₁ : t₂ ≤ t₃
h₂ : coinduced f t₁ ≤ t₂
⊢ coinduced f t₁ ≤ t₃

Tactic:
]

Goal state after:
α : Type u
β : Type v
f : α → β
t₁ : TopologicalSpace α
t₂ t₃ : TopologicalSpace β
h₁ : t₂ ≤ t₃
h₂ : coinduced f t₁ ≤ t₂
⊢ coinduced f t₁ ≤ t₃
Module: Mathlib.Topology.Homeomorph.Defs

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
h : X ≃ₜ Y
f : Z → X
hf : IsOpenMap (⇑h ∘ f)
⊢ IsOpenMap (⇑h.symm ∘ ⇑h ∘ f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
h : X ≃ₜ Y
f : Z → X
hf : IsOpenMap (⇑h ∘ f)
⊢ IsOpenMap (⇑h.symm ∘ ⇑h ∘ f)
Module: Mathlib.Topology.Homeomorph.Defs

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
h : X ≃ₜ Y
f : Y → Z
hf : IsOpenMap (f ∘ ⇑h)
⊢ IsOpenMap ((f ∘ ⇑h) ∘ ⇑h.symm)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
h : X ≃ₜ Y
f : Y → Z
hf : IsOpenMap (f ∘ ⇑h)
⊢ IsOpenMap ((f ∘ ⇑h) ∘ ⇑h.symm)
Module: Mathlib.Topology.Maps.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hf : Continuous[inst✝², inst✝¹] f
hg : Continuous[inst✝¹, inst✝] g
hgf : IsCoinducing (g ∘ f)
⊢ coinduced g inst✝¹ ≤ coinduced g inst✝¹

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Homeomorph.Defs

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h : X ≃ₜ Y
x : X
⊢ 𝓝 x = 𝓝 x

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order

Goal state before:
pp: case mpr
α : Type u
β : Type v
T : TopologicalSpace α
f : β → α
a : β
s : Set β
this : TopologicalSpace β := ⋯
⊢ (∃ u, (∃ t ⊆ u, IsOpen[T] t ∧ f a ∈ t) ∧ f ⁻¹' u ⊆ s) → ∃ t ⊆ s, (∃ t_1, IsOpen[T] t_1 ∧ f ⁻¹' t_1 = t) ∧ a ∈ t

Tactic:
rintro ⟨u, ⟨v, vsubu, openv, amem⟩, finvsub⟩
    exact ⟨f ⁻¹' v, (Set.preimage_mono vsubu).trans finvsub, ⟨⟨v, openv, rfl⟩, amem⟩⟩

Goal state after: