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

Goal state before:
pp: α : Type u
β : Type v
inst✝² : UniformSpace α
inst✝¹ : UniformSpace β
inst✝ : T0Space β
f : α → β
⊢ UniformContinuous (lift' f)

Tactic:
by
  by_cases hf : UniformContinuous f
  · rwa [lift', dif_pos hf, uniformContinuous_lift]
  · rw [lift', dif_neg hf]
    exact uniformContinuous_of_const fun a _ => rfl

Goal state after:
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: case mpr
α : Type u
uniformSpace : UniformSpace α
g : Filter α
d : SetRel α α
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), t.Finite → d.preimage t ∉ g
f : Filter α := ⨅ t, g ⊓ 𝓟 (d.preimage ↑t)ᶜ
hb : Antitone fun t ↦ g ⊓ 𝓟 (d.preimage ↑t)ᶜ
this✝ : f.NeBot
this : f ≤ g
⊢ ∃ f, f.NeBot ∧ f ≤ g ∧ ∀ c ≤ f, ¬Cauchy c

Tactic:
refine ⟨f, ‹_›, ‹_›, fun c hcf hc => ?_⟩

Goal state after:
case mpr
α : Type u
uniformSpace : UniformSpace α
g : Filter α
d : SetRel α α
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), t.Finite → d.preimage t ∉ g
f : Filter α := ⨅ t, g ⊓ 𝓟 (d.preimage ↑t)ᶜ
hb : Antitone fun t ↦ g ⊓ 𝓟 (d.preimage ↑t)ᶜ
this✝ : f.NeBot
this : f ≤ g
c : Filter α
hcf : c ≤ f
hc : Cauchy c
⊢ False
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
g : Filter α
d : SetRel α α
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), t.Finite → d.preimage t ∉ g
f : Filter α := ⨅ t, g ⊓ 𝓟 (d.preimage ↑t)ᶜ
hb : Antitone fun t ↦ g ⊓ 𝓟 (d.preimage ↑t)ᶜ
this✝ : f.NeBot
this : f ≤ g
c : Filter α
hcf : c ≤ f
hc : Cauchy c
m : Set α
hm : m ∈ c
hmd : m ×ˢ m ⊆ d
y : α
hym : y ∈ m
⊢ {x | (x, y) ∈ d}ᶜ ∈ c

Tactic:
simpa [SetRel.preimage] using hcf.trans <| (iInf_le _ {y}).trans inf_le_right

Goal state after:
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
inst✝¹ : CompleteSpace α
f : Filter α
inst✝ : f.NeBot
hf : f.TotallyBounded
m : Ultrafilter α := Ultrafilter.of f
hmf : ↑m ≤ f
hm : Cauchy ↑m
x : α
hx : ↑m ≤ 𝓝 x
⊢ ∃ x, ClusterPt x f

Tactic:
le_nhds_iff_adhp_of_cauchy hm

Goal state after:
α : Type u
uniformSpace : UniformSpace α
inst✝¹ : CompleteSpace α
f : Filter α
inst✝ : f.NeBot
hf : f.TotallyBounded
m : Ultrafilter α := Ultrafilter.of f
hmf : ↑m ≤ f
hm : Cauchy ↑m
x : α
hx : ClusterPt x ↑m
⊢ ∃ x, ClusterPt x f
Module: Mathlib.Topology.UniformSpace.Cauchy

Goal state before:
pp: α : Type u
uniformSpace : UniformSpace α
s : ℕ → α
hs : CauchySeq s
a : Set (α × α)
ha : a ∈ 𝓤 α
n : ℕ
hn : ∀ k ≥ n, ∀ l ≥ n, (s k, s l) ∈ a
⊢ ∃ t, t.Finite ∧ range s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ a}

Tactic:
refine ⟨s '' { k | k ≤ n }, (finite_le_nat _).image _, ?_⟩

Goal state after:
α : Type u
uniformSpace : UniformSpace α
s : ℕ → α
hs : CauchySeq s
a : Set (α × α)
ha : a ∈ 𝓤 α
n : ℕ
hn : ∀ k ≥ n, ∀ l ≥ n, (s k, s l) ∈ a
⊢ range s ⊆ ⋃ y ∈ s '' {k | k ≤ n}, {x | (x, y) ∈ a}
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: α : Type ua
inst✝¹ : UniformSpace α
inst✝ : CompactSpace α
⊢ 𝓝ˢ (diagonal α) = 𝓤 α

Tactic:
refine nhdsSet_diagonal_le_uniformity.antisymm ?_
  have :
    (𝓤 (α × α)).HasBasis (fun U => U ∈ 𝓤 α) fun U =>
      (fun p : (α × α) × α × α => ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' U ×ˢ U := by
    rw [uniformity_prod_eq_comap_prod]
    exact (𝓤 α).basis_sets.prod_self.comap _
  refine (isCompact_diagonal.nhdsSet_basis_uniformity this).ge_iff.2 fun U hU => ?_
  exact mem_of_superset hU fun ⟨x, y⟩ hxy => mem_iUnion₂.2
    ⟨(x, x), rfl, refl_mem_uniformity hU, hxy⟩

Goal state after:
Module: Mathlib.Topology.UniformSpace.LocallyUniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
p : Filter ι
h : TendstoUniformlyOn F f p s
u : Set (β × β)
hu : u ∈ 𝓤 β
x✝¹ : α
x✝ : x✝¹ ∈ s
⊢ ∀ᶠ (n : ι) in p, ∀ y ∈ s, (f y, F n y) ∈ u

Tactic:
simpa using h u hu

Goal state after:
Module: Mathlib.Topology.UniformSpace.LocallyUniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
h : TendstoUniformly F f p
u : Set (β × β)
hu : u ∈ 𝓤 β
x✝ : α
⊢ ∀ᶠ (n : ι) in p, ∀ y ∈ univ, (f y, F n y) ∈ u

Tactic:
simpa using h u hu

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equiv

Goal state before:
pp: α : Type u
β : Type u_1
γ : Type u_2
δ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : UniformSpace δ
hf : IsUniformInducing Equiv.ulift.toFun
⊢ UniformContinuous Equiv.ulift.invFun

Tactic:
hf.uniformContinuous_iff

Goal state after:
α : Type u
β : Type u_1
γ : Type u_2
δ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : UniformSpace δ
hf : IsUniformInducing Equiv.ulift.toFun
⊢ UniformContinuous (Equiv.ulift.toFun ∘ Equiv.ulift.invFun)
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: case refine_3
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
δ₁ : Type u_6
δ₂ : Type u_7
φ₁ : δ₁ → α
φ₂ : δ₂ → α
h_cover : range φ₁ ∪ range φ₂ = univ
⊢ 𝓤 (α →ᵤ β) ≤ 𝓤 (α →ᵤ β)

Tactic:
· refine
      (UniformFun.hasBasis_uniformity δ₁ β |>.comap _).inf
      (UniformFun.hasBasis_uniformity δ₂ β |>.comap _)
        |>.le_basis_iff (UniformFun.hasBasis_uniformity α β) |>.mpr fun U hU ↦
        ⟨⟨U, U⟩, ⟨hU, hU⟩, fun ⟨f, g⟩ hfg x ↦ ?_⟩
    rcases h_cover.ge <| mem_univ x with (⟨y, rfl⟩ | ⟨y, rfl⟩)
    · exact hfg.1 y
    · exact hfg.2 y

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α : Type u_6
β' : Type u_9
uα : UniformSpace α
u : κ → UniformSpace β'
F : ι → β' → α
k : κ
hk : UniformContinuous (⇑ofFun ∘ swap F)
⊢ UniformContinuous (⇑ofFun ∘ swap F)

Tactic:
exact uniformContinuous_iInf_dom hk

Goal state after:
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: α : Type u_1
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
a b c : α
⊢ (fun x ↦ a / x) ⁻¹' Ioo b c = Ioo (a / c) (a / b)

Tactic:
by
  simp [← Ioi_inter_Iio, inter_comm]

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

Goal state before:
pp: G₀ : Type u_5
X : Type u_6
inst✝⁶ : GroupWithZero G₀
inst✝⁵ : Zero X
inst✝⁴ : MulActionWithZero G₀ X
inst✝³ : TopologicalSpace G₀
inst✝² : (𝓝[≠] 0).NeBot
inst✝¹ : TopologicalSpace X
inst✝ : ContinuousSMul G₀ X
s : Set X
hs : s ∈ 𝓝 0
x : X
⊢ x ∈ univ • s

Tactic:
have : Tendsto (· • x) (𝓝 (0 : G₀)) (𝓝 0) :=
    zero_smul G₀ x ▸ tendsto_id.smul tendsto_const_nhds

Goal state after:
G₀ : Type u_5
X : Type u_6
inst✝⁶ : GroupWithZero G₀
inst✝⁵ : Zero X
inst✝⁴ : MulActionWithZero G₀ X
inst✝³ : TopologicalSpace G₀
inst✝² : (𝓝[≠] 0).NeBot
inst✝¹ : TopologicalSpace X
inst✝ : ContinuousSMul G₀ X
s : Set X
hs : s ∈ 𝓝 0
x : X
this : Tendsto (fun x_1 ↦ x_1 • x) (𝓝 0) (𝓝 0)
⊢ x ∈ univ • s
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: α : Type u_2
inst✝⁶ : TopologicalSpace α
M : Type u_4
β : Type u_5
inst✝⁵ : Semiring M
inst✝⁴ : AddCommMonoid α
inst✝³ : Module M α
inst✝² : ContinuousConstSMul M α
inst✝¹ : AddMonoid β
f : α →+ β
inst✝ : TopologicalSpace β
hf : IsQuotientMap ⇑f
n : ℕ
hc : IsUnit ↑n
⊢ IsUnit (n • 1)

Tactic:
rwa [nsmul_one]

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : Group α
inst✝ : ContinuousConstSMul αᵐᵒᵖ α
s t : Set α
hs : IsOpen[inst✝²] s
⊢ IsOpen[inst✝²] (s * t)

Tactic:
rw [← image_op_smul]

Goal state after:
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : Group α
inst✝ : ContinuousConstSMul αᵐᵒᵖ α
s t : Set α
hs : IsOpen[inst✝²] s
⊢ IsOpen[inst✝²] (op '' t • s)
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s t : Set G
ht : IsClosed[inst✝²] t
hs : IsCompact s
⊢ IsClosed[inst✝²] (t * s)

Tactic:
rw [← image_op_smul]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s t : Set G
ht : IsClosed[inst✝²] t
hs : IsCompact s
⊢ IsClosed[inst✝²] (op '' s • t)
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
H : Type x
α : Type u
β : Type v
inst✝⁵ : Group G
inst✝⁴ : Group H
inst✝³ : TopologicalSpace G
inst✝² : MulDistribMulAction H G
inst✝¹ : ContinuousConstSMul H G
𝒢 : Subgroup G
h : H
inst✝ : DiscreteTopology ↥𝒢
⊢ DiscreteTopology ↥(h • 𝒢)

Tactic:
simp only [← SetLike.coe_sort_coe, ← isDiscrete_iff_discreteTopology] at *
  refine IsDiscrete.image_of_isOpenMap ‹_› ?_ fun x y ↦ by simp
  apply IsOpenMap.of_inverse (f' := fun x ↦ h⁻¹ • x) (continuous_const_smul _) <;>
  · intro x
    simp

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
H : Type x
α : Type u
β : Type v
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
a : G
s : Set G
hs : s ∈ 𝓝 a
⊢ ∃ t ∈ 𝓝 a, IsClosed[inst✝²] t ∧ t ⊆ s

Tactic:
have : Tendsto (fun p : G × G => p.1 * p.2) (𝓝 (a, 1)) (𝓝 a) :=
    continuous_mul.tendsto' _ _ (mul_one a)

Goal state after:
G : Type w
H : Type x
α : Type u
β : Type v
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
a : G
s : Set G
hs : s ∈ 𝓝 a
this : Tendsto (fun p ↦ p.1 * p.2) (𝓝 (a, 1)) (𝓝 a)
⊢ ∃ t ∈ 𝓝 a, IsClosed[inst✝²] t ∧ t ⊆ s
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x y : G
⊢ Inseparable x y ↔ x / y ∈ closure[inst✝²] 1

Tactic:
← singleton_one,

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x y : G
⊢ Inseparable x y ↔ x / y ∈ closure[inst✝²] {1}
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
U : Set G
H : U ∈ 𝓝 1
V : Set G
V_nhds : V ∈ 𝓝 1
V_mul : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U
x y z : G
hz₁ : (x, z) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V
hz₂ : (z, y) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V
⊢ (x, y) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' U

Tactic:
simpa using V_mul _ hz₁ _ hz₂

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
hom : Type u_3
inst✝⁶ : TopologicalSpace α
inst✝⁵ : Group α
inst✝⁴ : IsTopologicalGroup α
inst✝³ : TopologicalSpace β
inst✝² : Group β
inst✝¹ : FunLike hom β α
inst✝ : MonoidHomClass hom β α
e : hom
de : IsDenseInducing ⇑e
x₀ : α
comm : ((fun x ↦ x.2 / x.1) ∘ fun t ↦ (e t.1, e t.2)) = ⇑e ∘ fun t ↦ t.2 / t.1
⊢ Tendsto (fun x ↦ x.2 / x.1) (𝓝 (x₀, x₀)) (𝓝 (e 1))

Tactic:
simpa using! (continuous_div'.comp (@continuous_swap α α _ _)).tendsto (x₀, x₀)

Goal state after:
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : m ≤ y i₀
this : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)
⊢ (univ.pi fun i ↦ Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ univ.pi fun i ↦ Ioc (x i) (y i)

Tactic:
simp_rw [univ_pi_update i₀ y m fun i z ↦ Ioc (x i) z, ← pi_inter_compl ({i₀} : Set ι),
    singleton_pi', ← inter_assoc, this]

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : m ≤ y i₀
this : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)
⊢ ({x_1 | x_1 i₀ ∈ Iic m ∩ Ioc (x i₀) (y i₀)} ∩ {i₀}ᶜ.pi fun j ↦ Ioc (x j) (y j)) =
    {z | z i₀ ≤ m} ∩ {x_1 | x_1 i₀ ∈ Ioc (x i₀) (y i₀)} ∩ {i₀}ᶜ.pi fun j ↦ Ioc (x j) (y j)
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: case right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : AddCommGroup α
inst✝¹² : IsTopologicalAddGroup α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : AddCommGroup β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : AddCommGroup γ
inst✝⁷ : IsTopologicalAddGroup γ
inst✝⁶ : TopologicalSpace δ
inst✝⁵ : AddCommGroup δ
inst✝⁴ : UniformSpace G
inst✝³ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝⁴.toTopologicalSpace] fun p ↦ (φ p.1) p.2
inst✝² : IsUniformAddGroup G
inst✝¹ : T0Space G
inst✝ : CompleteSpace G
x₀ : α
y₀ : γ
⊢ map (fun p ↦ (φ p.1) p.2) (comap (Prod.map ⇑e ⇑f) (𝓝 (x₀, y₀))) ×ˢ
      map (fun p ↦ (φ p.1) p.2) (comap (Prod.map ⇑e ⇑f) (𝓝 (x₀, y₀))) ≤
    𝓤 G

Tactic:
suffices map (fun p : (β × δ) × β × δ => (fun p : β × δ => φ p.1 p.2) p.2 -
      (fun p : β × δ => φ p.1 p.2) p.1)
        (comap (fun p : (β × δ) × β × δ => ((e p.1.1, f p.1.2), (e p.2.1, f p.2.2)))
        (𝓝 (x₀, y₀) ×ˢ 𝓝 (x₀, y₀))) ≤ 𝓝 0 by
      rwa [uniformity_eq_comap_nhds_zero G, prod_map_map_eq, ← map_le_iff_le_comap, Filter.map_map,
        prod_comap_comap_eq]

Goal state after:
case right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : AddCommGroup α
inst✝¹² : IsTopologicalAddGroup α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : AddCommGroup β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : AddCommGroup γ
inst✝⁷ : IsTopologicalAddGroup γ
inst✝⁶ : TopologicalSpace δ
inst✝⁵ : AddCommGroup δ
inst✝⁴ : UniformSpace G
inst✝³ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝⁴.toTopologicalSpace] fun p ↦ (φ p.1) p.2
inst✝² : IsUniformAddGroup G
inst✝¹ : T0Space G
inst✝ : CompleteSpace G
x₀ : α
y₀ : γ
⊢ map (fun p ↦ (fun p ↦ (φ p.1) p.2) p.2 - (fun p ↦ (φ p.1) p.2) p.1)
      (comap (fun p ↦ ((e p.1.1, f p.1.2), e p.2.1, f p.2.2)) (𝓝 (x₀, y₀) ×ˢ 𝓝 (x₀, y₀))) ≤
    𝓝 0
Module: Mathlib.Topology.Algebra.Order.Group

Goal state before:
pp: G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a b ε : G
hε : ε > 1
⊢ ∀ᶠ (b_1 : G × G) in 𝓝 (a, b), |b_1.1 * b_1.2 / (a * b)|ₘ < ε

Tactic:
rcases dense_or_discrete 1 ε with ⟨δ, hδ₁, hδε⟩ | ⟨-, hε_min⟩

Goal state after:
case inl
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a b ε : G
hε : ε > 1
δ : G
hδ₁ : 1 < δ
hδε : δ < ε
⊢ ∀ᶠ (b_1 : G × G) in 𝓝 (a, b), |b_1.1 * b_1.2 / (a * b)|ₘ < ε

case inr
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a b ε : G
hε : ε > 1
hε_min : ∀ a < ε, a ≤ 1
⊢ ∀ᶠ (b_1 : G × G) in 𝓝 (a, b), |b_1.1 * b_1.2 / (a * b)|ₘ < ε
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
⊢ 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b

Tactic:
by
  by_cases ha : IsTop a
  · simp [ha, ha.isMax.Ioi_eq]
  · simp only [ha, false_or]
    rw [isTop_iff_isMax, not_isMax_iff] at ha
    simp only [(nhdsGT_basis_of_exists_gt ha).eq_bot_iff, covBy_iff_Ioo_eq]

Goal state after:
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
x : α
hx : x ∈ s ∧ (∃ u ∈ 𝓝 x, u ∩ (s ∩ Ioi x) = ∅) ∧ ∃ x_1, x < x_1
u : Set α
hu : u ∈ 𝓝 x
y : α
hxy : x < y
hy : Ico x y ⊆ u
h'u : (s ∩ Ioo x y).Nonempty
⊢ (u ∩ (s ∩ Ioi x)).Nonempty

Tactic:
apply h'u.mono

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := ⋯
x : α
hx : x ∈ s ∧ (∃ u ∈ 𝓝 x, u ∩ (s ∩ Ioi x) = ∅) ∧ ∃ x_1, x < x_1
u : Set α
hu : u ∈ 𝓝 x
y : α
hxy : x < y
hy : Ico x y ⊆ u
h'u : (s ∩ Ioo x y).Nonempty
⊢ s ∩ Ioo x y ⊆ u ∩ (s ∩ Ioi x)
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : FirstCountableTopology G
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : CompleteSpace G
this✝¹ : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this✝ : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : (𝓤 (G ⧸ N)).IsCountablyGenerated
u : ℕ → Set G
hu : (𝓝 1).HasAntitoneBasis u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : (𝓝 1).HasBasis (fun x ↦ True) fun x ↦ mk '' u x
v_anti : Antitone fun x ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ : ∀ (i j : ℕ), ∃ M, j < M ∧ ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u i ∧ x a = ↑g'
φ : ℕ → ℕ := fun n ↦ Nat.recOn n (Classical.choose ⋯) fun k yk ↦ Classical.choose ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ), φ (n + 1) ≤ a → φ (n + 1) ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u (n + 1) ∧ x a = ↑g'
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = ↑g :=
  fun n ↦ Nat.recOn n ⟨Classical.choose ⋯, ⋯⟩ fun k hk ↦ ⟨Classical.choose ⋯, ⋯⟩
hx' : ∀ (n : ℕ), (x' n).fst / (x' (n + 1)).fst ∈ u (n + 1)
x'_cauchy : CauchySeq fun n ↦ (x' n).fst
x₀ : G
hx₀ : Tendsto (fun n ↦ (x' n).fst) atTop (𝓝 x₀)
⊢ Tendsto (x ∘ fun n ↦ φ (n + 1)) atTop (𝓝 ↑x₀)

Tactic:
convert! ((continuous_coinduced_rng : Continuous ((↑) : G → G ⧸ N)).tendsto x₀).comp hx₀

Goal state after:
G : Type u
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : FirstCountableTopology G
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : CompleteSpace G
this✝¹ : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this✝ : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : (𝓤 (G ⧸ N)).IsCountablyGenerated
u : ℕ → Set G
hu : (𝓝 1).HasAntitoneBasis u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : (𝓝 1).HasBasis (fun x ↦ True) fun x ↦ mk '' u x
v_anti : Antitone fun x ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ : ∀ (i j : ℕ), ∃ M, j < M ∧ ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u i ∧ x a = ↑g'
φ : ℕ → ℕ := fun n ↦ Nat.recOn n (Classical.choose ⋯) fun k yk ↦ Classical.choose ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ), φ (n + 1) ≤ a → φ (n + 1) ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u (n + 1) ∧ x a = ↑g'
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = ↑g :=
  fun n ↦ Nat.recOn n ⟨Classical.choose ⋯, ⋯⟩ fun k hk ↦ ⟨Classical.choose ⋯, ⋯⟩
hx' : ∀ (n : ℕ), (x' n).fst / (x' (n + 1)).fst ∈ u (n + 1)
x'_cauchy : CauchySeq fun n ↦ (x' n).fst
x₀ : G
hx₀ : Tendsto (fun n ↦ (x' n).fst) atTop (𝓝 x₀)
⊢ (x ∘ fun n ↦ φ (n + 1)) = mk ∘ fun n ↦ (x' n).fst
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CommGroup α
inst✝³ : LinearOrder α
inst✝² : IsOrderedMonoid α
inst✝¹ : OrderTopology α
inst✝ : NoMaxOrder α
a : α
⊢ (⨅ r, ⨅ (_ : r > 1), 𝓟 {b | |b / a|ₘ < r}).HasBasis (fun ε ↦ 1 < ε) fun ε ↦ {b | |b / a|ₘ < ε}

Tactic:
refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)

Goal state after:
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CommGroup α
inst✝³ : LinearOrder α
inst✝² : IsOrderedMonoid α
inst✝¹ : OrderTopology α
inst✝ : NoMaxOrder α
a x : α
hx : x > 1
y : α
hy : y > 1
⊢ ∃ k > 1, {b | |b / a|ₘ < k} ⊆ {b | |b / a|ₘ < x} ∧ {b | |b / a|ₘ < k} ⊆ {b | |b / a|ₘ < y}
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
e : PartialEquiv α β
e' : PartialEquiv α γ
x : α
he : x ∈ e.source
he' : x ∈ e'.source
⊢ ↑e.symm.symm.symm (↑e x) ∈ e'.source

Tactic:
PartialEquiv.symm_symm,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
e : PartialEquiv α β
e' : PartialEquiv α γ
x : α
he : x ∈ e.source
he' : x ∈ e'.source
⊢ ↑e.symm (↑e x) ∈ e'.source
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case left
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ (e.restr s).source = (e'.restr s).source

pp: case right
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ EqOn (↑(e.restr s)) (↑(e'.restr s)) (e.restr s).source

Tactic:
· simp [he.1]

Goal state after:
case right
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ EqOn (↑(e.restr s)) (↑(e'.restr s)) (e.restr s).source
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case inl
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
s : Set α
hs : IsOpen[inst✝³] s
x : α
hx : x ∈ s
y : α
hy : y ≠ x
H : x < y
⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s

pp: case inr.inl
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
s : Set α
hs : IsOpen[inst✝³] s
x : α
hx : x ∈ s
hy : x ≠ x
⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s

pp: case inr.inr
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
s : Set α
hs : IsOpen[inst✝³] s
x : α
hx : x ∈ s
y : α
hy : y ≠ x
H : y < x
⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s

Tactic:
· obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=
      exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩
    exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩

Goal state after:
case inr.inl
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
s : Set α
hs : IsOpen[inst✝³] s
x : α
hx : x ∈ s
hy : x ≠ x
⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s

case inr.inr
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
s : Set α
hs : IsOpen[inst✝³] s
x : α
hx : x ∈ s
y : α
hy : y ≠ x
H : y < x
⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : WellFoundedLT α
s : Set α
h : IsUpperSet s
⊢ IsClosed[inst✝³] s

Tactic:
by
  obtain rfl | ⟨a, rfl⟩ := h.eq_empty_or_Ici
  exacts [isClosed_empty, isClosed_Ici]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
y : 𝕜
hy : y ∈ Ioc a (a + p)
⊢ ↑((equivIoc p a) ↑y) = y

Tactic:
by
  have : equivIoc p a y = ⟨y, hy⟩ := (equivIoc p a).right_inv ⟨y, hy⟩
  simp [this]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
x y : 𝕜
hx : x ∈ Ioc a (a + p)
hy : y ∈ Ioc a (a + p)
h : (equivIoc p a) ↑x = (equivIoc p a) ↑y
⊢ ⟨x, hx⟩ = ⟨y, hy⟩

Tactic:
← (equivIoc p a).right_inv ⟨x, hx⟩,

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
x y : 𝕜
hx : x ∈ Ioc a (a + p)
hy : y ∈ Ioc a (a + p)
h : (equivIoc p a) ↑x = (equivIoc p a) ↑y
⊢ (equivIoc p a).toFun ((equivIoc p a).invFun ⟨x, hx⟩) = ⟨y, hy⟩
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
⊢ ↑(↑n * (p / ↑n)) = 0

pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
m : ℕ
hn : m < n
h0 : 0 < m
⊢ ¬↑(↑m * (p / ↑n)) = 0

Tactic:
· rw [mul_div_cancel₀ _ h.ne', coe_period]

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
m : ℕ
hn : m < n
h0 : 0 < m
⊢ ¬↑(↑m * (p / ↑n)) = 0
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
m : ℕ
hn : m < n
h0 : 0 < m
k : ℕ
hk : ↑(k * n) = ↑m
⊢ False

Tactic:
Nat.cast_inj,

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
m : ℕ
hn : m < n
h0 : 0 < m
k : ℕ
hk : k * n = m
⊢ False