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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s s' t : Set α
hs : IsOpen[inst✝] s
hs' : IsOpen[inst✝] s'
ht : s ∩ frontier t = s' ∩ frontier t
x : α
hx : x ∈ frontier t
⊢ x ∈ s ↔ x ∈ s'

Tactic:
by simpa [hx] using Set.ext_iff.1 ht x

Goal state after:
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: case hf
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn f (t.ite s s' ∩ closure[inst✝²] t)

Tactic:
rwa [ite_inter_closure_eq_of_inter_frontier_eq H]

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

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
H : ∀ (b : X), b ≠ x → b ∈ s → ContinuousWithinAt (update f x y) s b
z : X
hz : z ∈ s \ {x}
⊢ ContinuousWithinAt f (s \ {x}) z

Tactic:
specialize H z hz.2 hz.1
    rw [continuousWithinAt_update_of_ne hz.2] at H
    exact H.mono sdiff_subset

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
S : Set X
⊢ IsPreirreducible S ↔ ∀ x ∈ S, S ⊆ closure[inst✝¹] {x}

Tactic:
by
  constructor
  · intro h x hx y hy
    by_contra e
    obtain ⟨U, V, hU, hV, hxU, hyV, h'⟩ := r1_separation fun h => e h.specializes.mem_closure
    exact ((h U V hU hV ⟨x, hx, hxU⟩ ⟨y, hy, hyV⟩).mono inter_subset_right).not_disjoint h'
  · intro h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩
    exact ⟨x, hxs, hxu, (specializes_iff_mem_closure.mpr (h x hxs hys)).mem_open hv hyv⟩

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T1Space X
x y : X
s : Set X
f : X → Y
⊢ ContinuousWithinAt f (s \ {y}) x ↔ ContinuousWithinAt f s x

Tactic:
← continuousWithinAt_insert,

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T1Space X
x y : X
s : Set X
f : X → Y
⊢ ContinuousWithinAt f (insert ?m.22 (s \ {y})) x ↔ ContinuousWithinAt f s x

X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T1Space X
x y : X
s : Set X
f : X → Y
⊢ X
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
K : Set X
hK : IsCompact K
⊢ cofinite ⊓ 𝓟 K ≤ codiscreteWithin K

Tactic:
intro s hs
  simpa [mem_inf_principal, compl_ofPred] using! hK.finite_sdiff_of_mem_codiscreteWithin hs

Goal state after:
Module: Mathlib.Data.Set.BoolIndicator

Goal state before:
pp: α : Type u_1
s : Set α
x : α
⊢ x ∈ s ↔ s.boolIndicator x = true

Tactic:
unfold boolIndicator
  split_ifs with h <;> simp [h]

Goal state after:
Module: Mathlib.Data.Set.BoolIndicator

Goal state before:
pp: α : Type u_1
s : Set α
x : α
⊢ ¬x ∈ s ↔ s.boolIndicator x = false

Tactic:
unfold boolIndicator
  split_ifs with h <;> simp [h]

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

Goal state before:
pp: α : Type u
ι : Type u_1
F : Filter ι
inst✝³ : F.NeBot
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
a : α
f : ι → α
hlt : ∀ (i : ι), f i < a
hlim : Tendsto f F (𝓝 a)
⊢ a ∉ range f

Tactic:
mem_range

Goal state after:
α : Type u
ι : Type u_1
F : Filter ι
inst✝³ : F.NeBot
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
a : α
f : ι → α
hlt : ∀ (i : ι), f i < a
hlim : Tendsto f F (𝓝 a)
⊢ ¬∃ y, f y = a
Module: Mathlib.Order.Filter.CountableInter

Goal state before:
pp: case mp.sInter
α : Type u_2
g : Set (Set α)
s : Set α
S : Set (Set α)
Sct : S.Countable
a✝ : ∀ s ∈ S, CountableGenerateSets g s
T : (s : Set α) → s ∈ S → Set (Set α)
Tg : ∀ (s : Set α) (a : s ∈ S), T s a ⊆ g
Tct : ∀ (s : Set α) (a : s ∈ S), (T s a).Countable
hT : ∀ (s : Set α) (a : s ∈ S), ⋂₀ T s a ⊆ s
⊢ ∃ S_1 ⊆ g, S_1.Countable ∧ ⋂₀ S_1 ⊆ ⋂₀ S

Tactic:
refine ⟨⋃ (s) (H : s ∈ S), T s H, by simpa, Sct.biUnion Tct, ?_⟩

Goal state after:
case mp.sInter
α : Type u_2
g : Set (Set α)
s : Set α
S : Set (Set α)
Sct : S.Countable
a✝ : ∀ s ∈ S, CountableGenerateSets g s
T : (s : Set α) → s ∈ S → Set (Set α)
Tg : ∀ (s : Set α) (a : s ∈ S), T s a ⊆ g
Tct : ∀ (s : Set α) (a : s ∈ S), (T s a).Countable
hT : ∀ (s : Set α) (a : s ∈ S), ⋂₀ T s a ⊆ s
⊢ ⋂₀ ⋃ s, ⋃ (H : s ∈ S), T s H ⊆ ⋂₀ S
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : Set α
H : IsPreconnected s
f : α → β
hf : ContinuousOn f s
u v : Set β
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x : α
xs : x ∈ s
xu : f x ∈ u
y : α
ys : y ∈ s
yv : f y ∈ v
u' : Set α
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set α
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
huv : s ⊆ (u' ∪ v') ∩ s
⊢ s ⊆ u' ∪ v'

Tactic:
exact (subset_inter_iff.1 huv).1

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s : Set X
f : X → Y
hs : IsLindelof s
hf : ContinuousOn f s
l : Filter Y
lne : l.NeBot
inst✝ : CountableInterFilter l
ls : l ≤ 𝓟 (f '' s)
this : (comap f l ⊓ 𝓟 s).NeBot
⊢ ∃ x ∈ f '' s, ClusterPt x l

Tactic:
obtain ⟨x, hxs, hx⟩ : ∃ x ∈ s, ClusterPt x (l.comap f ⊓ 𝓟 s) := @hs _ this _ inf_le_right

Goal state after:
X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s : Set X
f : X → Y
hs : IsLindelof s
hf : ContinuousOn f s
l : Filter Y
lne : l.NeBot
inst✝ : CountableInterFilter l
ls : l ≤ 𝓟 (f '' s)
this : (comap f l ⊓ 𝓟 s).NeBot
x : X
hxs : x ∈ s
hx : ClusterPt x (comap f l ⊓ 𝓟 s)
⊢ ∃ x ∈ f '' s, ClusterPt x l
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
i : α → β
di : IsDenseInducing i
s : Set α
H : Dense (i '' s)
x : α
⊢ x ∈ closure[inst✝¹] s

Tactic:
rw [di.isInducing.closure_eq_preimage_closure_image, H.closure_eq, preimage_univ]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
i : α → β
di : IsDenseInducing i
s : Set α
H : Dense (i '' s)
x : α
⊢ x ∈ univ
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case mp
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
U : Set X
⊢ IsLindelof U ∧ IsOpen[inst✝] U → ∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i

Tactic:
rintro ⟨h₁, h₂⟩
    obtain ⟨Y, f, rfl, hf⟩ := hb.open_eq_iUnion h₂
    choose f' hf' using hf
    have : b ∘ f' = f := funext hf'
    subst this
    obtain ⟨t, ht⟩ :=
      h₁.elim_countable_subcover (b ∘ f') (fun i => hb.isOpen (Set.mem_range_self _)) Subset.rfl
    refine ⟨t.image f', Countable.image (ht.1) f', le_antisymm ?_ ?_⟩
    · refine Set.Subset.trans ht.2 ?_
      simp only [Set.iUnion_subset_iff]
      intro i hi
      rw [← Set.iUnion_subtype (fun x : ι => x ∈ t.image f') fun i => b i.1]
      exact Set.subset_iUnion (fun i : t.image f' => b i) ⟨_, mem_image_of_mem _ hi⟩
    · apply Set.iUnion₂_subset
      rintro i hi
      obtain ⟨j, -, rfl⟩ := (mem_image ..).mp hi
      exact Set.subset_iUnion (b ∘ f') j

Goal state after:
Module: Mathlib.Topology.Baire.Lemmas

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
s : Set X
hG : IsGδ s
hd : Dense s
f : ℕ → Set ↑s
hof : ∀ (n : ℕ), IsOpen[instTopologicalSpaceSubtype] (f n)
⊢ (∀ (n : ℕ), Dense (f n)) → Dense (⋂ n, f n)

Tactic:
hdf

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
s : Set X
hG : IsGδ s
hd : Dense s
f : ℕ → Set ↑s
hof : ∀ (n : ℕ), IsOpen[instTopologicalSpaceSubtype] (f n)
hdf : ∀ (n : ℕ), Dense (f n)
⊢ Dense (⋂ n, f n)
Module: Mathlib.Topology.Baire.Lemmas

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : BaireSpace X
Y : Type u_4
inst✝ : TopologicalSpace Y
p : Y → X
hp : IsOpenEmbedding p
f : ℕ → Set Y
hof : ∀ (n : ℕ), IsOpen[inst✝] (f n)
⊢ (∀ (n : ℕ), Dense (f n)) → Dense (⋂ n, f n)

Tactic:
hdf

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : BaireSpace X
Y : Type u_4
inst✝ : TopologicalSpace Y
p : Y → X
hp : IsOpenEmbedding p
f : ℕ → Set Y
hof : ∀ (n : ℕ), IsOpen[inst✝] (f n)
hdf : ∀ (n : ℕ), Dense (f n)
⊢ Dense (⋂ n, f n)
Module: Mathlib.Topology.Baire.Lemmas

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : BaireSpace X
Y : Type u_4
inst✝ : TopologicalSpace Y
p : Y → X
hp : IsOpenEmbedding p
f : ℕ → Set Y
hof : ∀ (n : ℕ), IsOpen[inst✝] (f n)
hdf : ∀ (n : ℕ), Dense (f n)
s : Set X := range p
c : ℕ → Set X := fun n ↦ p '' f n ∪ (closure[inst✝²] s)ᶜ
c_open : ∀ (n : ℕ), IsOpen[inst✝²] (c n)
c_dense : ∀ (n : ℕ), Dense (c n)
c_inter_dense : Dense (⋂ n, c n)
c_inter_eq : ⋂ n, f n = p ⁻¹' ⋂ n, c n
⊢ Dense (⋂ n, f n)

Tactic:
exact c_inter_eq ▸ Dense.preimage c_inter_dense hp.isOpenMap

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

Goal state before:
pp: α✝ : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
α : Type u
u : Core α
t : TopologicalSpace α
h : t = u.toTopologicalSpace
x : α
⊢ 𝓝 x = comap (Prod.mk x) u.uniformity

Tactic:
by rw [h, u.nhds_toTopologicalSpace]

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

Goal state before:
pp: α : Type ua
ι : Sort u_1
inst✝ : UniformSpace α
p : ι → Prop
s : ι → SetRel α α
h : (𝓤 α).HasBasis p s
x : α
⊢ (𝓝 x).HasBasis p fun i ↦ {y | (y, x) ∈ s i}

Tactic:
replace h := h.comap Prod.swap
  rw [comap_swap_uniformity] at h
  exact nhds_basis_uniformity' h

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

Goal state before:
pp: α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝¹ : SemilatticeSup β
inst✝ : Nonempty β
U : β → SetRel α α
hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s
f : β → α
hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N
s : Set (α × α)
hs : s ∈ 𝓤 α
N : β
hN : U N ⊆ s
⊢ ∃ a, ∀ (b : β × β), a ≤ b → b ∈ Prod.map f f ⁻¹' s

Tactic:
refine ⟨(N, N), fun mn hmn => ?_⟩

Goal state after:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝¹ : SemilatticeSup β
inst✝ : Nonempty β
U : β → SetRel α α
hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s
f : β → α
hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N
s : Set (α × α)
hs : s ∈ 𝓤 α
N : β
hN : U N ⊆ s
mn : β × β
hmn : (N, N) ≤ mn
⊢ mn ∈ Prod.map f f ⁻¹' s
Module: Mathlib.Topology.UniformSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
h : TendstoUniformlyOnFilter F f p ⊤
g : γ → α
⊢ TendstoUniformlyOnFilter (fun n ↦ F n ∘ g) (f ∘ g) p ⊤

Tactic:
simpa [principal_univ, comap_principal] using h.comp g

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

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
uγ : UniformSpace γ
f : α → β
g : β → γ
⊢ 𝓤 α = 𝓤 α

Tactic:
simp only [uniformity_comap, Filter.comap_comap, Prod.map_comp_map]

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

Goal state before:
pp: α : Type u
inst✝¹ : UniformSpace α
inst✝ : T0Space α
V₀ : Set (α × α)
V₀_in : V₀ ∈ 𝓤 α
s : Set α
hs : s.Pairwise fun x y ↦ (x, y) ∉ V₀
V₁ : Set (α × α)
V₁_in : V₁ ∈ 𝓤 α
V₁_symm : SetRel.IsSymm V₁
h_comp : SetRel.comp V₁ V₁ ⊆ V₀
x : α
hx : ∀ {V : Set (α × α)}, V ∈ 𝓤 α → (ball x V ∩ s).Nonempty
y : α
hy : y ∈ ball x V₁
hy' : y ∈ s
V : Set (α × α)
V_in : V ∈ 𝓤 α
a✝ : SetRel.IsSymm V
z : α
hz : z ∈ ball x (V₁ ∩ V)
hz' : z ∈ s
⊢ (x, y) ∈ V

Tactic:
obtain rfl : z = y := by
    by_contra hzy
    exact hs hz' hy' hzy (h_comp <| mem_comp_of_mem_ball (ball_inter_left x _ _ hz) hy)

Goal state after:
α : Type u
inst✝¹ : UniformSpace α
inst✝ : T0Space α
V₀ : Set (α × α)
V₀_in : V₀ ∈ 𝓤 α
s : Set α
hs : s.Pairwise fun x y ↦ (x, y) ∉ V₀
V₁ : Set (α × α)
V₁_in : V₁ ∈ 𝓤 α
V₁_symm : SetRel.IsSymm V₁
h_comp : SetRel.comp V₁ V₁ ⊆ V₀
x : α
hx : ∀ {V : Set (α × α)}, V ∈ 𝓤 α → (ball x V ∩ s).Nonempty
V : Set (α × α)
V_in : V ∈ 𝓤 α
a✝ : SetRel.IsSymm V
z : α
hz : z ∈ ball x (V₁ ∩ V)
hz' : z ∈ s
hy : z ∈ ball x V₁
hy' : z ∈ s
⊢ (x, z) ∈ V
Module: Mathlib.Topology.Algebra.Support

Goal state before:
pp: α : Type u_2
γ : Type u_5
inst✝² : TopologicalSpace α
inst✝¹ : One γ
f : α → γ
inst✝ : TopologicalSpace γ
h : HasCompactMulSupport f
N : Set γ
hN : N ∈ 𝓝 1
v : α
hv : v ∈ (mulTSupport f)ᶜ
⊢ f v ∈ N

Tactic:
image_eq_one_of_notMem_mulTSupport hv

Goal state after:
α : Type u_2
γ : Type u_5
inst✝² : TopologicalSpace α
inst✝¹ : One γ
f : α → γ
inst✝ : TopologicalSpace γ
h : HasCompactMulSupport f
N : Set γ
hN : N ∈ 𝓝 1
v : α
hv : v ∈ (mulTSupport f)ᶜ
⊢ 1 ∈ N
Module: Mathlib.Topology.UniformSpace.HeineCantor

Goal state before:
pp: case he
α : Type u_4
β : Type u_5
E : Type u_6
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : UniformSpace E
f : α → β → E
s : Set α
k : Set β
q : α
u : Set (E × E)
hk : IsCompact k
hf : ContinuousOn (Function.uncurry f) (s ×ˢ k)
hq : q ∈ s
hu : u ∈ 𝓤 E
⊢ ∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ ∅, (f p x, f q x) ∈ u

Tactic:
exact ⟨univ, univ_mem, by simp⟩

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

Goal state before:
pp: ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
inst✝ : Finite ι
F : ι → β → α
S : Set β
⊢ UniformEquicontinuousOn F S ↔ ∀ (i : ι), UniformContinuousOn (F i) S

Tactic:
simp only [UniformEquicontinuousOn, eventually_all, @forall_comm _ ι]

Goal state after:
ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
inst✝ : Finite ι
F : ι → β → α
S : Set β
⊢ (∀ (b : ι), ∀ a ∈ 𝓤 α, ∀ᶠ (x : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), (F b x.1, F b x.2) ∈ a) ↔
    ∀ (i : ι), UniformContinuousOn (F i) S
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
g : Filter β
⊢ map (fun p ↦ m p.1 p.2) (map Prod.swap (g ×ˢ f)) = map₂ (fun a b ↦ m b a) g f

Tactic:
map_map,

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
g : Filter β
⊢ map ((fun p ↦ m p.1 p.2) ∘ Prod.swap) (g ×ˢ f) = map₂ (fun a b ↦ m b a) g f
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
δ : Type u_7
f : Filter α
g : Filter β
m : γ → β → δ
n : α → γ
⊢ map (fun p ↦ m p.1 p.2) (map (fun p ↦ (n p.1, id p.2)) (f ×ˢ g)) = map (fun p ↦ m (n p.1) p.2) (f ×ˢ map id g)

Tactic:
map_map,

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
δ : Type u_7
f : Filter α
g : Filter β
m : γ → β → δ
n : α → γ
⊢ map ((fun p ↦ m p.1 p.2) ∘ fun p ↦ (n p.1, id p.2)) (f ×ˢ g) = map (fun p ↦ m (n p.1) p.2) (f ×ˢ map id g)
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: case mp
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T1Space Y
y : Y
⊢ ((Continuous[inst✝², inst✝¹] fun x ↦ y) ∧ (IsClosedMap fun x ↦ y) ∧ ∀ (y_1 : Y), IsCompact ((fun x ↦ y) ⁻¹' {y_1})) →
    CompactSpace X

Tactic:
rintro ⟨-, -, h⟩

Goal state after:
case mp
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T1Space Y
y : Y
h : ∀ (y_1 : Y), IsCompact ((fun x ↦ y) ⁻¹' {y_1})
⊢ CompactSpace X
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
S : Set G
hS : S ∈ 𝓝 1
⊢ S⁻¹ ∈ 𝓝 1

Tactic:
rwa [← nhds_one_symm'] at hS

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
n : ℤ
hn : 0 < n
⊢ Tendsto (fun x ↦ x ^ n) atTop atTop

Tactic:
lift n to ℕ using hn.le; simp [(Int.natCast_pos.mp hn).ne']

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
G₀ : Type u_3
inst✝⁵ : GroupWithZero G₀
inst✝⁴ : TopologicalSpace G₀
inst✝³ : ContinuousInv₀ G₀
inst✝² : ContinuousMul G₀
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
a : α
f g : α → G₀
h : α → G₀ → β
hf : ContinuousAt f a
hg : ContinuousAt g a
hh : g a ≠ 0 → ContinuousAt ↿h (a, f a / g a)
h2h : g a = 0 → Tendsto (↿h) (𝓝 a ×ˢ ⊤) (𝓝 (h a 0))
hga : g a = 0
⊢ ContinuousAt (↿h ∘ fun x ↦ (x, f x / g x)) a

Tactic:
rw [ContinuousAt]

Goal state after:
case pos
α : Type u_1
β : Type u_2
G₀ : Type u_3
inst✝⁵ : GroupWithZero G₀
inst✝⁴ : TopologicalSpace G₀
inst✝³ : ContinuousInv₀ G₀
inst✝² : ContinuousMul G₀
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
a : α
f g : α → G₀
h : α → G₀ → β
hf : ContinuousAt f a
hg : ContinuousAt g a
hh : g a ≠ 0 → ContinuousAt ↿h (a, f a / g a)
h2h : g a = 0 → Tendsto (↿h) (𝓝 a ×ˢ ⊤) (𝓝 (h a 0))
hga : g a = 0
⊢ Tendsto (↿h ∘ fun x ↦ (x, f x / g x)) (𝓝 a) (𝓝 ((↿h ∘ fun x ↦ (x, f x / g x)) a))
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: ι : Type u_1
α : Type u_2
G : Type u_3
inst✝¹ : Group G
u : UniformSpace G
inst✝ : IsTopologicalGroup G
F : ι → α → G
p : Filter ι
s : Set α
hu : rightUniformSpace G = u
⊢ UniformCauchySeqOn F p s ↔ ∀ u_1 ∈ 𝓝 1, ∀ᶠ (m : ι × ι) in p ×ˢ p, ∀ a ∈ s, F m.2 a / F m.1 a ∈ u_1

Tactic:
simp only [div_eq_mul_inv]
  exact hu ▸ ⟨fun h u hu ↦ h _ ⟨u, hu, fun _ ↦ id⟩,
    fun h _ ⟨u, hu, hv⟩ => mem_of_superset (h u hu) fun _ hi a ha => hv (hi a ha)⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝¹.toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
x₀ : α
y₁ : δ
Nx : Filter α := 𝓝 x₀
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
lim1 : Tendsto (fun a ↦ (a.2 - a.1, y₁)) (comap (⇑e) Nx ×ˢ comap (⇑e) Nx) (𝓝 (0, y₁))
lim2 : Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, y₁)) (𝓝 0)
⊢ ∃ U₂ ∈ comap (⇑e) (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂, (fun p ↦ (φ p.1) p.2) (x' - x, y₁) ∈ W'

Tactic:
have lim := lim2.comp lim1

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝¹.toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
x₀ : α
y₁ : δ
Nx : Filter α := 𝓝 x₀
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
lim1 : Tendsto (fun a ↦ (a.2 - a.1, y₁)) (comap (⇑e) Nx ×ˢ comap (⇑e) Nx) (𝓝 (0, y₁))
lim2 : Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, y₁)) (𝓝 0)
lim : Tendsto ((fun p ↦ (φ p.1) p.2) ∘ fun a ↦ (a.2 - a.1, y₁)) (comap (⇑e) Nx ×ˢ comap (⇑e) Nx) (𝓝 0)
⊢ ∃ U₂ ∈ comap (⇑e) (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂, (fun p ↦ (φ p.1) p.2) (x' - x, y₁) ∈ W'
Module: Mathlib.Order.Filter.AtTopBot.CompleteLattice

Goal state before:
pp: α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
this✝ : Nonempty α
this : Nonempty β
⊢ ⨆ a, f (g a) = ⨆ b, f b

Tactic:
← csInf_upperBounds_range,

Goal state after:
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
this✝ : Nonempty α
this : Nonempty β
⊢ sInf (upperBounds (range fun a ↦ f (g a))) = ⨆ b, f b

α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
this✝ : Nonempty α
this : Nonempty β
⊢ BddAbove (range fun a ↦ f (g a))
Module: Mathlib.Order.Filter.AtTopBot.CompleteLattice

Goal state before:
pp: α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
this✝ : Nonempty α
this : Nonempty β
⊢ sInf (upperBounds (range fun a ↦ f (g a))) = ⨆ b, f b

pp: α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
this✝ : Nonempty α
this : Nonempty β
⊢ BddAbove (range fun a ↦ f (g a))

Tactic:
← csInf_upperBounds_range,

Goal state after:
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
this✝ : Nonempty α
this : Nonempty β
⊢ sInf (upperBounds (range fun a ↦ f (g a))) = sInf (upperBounds (range f))

α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
this✝ : Nonempty α
this : Nonempty β
⊢ BddAbove (range f)

α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Preorder β
inst✝¹ : ConditionallyCompleteLattice γ
l : Filter α
inst✝ : l.NeBot
f : β → γ
hf : Monotone f
hb : BddAbove (range f)
g : α → β
hg : Tendsto g l atTop
this✝ : Nonempty α
this : Nonempty β
⊢ BddAbove (range fun a ↦ f (g a))
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a b : α i
⊢ update f i '' Ioo a b = Ioo (update f i a) (update f i b)

Tactic:
rw [← Ico_sdiff_left, ← Ico_sdiff_left, image_sdiff (update_injective _ _), image_singleton,
    image_update_Ico]

Goal state after:
Module: Mathlib.Order.Filter.Interval

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
l : Filter α
inst✝ : TendstoIxxClass Icc l l
s : Set (Set α)
hs : s ∈ l.smallSets
t : Set α
htl : t ∈ l
hts : ∀ p ∈ t ×ˢ t, Icc p.1 p.2 ∈ s
p : α × α
hp : p ∈ t ×ˢ t
⊢ p ∈ (fun p ↦ [[p.1, p.2]]) ⁻¹' s

Tactic:
rcases le_total p.1 p.2 with h | h

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
l : Filter α
inst✝ : TendstoIxxClass Icc l l
s : Set (Set α)
hs : s ∈ l.smallSets
t : Set α
htl : t ∈ l
hts : ∀ p ∈ t ×ˢ t, Icc p.1 p.2 ∈ s
p : α × α
hp : p ∈ t ×ˢ t
h : p.1 ≤ p.2
⊢ p ∈ (fun p ↦ [[p.1, p.2]]) ⁻¹' s

case inr
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
l : Filter α
inst✝ : TendstoIxxClass Icc l l
s : Set (Set α)
hs : s ∈ l.smallSets
t : Set α
htl : t ∈ l
hts : ∀ p ∈ t ×ˢ t, Icc p.1 p.2 ∈ s
p : α × α
hp : p ∈ t ×ˢ t
h : p.2 ≤ p.1
⊢ p ∈ (fun p ↦ [[p.1, p.2]]) ⁻¹' s
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e✝ : PartialEquiv α β
e' : PartialEquiv β γ
e : α ≃ β
⊢ ⇑e '' univ = univ

Tactic:
rw [image_univ, e.surjective.range_eq]

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

Goal state before:
pp: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e✝ : PartialHomeomorph X Y
e : X ≃ₜ Y
⊢ ⇑e '' univ = univ

Tactic:
rw [image_univ, e.surjective.range_eq]

Goal state after: