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

Goal state before:
pp: case refine_1
α : Type u_1
ts : TopologicalSpace α
inst✝ : SecondCountableTopology α
t : Set (Set α)
ht : ts = generateFrom t
t' : Set (Set α) := (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ t}
this : IsTopologicalBasis t'
s' : Set (Set α)
s't' : s' ⊆ t'
s'_count : s'.Countable
hs' : IsTopologicalBasis s'
f : Set α → Set (Set α)
f_fin : ∀ u ∈ s', (f u).Finite
ft : ∀ u ∈ s', f u ⊆ t
hf : ∀ u ∈ s', ⋂₀ f u = u
⊢ (⋃ u ∈ s', f u).Countable

Tactic:
apply s'_count.biUnion
    intro u hu
    exact Finite.countable (f_fin u hu)

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyCompactSpace X
x : X
U : Set X
hU : IsOpen[inst✝¹] U
hx : x ∈ U
⊢ ∃ K, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U

Tactic:
rcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyCompactSpace X
x : X
U : Set X
hU : IsOpen[inst✝¹] U
hx : x ∈ U
K : Set X
h1K : K ∈ 𝓝 x
h2K : K ⊆ U
h3K : IsCompact K
⊢ ∃ K, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U
Module: Mathlib.Topology.LocallyFinite

Goal state before:
pp: ι : Type u_1
X : Type u_4
inst✝ : TopologicalSpace X
f : ι → Set X
hf : LocallyFinite f
hc : ∀ (i : ι), IsClosed[inst✝] (f i)
⊢ IsClosed[inst✝] (⋃ i, f i)

Tactic:
simp only [← closure_eq_iff_isClosed, hf.closure_iUnion, (hc _).closure_eq]

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
f : Filter X
hf : sᶜ ∉ f
⊢ ∃ x ∈ s, sᶜ ∉ 𝓝 x ⊓ f

Tactic:
simp only [notMem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
f : Filter X
hf : (f ⊓ 𝓟 s).NeBot
⊢ ∃ x ∈ s, (𝓝 x ⊓ (f ⊓ 𝓟 s)).NeBot
Module: Mathlib.Topology.Compactness.SigmaCompact

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
K : CompactExhaustion X
s : Set X
hs : IsCompact s
⊢ ∃ n, s ⊆ K n

Tactic:
suffices ∃ n, s ⊆ interior (K n) from this.imp fun _ ↦ (Subset.trans · interior_subset)
  refine hs.elim_directed_cover (interior ∘ K) (fun _ ↦ isOpen_interior) ?_ ?_
  · intro x _
    rcases K.exists_mem x with ⟨k, hk⟩
    exact mem_iUnion.2 ⟨k + 1, K.subset_interior_succ _ hk⟩
  · exact Monotone.directed_le fun _ _ h ↦ interior_mono <| K.subset h

Goal state after:
Module: Mathlib.Order.UpperLower.Closure

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : Set α
h : ↑(upperClosure s) ∩ ↑(lowerClosure s) = s
⊢ s.OrdConnected

Tactic:
rw [← h]

Goal state after:
α : Type u_1
inst✝ : Preorder α
s : Set α
h : ↑(upperClosure s) ∩ ↑(lowerClosure s) = s
⊢ (↑(upperClosure s) ∩ ↑(lowerClosure s)).OrdConnected
Module: Mathlib.Order.UpperLower.CompleteLattice

Goal state before:
pp: α : Type u_1
inst✝ : LE α
S : Set (UpperSet α)
a : α
⊢ (∃ i, ∃ (_ : i ∈ S), a ∈ ↑i) ↔ ∃ s ∈ S, a ∈ s

Tactic:
by simp only [exists_prop, SetLike.mem_coe]

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
t : Set X
ht : IsCompact t
⊢ ∃ i, IsCompact i ∧ f ⁻¹' iᶜ ⊆ tᶜ

Tactic:
refine ⟨f '' t, ht.image hf, ?_⟩

Goal state after:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
t : Set X
ht : IsCompact t
⊢ f ⁻¹' (f '' t)ᶜ ⊆ tᶜ
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y : X
f : X → Y
s : Set X
h : x ⤳ y
hf : ContinuousWithinAt f s y
hx : x ∈ s
⊢ f x ⤳ f y

Tactic:
by
  rw [specializes_iff_pure] at h ⊢
  calc pure (f x)
    _ = map f (pure x) := (map_pure f x).symm
    _ ≤ map f (𝓝 y ⊓ 𝓟 s) := map_mono (le_inf h ((pure_le_principal x).mpr hx))
    _ = map f (𝓝[s] y) := rfl
    _ ≤ _ := hf.tendsto

Goal state after:
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s)

Tactic:
by
  rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]

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

Goal state before:
pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsClosed[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsClosed[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
this :
  ∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {s : Set X},
    IsClosed s →
      (∀ t ⊆ s, t.Nonempty → IsClosed t → t = s) →
        ∀ x ∈ s, ∀ y ∈ s, ¬x = y → ∀ (U : Set X), IsOpen U → Xor (x ∈ U) (y ∈ U) → x ∈ U ∧ y ∉ U → False
h : ¬(x ∈ U ∧ y ∉ U)
⊢ False

Tactic:
refine this hs hmin y hy x hx (Ne.symm hxy) U hUo hU.symm (hU.resolve_left h)

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

Goal state before:
pp: case a.inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
⊢ ∃ x ∈ ↑s, IsOpen[inst✝¹] {x}

Tactic:
· -- Porting note: was `rcases minimal_nonempty_open_eq_singleton ho hne _ with ⟨x, hx⟩`
    --               https://github.com/leanprover-community/batteries/issues/116
    rsuffices ⟨x, hx⟩ : ∃ x, (s : Set X) = {x}
    · exact ⟨x, hx.symm ▸ rfl, hx ▸ ho⟩
    refine minimal_nonempty_open_eq_singleton ho hne ?_
    refine fun t hts htne hto => of_not_not fun hts' => ht ?_
    lift t to Finset X using s.finite_toSet.subset hts
    exact ⟨t, ssubset_iff_subset_ne.2 ⟨hts, mt Finset.coe_inj.2 hts'⟩, htne, hto⟩

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

Goal state before:
pp: case inl
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T1Space X
x : X
s : Set X
f : X → Y
⊢ ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x

Tactic:
exact continuousWithinAt_insert_self

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
⊢ IsClosed[instTopologicalSpaceProd] {p | Inseparable p.1 p.2}

Tactic:
simp only [← specializes_iff_inseparable, isClosed_setOfPred_specializes]

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

Goal state after:
α : Type u_1
s : Set α
x : α
⊢ x ∈ s ↔ (if x ∈ s then true else false) = true
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:
by
  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

Goal state after:
α : Type u_1
s : Set α
x : α
⊢ ¬x ∈ s ↔ (if x ∈ s then true else false) = false
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:
by
  unfold boolIndicator
  split_ifs with h <;> simp [h]

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
s : Set α
hs : IsClopen s
h : ¬(s = ∅ ∨ s = univ)
⊢ s.Nonempty ∧ sᶜ.Nonempty

Tactic:
by simpa [nonempty_iff_ne_empty] using h

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

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
hs : IsLindelof s
f : Filter X
inst✝ : CountableInterFilter f
hf : sᶜ ∉ f
⊢ ∃ x ∈ s, sᶜ ∉ 𝓝 x ⊓ f

Tactic:
simp only [notMem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢

Goal state after:
X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
hs : IsLindelof s
f : Filter X
inst✝ : CountableInterFilter f
hf : (f ⊓ 𝓟 s).NeBot
⊢ ∃ x ∈ s, (𝓝 x ⊓ (f ⊓ 𝓟 s)).NeBot
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hs : IsLindelof s
ht : IsClosed[inst✝] t
⊢ IsLindelof (s ∩ t)

Tactic:
intro f hnf _ hstf
  rw [← inf_principal, le_inf_iff] at hstf
  obtain ⟨x, hsx, hx⟩ : ∃ x ∈ s, ClusterPt x f := hs hstf.1
  have hxt : x ∈ t := ht.mem_of_nhdsWithin_neBot <| hx.mono hstf.2
  exact ⟨x, ⟨hsx, hxt⟩, hx⟩

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

Goal state before:
pp: case inl
X : Type u_1
inst✝² : TopologicalSpace X
s t : Set X
inst✝¹ : LindelofSpace X
inst✝ : RegularSpace X
s_cl : IsClosed[inst✝²] s
t_cl : IsClosed[inst✝²] t
st_dis : Disjoint s t
empty_X : IsEmpty X
⊢ _root_.HasSeparatingCover s t

Tactic:
rw [subset_eq_empty (t := s) (fun ⦃_⦄ _ ↦ trivial) (univ_eq_empty_iff.mpr empty_X)]

Goal state after:
case inl
X : Type u_1
inst✝² : TopologicalSpace X
s t : Set X
inst✝¹ : LindelofSpace X
inst✝ : RegularSpace X
s_cl : IsClosed[inst✝²] s
t_cl : IsClosed[inst✝²] t
st_dis : Disjoint s t
empty_X : IsEmpty X
⊢ _root_.HasSeparatingCover ∅ t
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X → Y
hf : IsClosedEmbedding f
s t : Set X
hs : IsClosed[inst✝²] s
ht : IsClosed[inst✝²] t
hst : Disjoint s t
⊢ SeparatedNhds s t

Tactic:
have H : SeparatedNhds (f '' s) (f '' t) :=
      NormalSpace.normal (f '' s) (f '' t) (hf.isClosedMap s hs) (hf.isClosedMap t ht)
        (disjoint_image_of_injective hf.injective hst)
    exact (H.preimage hf.continuous).mono (subset_preimage_image _ _) (subset_preimage_image _ _)

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

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
x : α
h : connectedComponent x = univ
⊢ IsPreconnected univ

Tactic:
rw [← h]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
x : α
h : connectedComponent x = univ
⊢ IsPreconnected (connectedComponent x)
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
t : Set X
ht : IsLindelof t
⊢ ∃ i, IsLindelof i ∧ f ⁻¹' iᶜ ⊆ tᶜ

Tactic:
refine ⟨f '' t, ht.image hf, ?_⟩

Goal state after:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
t : Set X
ht : IsLindelof t
⊢ f ⁻¹' (f '' t)ᶜ ⊆ tᶜ
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed[inst✝²] (⋂ s, ↑s)
a b : Set X
ha : IsClosed[inst✝²] a
hb : IsClosed[inst✝²] b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen[inst✝²] u
hv : IsOpen[inst✝²] v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
H1 : Disjoint (u ∪ v)ᶜ (⋂ i, ↑i) → ∃ x_1, ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ x_1, ↑i).Nonempty
si : Finset { s // IsClopen s ∧ x ∈ s }
H2 : ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ si, ↑i).Nonempty
⊢ IsClopen (⋂ U ∈ si, ↑U)

Tactic:
exact isClopen_biInter_finset fun s _ => s.2.1

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
h : ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ
f : α → Bool
hf : Continuous[inst✝, _] f
x y : α
⊢ f ⁻¹' {false} = (f ⁻¹' {true})ᶜ

Tactic:
← Set.preimage_compl,

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ
f : α → Bool
hf : Continuous[inst✝, _] f
x y : α
⊢ f ⁻¹' {false} = f ⁻¹' {true}ᶜ
Module: Mathlib.Topology.UniformSpace.Defs

Goal state before:
pp: α : Type ua
inst✝ : UniformSpace α
s : SetRel α α
hs : s ∈ 𝓤 α
w : Set (α × α)
w_in : w ∈ 𝓤 α
left✝ : SetRel.IsSymm w
w_sub : w ○ w ⊆ s
t : Set (α × α)
t_in : t ∈ 𝓤 α
t_symm : SetRel.IsSymm t
t_sub : t ○ t ⊆ w
⊢ ∃ t ∈ 𝓤 α, SetRel.IsSymm t ∧ t ○ t ○ t ⊆ s

Tactic:
use t, t_in, t_symm

Goal state after:
case right
α : Type ua
inst✝ : UniformSpace α
s : SetRel α α
hs : s ∈ 𝓤 α
w : Set (α × α)
w_in : w ∈ 𝓤 α
left✝ : SetRel.IsSymm w
w_sub : w ○ w ⊆ s
t : Set (α × α)
t_in : t ∈ 𝓤 α
t_symm : SetRel.IsSymm t
t_sub : t ○ t ⊆ w
⊢ t ○ t ○ t ⊆ s
Module: Mathlib.Topology.UniformSpace.Defs

Goal state before:
pp: case right
α : Type ua
inst✝ : UniformSpace α
s : SetRel α α
hs : s ∈ 𝓤 α
w : Set (α × α)
w_in : w ∈ 𝓤 α
left✝ : SetRel.IsSymm w
w_sub : w ○ w ⊆ s
t : Set (α × α)
t_in : t ∈ 𝓤 α
t_symm : SetRel.IsSymm t
t_sub : t ○ t ⊆ w
⊢ t ○ t ○ t ⊆ s

Tactic:
have : t ⊆ t ○ t := subset_comp_self_of_mem_uniformity t_in

Goal state after:
case right
α : Type ua
inst✝ : UniformSpace α
s : SetRel α α
hs : s ∈ 𝓤 α
w : Set (α × α)
w_in : w ∈ 𝓤 α
left✝ : SetRel.IsSymm w
w_sub : w ○ w ⊆ s
t : Set (α × α)
t_in : t ∈ 𝓤 α
t_symm : SetRel.IsSymm t
t_sub : t ○ t ⊆ w
this : t ⊆ t ○ t
⊢ t ○ t ○ t ⊆ s
Module: Mathlib.Topology.UniformSpace.Defs

Goal state before:
pp: case mp
β : Type ub
V W M : SetRel β β
inst✝ : W.IsSymm
x y : β
⊢ (x, y) ∈ V ○ M ○ W → (ball (x, y).1 V ×ˢ ball (x, y).2 W ∩ M).Nonempty

Tactic:
rintro ⟨z, ⟨w, hpw, hwz⟩, hzy⟩

Goal state after:
case mp
β : Type ub
V W M : SetRel β β
inst✝ : W.IsSymm
x y z : β
hzy : (z, y) ∈ W
w : β
hpw : (x, w) ∈ V
hwz : (w, z) ∈ M
⊢ (ball (x, y).1 V ×ˢ ball (x, y).2 W ∩ M).Nonempty
Module: Mathlib.Topology.UniformSpace.Basic

Goal state before:
pp: α✝ : Type ua
β✝ : Type ub
γ✝ : Type uc
δ : Type ud
ι : Sort u_1
inst✝² : UniformSpace α✝
inst✝¹ : UniformSpace β✝
inst✝ : UniformSpace γ✝
α : Type ?u.21
β : Type ?u.25
γ : Type ?u.29
f : α → β → γ
ua1 ua2 : UniformSpace α
ub1 ub2 : UniformSpace β
uc1 : UniformSpace γ
⊢ Sort ?u.30

Tactic:
haveI := ua1

Goal state after:
α✝ : Type ua
β✝ : Type ub
γ✝ : Type uc
δ : Type ud
ι : Sort u_1
inst✝² : UniformSpace α✝
inst✝¹ : UniformSpace β✝
inst✝ : UniformSpace γ✝
α : Type ?u.21
β : Type ?u.25
γ : Type ?u.29
f : α → β → γ
ua1 ua2 : UniformSpace α
ub1 ub2 : UniformSpace β
uc1 : UniformSpace γ
this : UniformSpace α
⊢ Sort ?u.30
Module: Mathlib.Topology.UniformSpace.Compact

Goal state before:
pp: α : Type ua
ι : Sort u_1
inst✝ : UniformSpace α
p : ι → Prop
s : ι → Set (α × α)
hU : (𝓤 α).HasBasis p s
A B : Set α
hA : IsCompact A
hB : IsClosed[inst✝.toTopologicalSpace] B
h : Disjoint A B
V : Set (α × α)
hV : V ∈ 𝓤 α
hVAB : Disjoint (⋃ x ∈ A, ball x V) (⋃ x ∈ B, ball x V)
i : ι
hi : p i
hiV : s i ⊆ V
⊢ ∃ i, p i ∧ Disjoint (⋃ x ∈ A, ball x (s i)) (⋃ x ∈ B, ball x (s i))

Tactic:
exact ⟨i, hi, hVAB.mono (iUnion₂_mono fun a _ => ball_mono hiV a)
    (iUnion₂_mono fun b _ => ball_mono hiV b)⟩

Goal state after:
Module: Mathlib.Algebra.Order.Group.MinMax

Goal state before:
pp: α : Type u_1
inst✝² : CommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedMonoid α
a b c : α
⊢ min (a / b) (a / c) = a / max b c

Tactic:
by
  simp only [div_eq_mul_inv, min_mul_mul_left, min_inv_inv']

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

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X' : Type u_4
α : Type u_6
uα : UniformSpace α
t : κ → TopologicalSpace X'
F : ι → X' → α
x₀ : X'
k : κ
hk : EquicontinuousAt F x₀
⊢ EquicontinuousAt F x₀

Tactic:
← equicontinuousWithinAt_univ (tX := _)

Goal state after:
ι : Type u_1
κ : Type u_2
X' : Type u_4
α : Type u_6
uα : UniformSpace α
t : κ → TopologicalSpace X'
F : ι → X' → α
x₀ : X'
k : κ
hk : EquicontinuousWithinAt F univ x₀
⊢ EquicontinuousWithinAt F univ x₀
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
f₁ f₂ : Filter α
g : Filter β
⊢ map (fun p ↦ m p.1 p.2) (f₁ ×ˢ g ⊔ f₂ ×ˢ g) = map (fun p ↦ m p.1 p.2) (f₁ ×ˢ g) ⊔ map (fun p ↦ m p.1 p.2) (f₂ ×ˢ g)

Tactic:
map_sup

Goal state after:
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
g₁ g₂ : Filter β
⊢ map (fun p ↦ m p.1 p.2) (f ×ˢ g₁ ⊔ f ×ˢ g₂) = map (fun p ↦ m p.1 p.2) (f ×ˢ g₁) ⊔ map (fun p ↦ m p.1 p.2) (f ×ˢ g₂)

Tactic:
map_sup

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
ε : Type u_9
f : Filter α
g : Filter β
h : Filter γ
m : α → δ → ε
n : β → γ → δ
m₁ : α → β → β'
m₂ : α → γ → γ'
n' : β' → γ' → ε
h_distrib : ∀ (a : α) (b : β) (c : γ), m a (n b c) = n' (m₁ a b) (m₂ a c)
s : Set ε
t₁ : Set β'
u₁ : Set α
hu₁ : u₁ ∈ f
v : Set β
hv : v ∈ g
ht₁ : image2 m₁ u₁ v ⊆ t₁
t₂ : Set γ'
hs : image2 n' t₁ t₂ ⊆ s
u₂ : Set α
hu₂ : u₂ ∈ f
w : Set γ
hw : w ∈ h
ht₂ : image2 m₂ u₂ w ⊆ t₂
⊢ image2 m₁ (u₁ ∩ u₂) v ⊆ t₁

Tactic:
exact (image2_subset_right inter_subset_left).trans ht₁

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

Goal state before:
pp: case inr
α : Type u_1
inst✝⁴ : CommMonoid α
inst✝³ : LinearOrder α
inst✝² : IsOrderedMonoid α
inst✝¹ : MulLeftReflectLE α
inst✝ : ExistsMulOfLE α
a b c d : α
hab : a ≤ b
hcd : c ≤ d
x : α
x✝ : x ∈ Icc (a * c) (b * d)
hacx : a * c ≤ x
hxbd : x ≤ b * d
hbcx : b * c ≤ x
⊢ x ∈ Icc a b * Icc c d

Tactic:
obtain ⟨y, hy, rfl⟩ := exists_one_le_mul_of_le hbcx
    refine ⟨b, right_mem_Icc.2 hab, c * y, ⟨le_mul_of_one_le_right' hy, ?_⟩, (mul_assoc ..).symm⟩
    rwa [mul_assoc, mul_le_mul_iff_left] at hxbd

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

Goal state before:
pp: case refine_2
α : Type u_2
inst✝ : DivisionMonoid α
a b : α
h : a * b = 1
⊢ pure a * pure b = 1

Tactic:
pure_mul_pure,

Goal state after:
case refine_2
α : Type u_2
inst✝ : DivisionMonoid α
a b : α
h : a * b = 1
⊢ pure (a * b) = 1
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a c : α
h : c < 0
⊢ (fun x ↦ c * x) ⁻¹' Iic a = Ici (a / c)

Tactic:
by
  simpa only [mul_comm] using preimage_mul_const_Iic_of_neg a h

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
p : Filter ι
s : Set β
F : ι → α → β
f : α → β
g : β → γ
hF : ∀ (i : ι) (x : α), F i x ∈ s
hf : ∀ (x : α), f x ∈ s
hg : UniformContinuousOn g s
h : TendstoUniformly F f p
⊢ TendstoUniformly (fun i x ↦ g (F i x)) (fun x ↦ g (f x)) p

Tactic:
rw [uniformContinuousOn_iff_restrict] at hg

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
p : Filter ι
s : Set β
F : ι → α → β
f : α → β
g : β → γ
hF : ∀ (i : ι) (x : α), F i x ∈ s
hf : ∀ (x : α), f x ∈ s
hg : UniformContinuous (s.domRestrict g)
h : TendstoUniformly F f p
⊢ TendstoUniformly (fun i x ↦ g (F i x)) (fun x ↦ g (f x)) p
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: α : Type u_2
G : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : ContinuousConstSMul G α
c : G
s : Set α
hs : Dense s
⊢ Dense (c • s)

Tactic:
by
  rw [dense_iff_closure_eq] at hs ⊢; rw [closure_smul, hs, smul_set_univ]

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

Goal state before:
pp: case right
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : Continuous[inst✝³, inst✝²] f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
hg : Continuous[inst✝¹, inst✝] g ∧ ∀ ⦃𝒰 : Ultrafilter Z⦄ ⦃y : W⦄, Tendsto g (↑𝒰) (𝓝 y) → ∃ x, g x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter (X × Z)
y : Y
w : W
hyw : Tendsto (fun n ↦ (Prod.map f g n).1) (↑𝒰) (𝓝 y) ∧ Tendsto (fun n ↦ (Prod.map f g n).2) (↑𝒰) (𝓝 w)
⊢ ∃ x, Prod.map f g x = (y, w) ∧ ↑𝒰 ≤ 𝓝 x

Tactic:
rcases hf.2 (show Tendsto f (Ultrafilter.map fst 𝒰) (𝓝 y) by simpa using! hyw.1) with
      ⟨x, hxy, hx⟩

Goal state after:
case right
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : Continuous[inst✝³, inst✝²] f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
hg : Continuous[inst✝¹, inst✝] g ∧ ∀ ⦃𝒰 : Ultrafilter Z⦄ ⦃y : W⦄, Tendsto g (↑𝒰) (𝓝 y) → ∃ x, g x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter (X × Z)
y : Y
w : W
hyw : Tendsto (fun n ↦ (Prod.map f g n).1) (↑𝒰) (𝓝 y) ∧ Tendsto (fun n ↦ (Prod.map f g n).2) (↑𝒰) (𝓝 w)
x : X
hxy : f x = y
hx : ↑(Ultrafilter.map fst 𝒰) ≤ 𝓝 x
⊢ ∃ x, Prod.map f g x = (y, w) ∧ ↑𝒰 ≤ 𝓝 x
Module: Mathlib.Topology.Algebra.Group.Quotient

Goal state before:
pp: G✝ : Type u_1
inst✝⁵ : TopologicalSpace G✝
inst✝⁴ : Group G✝
inst✝³ : SeparatelyContinuousMul G✝
N✝ : Subgroup G✝
G : Type u_2
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
N : Subgroup G
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpace N] fun p ↦ p.1 • p.2

Tactic:
← (IsOpenQuotientMap.id.prodMap isOpenQuotientMap_mk).continuous_comp_iff

Goal state after:
G✝ : Type u_1
inst✝⁵ : TopologicalSpace G✝
inst✝⁴ : Group G✝
inst✝³ : SeparatelyContinuousMul G✝
N✝ : Subgroup G✝
G : Type u_2
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
N : Subgroup G
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpace N] ((fun p ↦ p.1 • p.2) ∘ Prod.map id mk)
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
U : Set G
hU : U ∈ 𝓝 1
V : Set G
V_open : IsOpen[inst✝²] V
V_mem : 1 ∈ V
hV : V * V ⊆ U
W : Set G
W_mem : W ∈ 𝓝 1
W_closed : IsClosed[inst✝²] W
hW : W ⊆ V
⊢ (W ∩ W⁻¹)⁻¹ = W ∩ W⁻¹

Tactic:
simp [inter_comm]

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

Goal state before:
pp: G : Type w
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
H : Type u_1
inst✝⁴ : Monoid H
inst✝³ : TopologicalSpace H
inst✝² : ContinuousConstSMul H H
F : Type u_2
inst✝¹ : FunLike F G H
inst✝ : MonoidHomClass F G H
f : F
h : 𝓝 1 ≤ map (⇑f) (𝓝 1)
x : G
this : map (fun x_1 ↦ f x * x_1) (𝓝 1) = 𝓝 (f x)
⊢ map (fun x_1 ↦ f x * x_1) (𝓝 1) ≤ map (fun x_1 ↦ f (x * x_1)) (𝓝 1)

Tactic:
refine (Filter.map_mono h).trans ?_

Goal state after:
G : Type w
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
H : Type u_1
inst✝⁴ : Monoid H
inst✝³ : TopologicalSpace H
inst✝² : ContinuousConstSMul H H
F : Type u_2
inst✝¹ : FunLike F G H
inst✝ : MonoidHomClass F G H
f : F
h : 𝓝 1 ≤ map (⇑f) (𝓝 1)
x : G
this : map (fun x_1 ↦ f x * x_1) (𝓝 1) = 𝓝 (f x)
⊢ map (fun x_1 ↦ f x * x_1) (map (⇑f) (𝓝 1)) ≤ map (fun x_1 ↦ f (x * x_1)) (𝓝 1)
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
hr : 0 < r
⊢ Tendsto (fun x ↦ r * f x) l atBot ↔ Tendsto f l atBot

Tactic:
by
  simpa only [← mul_neg, ← tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr

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

Goal state before:
pp: case refine_2
G✝ : Type u_1
inst✝⁵ : Group G✝
inst✝⁴ : TopologicalSpace G✝
inst✝³ : IsTopologicalGroup G✝
G : Type u_2
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
this✝ : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : UniformSpace Gᵐᵒᵖ := IsTopologicalGroup.leftUniformSpace Gᵐᵒᵖ
⊢ UniformContinuous opEquiv.invFun

Tactic:
· simp [uniformContinuous_iff_le_comap, ← comap_op_leftUniformSpace, ← UniformSpace.comap_comap]

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

Goal state before:
pp: K : Type u_2
inst✝⁴ : DivisionRing K
inst✝³ : TopologicalSpace K
inst✝² : IsTopologicalRing K
inst✝¹ : CompactSpace K
inst✝ : T2Space K
⊢ IsOpen {0}

Tactic:
exact GroupWithZero.isOpen_singleton_zero

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 : x i₀ ≤ m
⊢ Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀)

Tactic:
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, ← inter_assoc,
      inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]

Goal state after: