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

Goal state before:
pp: case hunion.inl
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
t t' : Set β
v : Set α
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ t, (f p x, f q x) ∈ u
v' : Set α
v'_mem : v' ∈ 𝓝[s] q
hv' : ∀ p ∈ v', ∀ x ∈ t', (f p x, f q x) ∈ u
p : α
hp : p ∈ v ∩ v'
x : β
h'x : x ∈ t
⊢ (f p x, f q x) ∈ u

pp: case hunion.inr
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
t t' : Set β
v : Set α
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ t, (f p x, f q x) ∈ u
v' : Set α
v'_mem : v' ∈ 𝓝[s] q
hv' : ∀ p ∈ v', ∀ x ∈ t', (f p x, f q x) ∈ u
p : α
hp : p ∈ v ∩ v'
x : β
h'x : x ∈ t'
⊢ (f p x, f q x) ∈ u

Tactic:
· exact hv p hp.1 x h'x

Goal state after:
case hunion.inr
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
t t' : Set β
v : Set α
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ t, (f p x, f q x) ∈ u
v' : Set α
v'_mem : v' ∈ 𝓝[s] q
hv' : ∀ p ∈ v', ∀ x ∈ t', (f p x, f q x) ∈ u
p : α
hp : p ∈ v ∩ v'
x : β
h'x : x ∈ t'
⊢ (f p x, f q x) ∈ u
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
S : Set X
x₀ : X
⊢ EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt Subtype.val S x₀

Tactic:
by
  simp only [EquicontinuousWithinAt, forall_subtype_range_iff]

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

Goal state before:
pp: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
m✝ : α → β → γ
f✝ f₁ f₂ : Filter α
g✝ g₁ g₂ : Filter β
h : Filter γ
s : Set α
t : Set β
u : Set γ
a : α
b : β
m : α → β → γ
f : Filter α
g : Filter β
x✝ : Set γ
⊢ x✝ ∈ {s | ∃ u ∈ f, ∃ v ∈ g, image2 m u v ⊆ s} ↔ x✝ ∈ map (uncurry m) (f ×ˢ g)

Tactic:
simp only [mem_map, mem_prod_iff, image2_subset_iff, prod_subset_iff]; rfl

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_8
uα : UniformSpace α
uβ : UniformSpace β
p : κ → Prop
s : κ → Set (β × β)
F : ι → β → α
hβ : (𝓤 β).HasBasis p s
⊢ UniformEquicontinuous F ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ (x y : β), (x, y) ∈ s k → ∀ (i : ι), (F i x, F i y) ∈ U

Tactic:
rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
    hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
  simp only [Prod.forall]
  rfl

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

Goal state before:
pp: case pos
M : Type u_1
X : Type u_2
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace X
inst✝³ : Group M
inst✝² : IsTopologicalGroup M
inst✝¹ : MulAction M X
inst✝ : DiscreteTopology X
h : ∀ (x : X), IsOpen[inst✝⁵] ↑(MulAction.stabilizer M x)
y x : X
U : Set M := {m' | m' • y = x}
hU' : U ≠ ∅
m : M
hm : m • y = x
⊢ IsOpen[inst✝⁵] U

Tactic:
convert! (h x).preimage (by fun_prop : Continuous fun m' : M ↦ m' * m⁻¹)

Goal state after:
M : Type u_1
X : Type u_2
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace X
inst✝³ : Group M
inst✝² : IsTopologicalGroup M
inst✝¹ : MulAction M X
inst✝ : DiscreteTopology X
h : ∀ (x : X), IsOpen[inst✝⁵] ↑(MulAction.stabilizer M x)
y x : X
U : Set M := {m' | m' • y = x}
hU' : U ≠ ∅
m : M
hm : m • y = x
⊢ U = (fun m' ↦ m' * m⁻¹) ⁻¹' ↑(MulAction.stabilizer M x)
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
⊢ univ • s = univ

Tactic:
by
  refine Set.eq_univ_of_forall fun x ↦ ?_
  have : Tendsto (· • x) (𝓝 (0 : G₀)) (𝓝 0) :=
    zero_smul G₀ x ▸ tendsto_id.smul tendsto_const_nhds
  rcases Filter.nonempty_of_mem (inter_mem_nhdsWithin {0}ᶜ <| mem_map.1 <| this hs)
    with ⟨c, hc₀, hc⟩
  simp only [mem_compl_iff, mem_singleton_iff] at hc₀
  simp only [mem_smul, mem_univ, true_and]
  exact ⟨c⁻¹, c • x, hc, inv_smul_smul₀ hc₀ _⟩

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

Goal state before:
pp: case inl.inr
G₀ : Type u_4
inst✝⁵ : GroupWithZero G₀
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
s : Set E
hs : s.Nonempty
⊢ closure[inst✝²] 0 = 0

Tactic:
exact closure_singleton

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 : X
hxy : f x = y
hx : ↑(Ultrafilter.map fst 𝒰) ≤ 𝓝 x
z : Z
hzw : g z = w
hz : ↑(Ultrafilter.map snd 𝒰) ≤ 𝓝 z
⊢ Tendsto fst (↑𝒰) (𝓝 x) ∧ Tendsto snd (↑𝒰) (𝓝 z)

Tactic:
exact ⟨hx, hz⟩

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⦄, Tendsto (Prod.map f g) (↑𝒰) (𝓝 y) → ∃ x, Prod.map f g x = y ∧ ↑𝒰 ≤ 𝓝 x

Tactic:
· intro 𝒰 ⟨y, w⟩ hyw
  -- That means that `f` tends to `y` along `map fst 𝒰` and `g` tends to `w` along `map snd 𝒰`.
    simp_rw [nhds_prod_eq, tendsto_prod_iff'] at hyw
  -- Thus, by properness of `f` and `g`, we get some `x : X` and `z : Z` such that `f x = y`,
  -- `g z = w`, `map fst 𝒰` tends to  `x`, and `map snd 𝒰` tends to `y`.
    rcases hf.2 (show Tendsto f (Ultrafilter.map fst 𝒰) (𝓝 y) by simpa using! hyw.1) with
      ⟨x, hxy, hx⟩
    rcases hg.2 (show Tendsto g (Ultrafilter.map snd 𝒰) (𝓝 w) by simpa using! hyw.2) with
      ⟨z, hzw, hz⟩
  -- By the properties of the product topology, that means that `𝒰` tends to `(x, z)`,
  -- which completes the proof since `(f × g)(x, z) = (y, w)`.
    refine ⟨⟨x, z⟩, Prod.ext hxy hzw, ?_⟩
    rw [nhds_prod_eq, le_prod]
    exact ⟨hx, hz⟩

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
c a : G
⊢ map (fun x ↦ c * x) (𝓝[≠] a) = 𝓝[≠] (c * a)

Tactic:
convert! (Homeomorph.mulLeft c).isEmbedding.map_nhdsWithin_eq .. using 2
  simp

Goal state after:
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 : Subgroup G
hS : IsDiscrete ↑S
V : Set G
hV : V ∈ 𝓝 1 ∧ V ∩ ↑S = {1}
U : Set G
hU : U ∈ 𝓝 1
hUinv : U⁻¹ = U
hUV : U * U ⊆ V
g : G
hgS : g ∈ S
x : G
hx : x ∈ U
hgx : (fun x ↦ g * x) x ∈ U
⊢ g = 1

Tactic:
refine hV.2.subset ⟨hUV ?_, hgS⟩

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
S : Subgroup G
hS : IsDiscrete ↑S
V : Set G
hV : V ∈ 𝓝 1 ∧ V ∩ ↑S = {1}
U : Set G
hU : U ∈ 𝓝 1
hUinv : U⁻¹ = U
hUV : U * U ⊆ V
g : G
hgS : g ∈ S
x : G
hx : x ∈ U
hgx : (fun x ↦ g * x) x ∈ U
⊢ g ∈ U * U
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
𝓕 : Filter (α × α)
⊢ 𝓕 ≤ 𝓤 α ↔ 𝓕 ≤ comap (fun x ↦ x.2 * x.1⁻¹) (𝓝 1)

Tactic:
rw [nhds_eq_comap_uniformity, comap_comap, ← tendsto_iff_comap,
      ← (tendsto_diag_uniformity Prod.fst 𝓕).uniformity_mul_iff_left, ← tendsto_id']

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
𝓕 : Filter (α × α)
⊢ Tendsto id 𝓕 (𝓤 α) ↔ Tendsto ((Prod.mk 1 ∘ fun x ↦ x.2 * x.1⁻¹) * fun x ↦ (x.1, x.1)) 𝓕 (𝓤 α)
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:
simpa only [← mul_neg, ← tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr

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

Goal state before:
pp: α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : T1Space 𝕜
inst✝² : Field 𝕜
inst✝¹ : ContinuousInv₀ 𝕜
inst✝ : ContinuousMul 𝕜
hS : IsPreconnected S
hf : ContinuousOn f S
hg : ContinuousOn g S
hsq : EqOn (f ^ 2) (g ^ 2) S
hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0
⊢ EqOn f g S ∨ EqOn f (-g) S

Tactic:
have hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by
    simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx)), div_pow] using hsq hx
  simpa +contextual [EqOn, div_eq_iff (hg_ne _)]
    using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopologicalAddGroup γ
inst✝⁴ : TopologicalSpace δ
inst✝³ : AddCommGroup δ
inst✝² : UniformSpace G
inst✝¹ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝².toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
inst✝ : IsUniformAddGroup G
x₀ : α
y₀ : γ
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
ff : δ × δ → γ × γ := fun u ↦ (f u.1, f u.2)
lim_φ : Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, 0)) (𝓝 0)
lim_φ_sub_sub :
  Tendsto (fun p ↦ (fun p ↦ (φ p.1) p.2) (p.1.2 - p.1.1, p.2.2 - p.2.1))
    (comap ee (𝓝 (x₀, x₀)) ×ˢ comap ff (𝓝 (y₀, y₀))) (𝓝 0)
W : Set G
W_nhds : W ∈ 𝓝 0
W4 : ∀ {v w s t : G}, v ∈ W → w ∈ W → s ∈ W → t ∈ W → v + w + s + t ∈ W'
this :
  ∃ U₁ ∈ comap (⇑e) (𝓝 x₀),
    ∃ V₁ ∈ comap (⇑f) (𝓝 y₀), ∀ x ∈ U₁, ∀ x' ∈ U₁, ∀ y ∈ V₁, ∀ y' ∈ V₁, (fun p ↦ (φ p.1) p.2) (x' - x, y' - y) ∈ W
⊢ ∃ U ∈ comap (⇑e) (𝓝 x₀),
    ∃ V ∈ comap (⇑f) (𝓝 y₀),
      ∀ x ∈ U, ∀ x' ∈ U, ∀ y ∈ V, ∀ y' ∈ V, (fun p ↦ (φ p.1) p.2) (x', y') - (fun p ↦ (φ p.1) p.2) (x, y) ∈ W'

Tactic:
rcases this with ⟨U₁, U₁_nhds, V₁, V₁_nhds, H⟩

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopologicalAddGroup γ
inst✝⁴ : TopologicalSpace δ
inst✝³ : AddCommGroup δ
inst✝² : UniformSpace G
inst✝¹ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝².toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
inst✝ : IsUniformAddGroup G
x₀ : α
y₀ : γ
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
ff : δ × δ → γ × γ := fun u ↦ (f u.1, f u.2)
lim_φ : Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, 0)) (𝓝 0)
lim_φ_sub_sub :
  Tendsto (fun p ↦ (fun p ↦ (φ p.1) p.2) (p.1.2 - p.1.1, p.2.2 - p.2.1))
    (comap ee (𝓝 (x₀, x₀)) ×ˢ comap ff (𝓝 (y₀, y₀))) (𝓝 0)
W : Set G
W_nhds : W ∈ 𝓝 0
W4 : ∀ {v w s t : G}, v ∈ W → w ∈ W → s ∈ W → t ∈ W → v + w + s + t ∈ W'
U₁ : Set β
U₁_nhds : U₁ ∈ comap (⇑e) (𝓝 x₀)
V₁ : Set δ
V₁_nhds : V₁ ∈ comap (⇑f) (𝓝 y₀)
H : ∀ x ∈ U₁, ∀ x' ∈ U₁, ∀ y ∈ V₁, ∀ y' ∈ V₁, (fun p ↦ (φ p.1) p.2) (x' - x, y' - y) ∈ W
⊢ ∃ U ∈ comap (⇑e) (𝓝 x₀),
    ∃ V ∈ comap (⇑f) (𝓝 y₀),
      ∀ x ∈ U, ∀ x' ∈ U, ∀ y ∈ V, ∀ y' ∈ V, (fun p ↦ (φ p.1) p.2) (x', y') - (fun p ↦ (φ p.1) p.2) (x, y) ∈ W'
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: case left
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)))).NeBot

Tactic:
apply NeBot.map

Goal state after:
case left
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
⊢ (comap (Prod.map ⇑e ⇑f) (𝓝 (x₀, y₀))).NeBot
Module: Mathlib.Topology.Algebra.Order.Group

Goal state before:
pp: 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

Tactic:
have (x : G) : ∀ᶠ y in 𝓝 x, y = x :=
        (eventually_mabs_div_lt _ hε).mono fun y hy ↦ mabs_div_le_one.mp <| hε_min _ hy
      filter_upwards [(this _).prod_nhds (this _)]
      simp [hε]

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : 0 < C
hf : Tendsto f l atTop
hg : Tendsto g l (𝓝 C)
⊢ Tendsto (fun x ↦ f x * g x) l atTop

Tactic:
refine tendsto_atTop_mono' _ ?_ (hf.atTop_mul_const (half_pos hC))

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : 0 < C
hf : Tendsto f l atTop
hg : Tendsto g l (𝓝 C)
⊢ (fun x ↦ f x * (C / 2)) ≤ᶠ[l] fun x ↦ f x * g x
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hf : ∀ᶠ (x : α) in l, |f x| ≤ C
hg : Tendsto g l (𝓝 0)
⊢ 0 = |0|

Tactic:
abs_zero

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

Goal state before:
pp: case refine_1.ofNat
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
c d : 𝕜
hc : c ≠ 0
n : ℕ
h : Tendsto (fun x ↦ c * x ^ Int.ofNat n) atTop (𝓝 d)
⊢ Int.ofNat n = 0 ∧ c = d

Tactic:
simpa [tendsto_const_mul_pow_nhds_iff hc] using h

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0

Tactic:
by
  refine ⟨fun h => ?_, fun h => ?_⟩
  · cases n with
    | ofNat n =>
      left
      simpa [tendsto_const_mul_pow_nhds_iff hc] using h
    | negSucc n =>
      have hn := Int.negSucc_lt_zero n
      exact Or.inr ⟨hn, tendsto_nhds_unique h (tendsto_const_mul_zpow_atTop_zero hn)⟩
  · rcases h with h | h
    · simp only [h.left, h.right, zpow_zero, mul_one]
      exact tendsto_const_nhds
    · exact h.2.symm ▸ tendsto_const_mul_zpow_atTop_zero h.1

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
hx : ¬x ≡ a [PMOD p]
⊢ toIcoDiv hp a =ᶠ[𝓝[<] x ⊔ 𝓝[≥] x] fun x_1 ↦ toIcoDiv hp a x

Tactic:
Filter.EventuallyEq,

Goal state after:
𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
hx : ¬x ≡ a [PMOD p]
⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[<] x ⊔ 𝓝[≥] x, toIcoDiv hp a x_1 = toIcoDiv hp a x
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
hx : ¬x ≡ a [PMOD p]
⊢ toIocDiv hp a =ᶠ[𝓝[≤] x ⊔ 𝓝[>] x] fun x_1 ↦ toIocDiv hp a x

Tactic:
Filter.EventuallyEq,

Goal state after:
𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
hx : ¬x ≡ a [PMOD p]
⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[≤] x ⊔ 𝓝[>] x, toIocDiv hp a x_1 = toIocDiv hp a x
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✝ : Nontrivial α
s : Set α
hs : IsOpen[inst✝³] s
h : s.Nonempty
⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s

Tactic:
obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h
  obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x
  rcases lt_trichotomy x y with (H | rfl | H)
  · 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⟩
  · exact (hy rfl).elim
  · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=
      exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩
    exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp

Goal state before:
pp: A : GrpCat
⊢ CartesianMonoidalCategory.lift (𝟙 (MonTypeEquivalenceMon.inverse.obj ((forget₂ GrpCat MonCat).obj A)).X)
        (↾fun x ↦ x⁻¹) ≫
      μ =
    SemiCartesianMonoidalCategory.toUnit (MonTypeEquivalenceMon.inverse.obj ((forget₂ GrpCat MonCat).obj A)).X ≫ η

Tactic:
ext x
            exact mul_inv_cancel (G := A) x

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
⊢ ⇑(equivIccQuot p a) ∘ Quotient.mk'' = fun x ↦ Quot.mk (EndpointIdent p a) ⟨toIocMod ⋯ a x, ⋯⟩

Tactic:
rw [equivIccQuot_comp_mk_eq_toIcoMod]

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
⊢ (fun x ↦ Quot.mk (EndpointIdent p a) ⟨toIcoMod ⋯ a x, ⋯⟩) = fun x ↦ Quot.mk (EndpointIdent p a) ⟨toIocMod ⋯ a x, ⋯⟩
Module: Mathlib.Algebra.Category.ModuleCat.Subobject

Goal state before:
pp: R : Type u
inst✝ : Ring R
M : ModuleCat R
S : Subobject M
⊢ S = (fun N ↦ mk (↟N.subtype)) ((fun S ↦ (Hom.hom S.arrow).range) S)

Tactic:
by
        fapply eq_mk_of_comm
        · apply LinearEquiv.toModuleIso
          apply LinearEquiv.ofBijective (LinearMap.codRestrict
            (LinearMap.range S.arrow.hom) S.arrow.hom _)
          constructor
          · simp [← LinearMap.ker_eq_bot, ker_eq_bot_of_mono]
          · rw [← LinearMap.range_eq_top, LinearMap.range_codRestrict,
              Submodule.comap_subtype_self]
            exact LinearMap.mem_range_self _
        · ext x
          rfl

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
A : C
x₂ x₂' : A ⟶ S.X₂
hx₂ : x₂ ≫ S.g = 0
hx₂' : x₂' ≫ S.g = 0
⊢ S.liftCycles x₂ hx₂ ≫ S.homologyπ = S.liftCycles x₂' hx₂' ≫ S.homologyπ ↔ S.liftCycles (x₂ - x₂') ⋯ ≫ S.homologyπ = 0

Tactic:
← sub_eq_zero,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
A : C
x₂ x₂' : A ⟶ S.X₂
hx₂ : x₂ ≫ S.g = 0
hx₂' : x₂' ≫ S.g = 0
⊢ S.liftCycles x₂ hx₂ ≫ S.homologyπ - S.liftCycles x₂' hx₂' ≫ S.homologyπ = 0 ↔
    S.liftCycles (x₂ - x₂') ⋯ ≫ S.homologyπ = 0
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h :
  ∀ ⦃A : C⦄ (y₂ : A ⟶ S₂.X₂),
    y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f
A : C
γ : A ⟶ S₂.homology
A₁ : C
π₁ : A₁ ⟶ A
hπ₁ : Epi π₁
y₂ : A₁ ⟶ S₂.X₂
hy₂ : y₂ ≫ S₂.g = 0
fac : π₁ ≫ γ = S₂.liftCycles y₂ hy₂ ≫ S₂.homologyπ
A₂ : C
π₂ : A₂ ⟶ A₁
hπ₂ : Epi π₂
x₂ : A₂ ⟶ S₁.X₂
hx₂ : x₂ ≫ S₁.g = 0
y₁ : A₂ ⟶ S₂.X₁
hy₁ : π₂ ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ π₂ ≫ y₂ = π ≫ x₂ ≫ φ.τ₂ + x₁ ≫ S₂.f

Tactic:
exact ⟨A₂, 𝟙 _, inferInstance, y₁, by simpa only [id_comp] using hy₁⟩

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Adjunctions

Goal state before:
pp: case single
R : Type u_1
inst✝⁸ : CommRing R
C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u
inst✝⁶ : Category.{v, u} D
inst✝⁵ : Preadditive D
inst✝⁴ : Linear R D
F G : Free R C ⥤ D
inst✝³ : F.Additive
inst✝² : Functor.Linear R F
inst✝¹ : G.Additive
inst✝ : Functor.Linear R G
α : embedding R C ⋙ F ≅ embedding R C ⋙ G
X Y : Free R C
f' : X ⟶ Y
r : R
⊢ r • F.map (single f' 1) ≫ α.hom.app Y = α.hom.app X ≫ (r • G.map (single f' 1))

Tactic:
comp_smul

Goal state after:
case single
R : Type u_1
inst✝⁸ : CommRing R
C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u
inst✝⁶ : Category.{v, u} D
inst✝⁵ : Preadditive D
inst✝⁴ : Linear R D
F G : Free R C ⥤ D
inst✝³ : F.Additive
inst✝² : Functor.Linear R F
inst✝¹ : G.Additive
inst✝ : Functor.Linear R G
α : embedding R C ⋙ F ≅ embedding R C ⋙ G
X Y : Free R C
f' : X ⟶ Y
r : R
⊢ r • F.map (single f' 1) ≫ α.hom.app Y = r • α.hom.app X ≫ G.map (single f' 1)
Module: Mathlib.CategoryTheory.Limits.Types.Pullbacks

Goal state before:
pp: X Y Z : Type u
f : X ⟶ Z
g : Y ⟶ Z
x✝ : PullbackObj f g
⊢ (hom ((pullbackIsoPullback f g).inv ≫ pullback.snd f g)).toFun x✝ = (hom (↾fun p ↦ (↑p).2)).toFun x✝

Tactic:
exact PullbackCone.IsLimit.equivPullbackObj_symm_apply_snd (pullbackIsPullback f g) _

Goal state after: