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

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure[inst✝] s
hx : x ∉ s
⊢ ∃ u, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ (x : ℕ) in atTop, u x ∈ s

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure[inst✝] s
hx : x ∉ s
⊢ ∃ u, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ (x : ℕ) in atTop, u x ∈ s
Module: Mathlib.Topology.Sequences

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
x : ℕ → X
inst✝¹ : SequentialSpace X
inst✝ : T1Space X
hx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)
⊢ IsClosed[inst✝²] (range x)

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
x : ℕ → X
inst✝¹ : SequentialSpace X
inst✝ : T1Space X
hx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)
⊢ IsClosed[inst✝²] (range x)
Module: Mathlib.Topology.Sequences

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : SeqCompactSpace X
f_cont : SeqContinuous f
⊢ IsSeqCompact (Set.range f)

Tactic:
simpa using

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : SeqCompactSpace X
f_cont : SeqContinuous f
⊢ IsSeqCompact (Set.range f)
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: α : Type u_1
m : α → ℝ≥0∞
f : Filter α
h : ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a
n : ℕ
⊢ ∀ᶠ (a : α) in f, ↑n < m a

Tactic:
simpa using

Goal state after:
α : Type u_1
m : α → ℝ≥0∞
f : Filter α
h : ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a
n : ℕ
⊢ ∀ᶠ (a : α) in f, ↑n < m a
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
inst✝ : SeminormedGroup E
⊢ comap norm (𝓝 0) = 𝓝 1

Tactic:
simpa using

Goal state after:
E : Type u_4
inst✝ : SeminormedGroup E
⊢ comap norm (𝓝 0) = 𝓝 1
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
inst✝ : SeminormedGroup E
x : E
⊢ Tendsto (fun a ↦ ‖a⁻¹ * x‖) (𝓝 x) (𝓝 0)

Tactic:
simpa using

Goal state after:
E : Type u_4
inst✝ : SeminormedGroup E
x : E
⊢ Tendsto (fun a ↦ ‖a⁻¹ * x‖) (𝓝 x) (𝓝 0)
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
inst✝ : SeminormedGroup E
x : E
⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 x) (𝓝 ‖x‖)

Tactic:
simpa using

Goal state after:
E : Type u_4
inst✝ : SeminormedGroup E
x : E
⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 x) (𝓝 ‖x‖)
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
inst✝ : SeminormedGroup E
⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 1) (𝓝 0)

Tactic:
simpa using

Goal state after:
E : Type u_4
inst✝ : SeminormedGroup E
⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 1) (𝓝 0)
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
inst✝ : SeminormedGroup E
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun a ↦ ‖a‖

Tactic:
simpa using

Goal state after:
E : Type u_4
inst✝ : SeminormedGroup E
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun a ↦ ‖a‖
Module: Mathlib.Topology.Algebra.InfiniteSum.Constructions

Goal state before:
pp: case right
α : Type u_1
β : Type u_2
inst✝³ : CommMonoid α
inst✝² : TopologicalSpace α
inst✝¹ : ContinuousMul α
inst✝ : RegularSpace α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
ha : HasProd f a
hf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)
s : Set α
hs : s ∈ 𝓝 a
hsc : IsClosed[inst✝²] s
u : Finset ((b : β) × γ b)
hu : ∀ (b : Finset ((b : β) × γ b)), u ⊆ b → b ∈ (fun s ↦ ∏ b ∈ s, f b) ⁻¹' s
⊢ ∀ x ∈ Set.Ici (image Sigma.fst u), ∏ b ∈ x, g b ∈ id s

Tactic:
intro bs

Goal state after:
case right
α : Type u_1
β : Type u_2
inst✝³ : CommMonoid α
inst✝² : TopologicalSpace α
inst✝¹ : ContinuousMul α
inst✝ : RegularSpace α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
ha : HasProd f a
hf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)
s : Set α
hs : s ∈ 𝓝 a
hsc : IsClosed[inst✝²] s
u : Finset ((b : β) × γ b)
hu : ∀ (b : Finset ((b : β) × γ b)), u ⊆ b → b ∈ (fun s ↦ ∏ b ∈ s, f b) ⁻¹' s
bs : Finset β
⊢ bs ∈ Set.Ici (image Sigma.fst u) → ∏ b ∈ bs, g b ∈ id s
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: case refine_1
ι : Type u_2
κ : Type u_3
G : Type u_6
inst✝ : SeminormedGroup G
f : ι → κ → G
l : Filter ι
l' : Filter κ
hf : UniformCauchySeqOnFilter f l l'
u : Set (G × G)
hu : u ∈ 𝓤 G
ε : ℝ
hε : 0 < ε
H : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 < ε} → (a, b) ∈ u
x : (ι × ι) × κ
hx : (f x.1.1 x.2, f x.1.2 x.2) ∈ {p | dist p.1 p.2 < ε}
⊢ (1, (f x.1.1 x.2)⁻¹ * f x.1.2 x.2) ∈ {p | dist p.1 p.2 < ε}

Tactic:
simpa using

Goal state after:
case refine_1
ι : Type u_2
κ : Type u_3
G : Type u_6
inst✝ : SeminormedGroup G
f : ι → κ → G
l : Filter ι
l' : Filter κ
hf : UniformCauchySeqOnFilter f l l'
u : Set (G × G)
hu : u ∈ 𝓤 G
ε : ℝ
hε : 0 < ε
H : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 < ε} → (a, b) ∈ u
x : (ι × ι) × κ
hx : (f x.1.1 x.2, f x.1.2 x.2) ∈ {p | dist p.1 p.2 < ε}
⊢ ‖(f x.1.1 x.2)⁻¹ * f x.1.2 x.2‖ < ε
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: case refine_2
ι : Type u_2
κ : Type u_3
G : Type u_6
inst✝ : SeminormedGroup G
f : ι → κ → G
l : Filter ι
l' : Filter κ
hf : TendstoUniformlyOnFilter (fun n z ↦ (f n.1 z)⁻¹ * f n.2 z) 1 (l ×ˢ l) l'
u : Set (G × G)
hu : u ∈ 𝓤 G
ε : ℝ
hε : 0 < ε
H : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 < ε} → (a, b) ∈ u
x : (ι × ι) × κ
hx : (1 x.2, (fun n z ↦ (f n.1 z)⁻¹ * f n.2 z) x.1 x.2) ∈ {p | dist p.1 p.2 < ε}
⊢ (f x.1.1 x.2, f x.1.2 x.2) ∈ {p | dist p.1 p.2 < ε}

Tactic:
simpa using

Goal state after:
case refine_2
ι : Type u_2
κ : Type u_3
G : Type u_6
inst✝ : SeminormedGroup G
f : ι → κ → G
l : Filter ι
l' : Filter κ
hf : TendstoUniformlyOnFilter (fun n z ↦ (f n.1 z)⁻¹ * f n.2 z) 1 (l ×ˢ l) l'
u : Set (G × G)
hu : u ∈ 𝓤 G
ε : ℝ
hε : 0 < ε
H : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 < ε} → (a, b) ∈ u
x : (ι × ι) × κ
hx : (1 x.2, (fun n z ↦ (f n.1 z)⁻¹ * f n.2 z) x.1 x.2) ∈ {p | dist p.1 p.2 < ε}
⊢ ‖(f x.1.1 x.2)⁻¹ * f x.1.2 x.2‖ < ε
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: x : ℝ≥0∞
xt : x ≠ ∞
⊢ (𝓝 x).HasBasis (fun x ↦ 0 < x) fun ε ↦ Icc (x - ε) (x + ε)

Tactic:
simpa using

Goal state after:
x : ℝ≥0∞
xt : x ≠ ∞
⊢ (𝓝 x).HasBasis (fun x ↦ x ≠ 0) fun ε ↦ Icc (x - ε) (x + ε)
Module: Mathlib.Topology.Algebra.InfiniteSum.Constructions

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : CommMonoid α
inst✝² : TopologicalSpace α
inst✝¹ : ContinuousMul α
inst✝ : T3Space α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
ha : HasProd g a
hf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)
hf' : Multipliable f
⊢ HasProd f a

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : CommMonoid α
inst✝² : TopologicalSpace α
inst✝¹ : ContinuousMul α
inst✝ : T3Space α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
ha : HasProd g a
hf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)
hf' : Multipliable f
⊢ HasProd f a
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: ⊢ 𝓟 (Icc (∞ - 1) (∞ + 1)) ≤ 𝓝 ∞

Tactic:
simpa using

Goal state after:
⊢ pure ∞ ≤ 𝓝 ∞
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
inst✝ : SeminormedCommGroup E
x : E
⊢ Tendsto (fun a ↦ ‖a / x‖) (𝓝 x) (𝓝 0)

Tactic:
simpa using

Goal state after:
E : Type u_4
inst✝ : SeminormedCommGroup E
x : E
⊢ Tendsto (fun a ↦ ‖a / x‖) (𝓝 x) (𝓝 0)
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
inst✝ : SeminormedCommGroup E
a : E
s : Subgroup E
hg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖(u n)⁻¹ * a‖ < b 0
z : ℕ → E := fun n ↦ u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
n : ℕ
⊢ {p | ‖p.1⁻¹ * p.2‖ < b (n + 1)} ∈ 𝓤 E

Tactic:
simpa using

Goal state after:
E : Type u_4
inst✝ : SeminormedCommGroup E
a : E
s : Subgroup E
hg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖(u n)⁻¹ * a‖ < b 0
z : ℕ → E := fun n ↦ u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
n : ℕ
⊢ {p | dist p.1 p.2 < b (n + 1)} ∈ 𝓤 E
Module: Mathlib.Topology.Algebra.InfiniteSum.Constructions

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : CommGroup α
inst✝¹ : UniformSpace α
inst✝ : IsUniformGroup α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
hf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)
hg : HasProd g a
h : CauchySeq fun s ↦ ∏ i ∈ s, f i
u : Set α
hu : u ∈ 𝓝 a
s : Finset ((b : β) × γ b)
v : Set α
vu : v ⊆ u
v_open : IsOpen[inst✝¹.toTopologicalSpace] v
hv : a ∈ v
t0 : Finset β
st0 : ∏ i ∈ t0, g i ∈ v
ht0 : image Sigma.fst s ⊆ t0
L : Tendsto (fun t ↦ ∏ p ∈ t with p.fst ∈ t0, f p) atTop (𝓝 (∏ b ∈ t0, g b))
t : Finset ((b : β) × γ b)
tv : ∏ p ∈ t with p.fst ∈ t0, f p ∈ v
st : s ⊆ t
x : (b : β) × γ b
hx : x ∈ s
⊢ x ∈ {p ∈ t | p.fst ∈ t0}

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : CommGroup α
inst✝¹ : UniformSpace α
inst✝ : IsUniformGroup α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
hf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)
hg : HasProd g a
h : CauchySeq fun s ↦ ∏ i ∈ s, f i
u : Set α
hu : u ∈ 𝓝 a
s : Finset ((b : β) × γ b)
v : Set α
vu : v ⊆ u
v_open : IsOpen[inst✝¹.toTopologicalSpace] v
hv : a ∈ v
t0 : Finset β
st0 : ∏ i ∈ t0, g i ∈ v
ht0 : image Sigma.fst s ⊆ t0
L : Tendsto (fun t ↦ ∏ p ∈ t with p.fst ∈ t0, f p) atTop (𝓝 (∏ b ∈ t0, g b))
t : Finset ((b : β) × γ b)
tv : ∏ p ∈ t with p.fst ∈ t0, f p ∈ v
st : s ⊆ t
x : (b : β) × γ b
hx : x ∈ s
⊢ x.fst ∈ t0
Module: Mathlib.Topology.Algebra.InfiniteSum.Constructions

Goal state before:
pp: α : Type u_1
β : Type u_2
L : SummationFilter β
inst✝³ : AddCommMonoid α
inst✝² : TopologicalSpace α
inst✝¹ : StarAddMonoid α
inst✝ : ContinuousStar α
f : β → α
hf : Summable (fun b ↦ Star.star (f b)) L
⊢ Summable f L

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
L : SummationFilter β
inst✝³ : AddCommMonoid α
inst✝² : TopologicalSpace α
inst✝¹ : StarAddMonoid α
inst✝ : ContinuousStar α
f : β → α
hf : Summable (fun b ↦ Star.star (f b)) L
⊢ Summable f L
Module: Mathlib.Topology.UniformSpace.AbstractCompletion

Goal state before:
pp: α : Type uα
inst✝² : UniformSpace α
pkg : AbstractCompletion.{vα, uα} α
β : Type uβ
inst✝¹ : UniformSpace β
f : α → β
inst✝ : CompleteSpace β
h : IsUniformInducing f
⊢ IsUniformInducing (pkg.extend f)

Tactic:
rw [extend_def _ h.uniformContinuous]
  exact pkg.isDenseInducing.isUniformInducing_extend pkg.isUniformInducing h

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

Goal state before:
pp: α : Type uα
inst✝³ : UniformSpace α
pkg : AbstractCompletion.{vα, uα} α
β : Type uβ
inst✝² : UniformSpace β
f : α → β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf : UniformContinuous f
g : pkg.space → β
hg : UniformContinuous g
h : ∀ (a : α), f a = g (pkg.coe a)
⊢ ∀ (a : α), pkg.extend f (pkg.coe a) = g (pkg.coe a)

Tactic:
simpa using

Goal state after:
α : Type uα
inst✝³ : UniformSpace α
pkg : AbstractCompletion.{vα, uα} α
β : Type uβ
inst✝² : UniformSpace β
f : α → β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf : UniformContinuous f
g : pkg.space → β
hg : UniformContinuous g
h : ∀ (a : α), f a = g (pkg.coe a)
⊢ ∀ (a : α), f a = g (pkg.coe a)
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: b : ℝ≥0
x✝ : ∞ ≠ ∞ ∨ ↑b ≠ ∞
x : ℝ≥0
y : ℝ≥0∞ × ℝ≥0∞
hy : ↑(b + 1 + x) < y.1 ∧ y.2 ≤ ↑(b + 1)
⊢ y.2 + ↑x < y.1

Tactic:
hy.2

Goal state after:
b : ℝ≥0
x✝ : ∞ ≠ ∞ ∨ ↑b ≠ ∞
x : ℝ≥0
y : ℝ≥0∞ × ℝ≥0∞
hy : ↑(b + 1 + x) < y.1 ∧ y.2 ≤ ↑(b + 1)
⊢ ↑(b + 1) + ↑x < y.1
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: case coe.top
a✝ b : ℝ≥0∞
ht : ∀ (b : ℝ≥0∞), b ≠ 0 → Tendsto (fun p ↦ p.1 * p.2) (𝓝 (∞, b)) (𝓝 ∞)
a : ℝ≥0
hb : ∞ ≠ 0 ∨ ↑a ≠ ∞
ha : ¬↑a = 0
⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 (↑a, ∞)) (𝓝 (↑a * ∞))

Tactic:
simpa using

Goal state after:
case coe.top
a✝ b : ℝ≥0∞
ht : ∀ (b : ℝ≥0∞), b ≠ 0 → Tendsto (fun p ↦ p.1 * p.2) (𝓝 (∞, b)) (𝓝 ∞)
a : ℝ≥0
hb : ∞ ≠ 0 ∨ ↑a ≠ ∞
ha : ¬↑a = 0
⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 (↑a, ∞)) (𝓝 ∞)
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: α : Type u_1
f : Filter α
m : α → ℝ≥0∞
a b : ℝ≥0∞
hm : Tendsto m f (𝓝 a)
ha : a ≠ 0 ∨ b ≠ ∞
⊢ Tendsto (fun x ↦ m x * b) f (𝓝 (a * b))

Tactic:
simpa using

Goal state after:
α : Type u_1
f : Filter α
m : α → ℝ≥0∞
a b : ℝ≥0∞
hm : Tendsto m f (𝓝 a)
ha : a ≠ 0 ∨ b ≠ ∞
⊢ Tendsto (fun x ↦ b * m x) f (𝓝 (a * b))
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: n : ℕ
⊢ Continuous fun x ↦ x ^ Int.negSucc n

Tactic:
simpa using

Goal state after:
n : ℕ
⊢ Continuous fun x ↦ (x ^ (n + 1))⁻¹
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
j : E →* F
b : F
hb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖(v 0)⁻¹ * b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
⊢ Tendsto (fun n ↦ ∏ i ∈ range (n + 1), j (g i)) atTop (𝓝 b)

Tactic:
simpa using

Goal state after:
E : Type u_4
F : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
j : E →* F
b : F
hb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖(v 0)⁻¹ * b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
⊢ Tendsto (fun n ↦ ∏ i ∈ range (n + 1), j (g i)) atTop (𝓝 b)
Module: Mathlib.Topology.UniformSpace.Completion

Goal state before:
pp: α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : T0Space α
f g : CauchyFilter α
⊢ (↑f).lim = (↑g).lim ↔ Inseparable f g

Tactic:
← inseparable_iff_eq,

Goal state after:
α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : T0Space α
f g : CauchyFilter α
⊢ Inseparable (↑f).lim (↑g).lim ↔ Inseparable f g
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
j : E →* F
b : F
hb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖(v 0)⁻¹ * b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
⊢ ‖(j (g 0))⁻¹ * b‖ < f 0

Tactic:
simpa using

Goal state after:
E : Type u_4
F : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
j : E →* F
b : F
hb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖(v 0)⁻¹ * b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
⊢ ‖(v 0)⁻¹ * b‖ < f 0
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
j : E →* F
b : F
hb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖(v 0)⁻¹ * b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
n : ℕ
hn : 0 < n
⊢ ‖j (g n)‖ < f n

Tactic:
simpa using

Goal state after:
E : Type u_4
F : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
j : E →* F
b : F
hb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖(v 0)⁻¹ * b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
n : ℕ
hn : 0 < n
⊢ ‖v n‖ < f n
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
inst✝ : NormedGroup E
a _x : E
hx : _x ∈ {a}ᶜ
⊢ _x⁻¹ * a ≠ 1

Tactic:
simpa using

Goal state after:
E : Type u_4
inst✝ : NormedGroup E
a _x : E
hx : _x ∈ {a}ᶜ
⊢ ¬_x = a
Module: Mathlib.Topology.Algebra.UniformMulAction

Goal state before:
pp: R : Type u
M : Type v
N : Type w
X : Type x
Y : Type y
inst✝⁵ : UniformSpace X
inst✝⁴ : UniformSpace Y
inst✝³ : SMul M X
inst✝² : SMul Mᵐᵒᵖ X
inst✝¹ : IsCentralScalar M X
inst✝ : UniformContinuousConstSMul M X
c : M
⊢ UniformContinuous fun x ↦ MulOpposite.op c • x

Tactic:
simpa using

Goal state after:
R : Type u
M : Type v
N : Type w
X : Type x
Y : Type y
inst✝⁵ : UniformSpace X
inst✝⁴ : UniformSpace Y
inst✝³ : SMul M X
inst✝² : SMul Mᵐᵒᵖ X
inst✝¹ : IsCentralScalar M X
inst✝ : UniformContinuousConstSMul M X
c : M
⊢ UniformContinuous fun x ↦ c • x
Module: Mathlib.Topology.Algebra.UniformMulAction

Goal state before:
pp: R : Type u_3
β : Type u_4
inst✝³ : DivisionRing R
inst✝² : UniformSpace R
inst✝¹ : UniformContinuousConstSMul Rᵐᵒᵖ R
inst✝ : UniformSpace β
f : β → R
hf : UniformContinuous f
a : R
⊢ UniformContinuous fun x ↦ f x / a

Tactic:
simpa using

Goal state after:
R : Type u_3
β : Type u_4
inst✝³ : DivisionRing R
inst✝² : UniformSpace R
inst✝¹ : UniformContinuousConstSMul Rᵐᵒᵖ R
inst✝ : UniformSpace β
f : β → R
hf : UniformContinuous f
a : R
⊢ UniformContinuous fun x ↦ f x * a⁻¹
Module: Mathlib.Topology.Algebra.UniformMulAction

Goal state before:
pp: M : Type v
X : Type x
inst✝³ : UniformSpace X
inst✝² : Monoid M
inst✝¹ : MulAction M X
inst✝ : UniformContinuousConstSMul M X
c : M
hc : IsUnit c
d : M
hcd : c * d = 1
cU : c • 𝓤 X ≤ 𝓤 X
dU : d • 𝓤 X ≤ 𝓤 X
⊢ 𝓤 X ≤ c • 𝓤 X

Tactic:
simpa using

Goal state after:
M : Type v
X : Type x
inst✝³ : UniformSpace X
inst✝² : Monoid M
inst✝¹ : MulAction M X
inst✝ : UniformContinuousConstSMul M X
c : M
hc : IsUnit c
d : M
hcd : c * d = 1
cU : c • 𝓤 X ≤ 𝓤 X
dU : d • 𝓤 X ≤ 𝓤 X
⊢ 𝓤 X ≤ c • 𝓤 X
Module: Mathlib.Topology.Metrizable.Uniformity

Goal state before:
pp: case neg
X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → SetRel X X
hU_symm : ∀ (n : ℕ), (U n).IsSymm
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
x y : X
h : ¬∃ n, (x, y) ∉ U n
⊢ 0 = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i

Tactic:
simpa using

Goal state after:
case neg
X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → SetRel X X
hU_symm : ∀ (n : ℕ), (U n).IsSymm
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
x y : X
h : ¬∃ n, (x, y) ∉ U n
⊢ ∀ (i : ℕ), (x, y) ∈ U i
Module: Mathlib.Topology.Metrizable.Uniformity

Goal state before:
pp: X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → SetRel X X
hU_symm : ∀ (n : ℕ), (U n).IsSymm
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹

Tactic:
have hd₀ : ∀ {x y}, d x y = 0 ↔ Inseparable x y := by
    intro x y
    refine Iff.trans ?_ hB.inseparable_iff_uniformity.symm
    simp only [d, true_imp_iff]
    split_ifs with h
    · simp [h, pow_eq_zero_iff']
    · simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff] using h

Goal state after:
X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → SetRel X X
hU_symm : ∀ (n : ℕ), (U n).IsSymm
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
hd₀ : ∀ {x y : X}, d x y = 0 ↔ Inseparable x y
⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹
Module: Mathlib.Topology.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
h : ∀ (x y : α), dist (f x) (f y) ≤ dist x y
⊢ ∀ (x y : α), dist (f x) (f y) ≤ ↑1 * dist x y

Tactic:
simpa using

Goal state after:
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
h : ∀ (x y : α), dist (f x) (f y) ≤ dist x y
⊢ ∀ (x y : α), dist (f x) (f y) ≤ dist x y
Module: Mathlib.Topology.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
f : α → ℝ
K : ℝ≥0
h : ∀ (x y : α), f x ≤ f y + ↑K * dist x y
⊢ LipschitzWith K f

Tactic:
simpa using

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
f : α → ℝ
K : ℝ≥0
h : ∀ (x y : α), f x ≤ f y + ↑K * dist x y
⊢ LipschitzWith K f
Module: Mathlib.Topology.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : α → ℝ
Kf : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
⊢ LipschitzWith Kf fun x ↦ max (f x) a

Tactic:
simpa using

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
f : α → ℝ
Kf : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
⊢ LipschitzWith Kf fun x ↦ max (f x) a
Module: Mathlib.Topology.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : α → ℝ
Kf : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
⊢ LipschitzWith Kf fun x ↦ max a (f x)

Tactic:
simpa using

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
f : α → ℝ
Kf : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
⊢ LipschitzWith Kf fun x ↦ max a (f x)
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: ι : Type u_4
f : Filter ι
u : ι → ℝ≥0∞
inst✝ : f.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ∞
le_b : ∀ᶠ (i : ι) in f, u i ≤ b
liminf_le : liminf u f ≤ b
aux : ∀ᶠ (i : ι) in f, (u i).toReal = b.truncateToReal (u i)
aux' : (liminf u f).toReal = b.truncateToReal (liminf u f)
⊢ ∀ᶠ (x : ℝ≥0∞) in map u f, (fun x1 x2 ↦ x1 ≤ x2) x b

Tactic:
simpa using

Goal state after:
ι : Type u_4
f : Filter ι
u : ι → ℝ≥0∞
inst✝ : f.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ∞
le_b : ∀ᶠ (i : ι) in f, u i ≤ b
liminf_le : liminf u f ≤ b
aux : ∀ᶠ (i : ι) in f, (u i).toReal = b.truncateToReal (u i)
aux' : (liminf u f).toReal = b.truncateToReal (liminf u f)
⊢ ∀ᶠ (a : ι) in f, u a ≤ b
Module: Mathlib.Topology.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : α → ℝ
Kf : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
⊢ LipschitzWith Kf fun x ↦ min (f x) a

Tactic:
simpa using

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
f : α → ℝ
Kf : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
⊢ LipschitzWith Kf fun x ↦ min (f x) a
Module: Mathlib.Topology.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : α → ℝ
Kf : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
⊢ LipschitzWith Kf fun x ↦ min a (f x)

Tactic:
simpa using

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
f : α → ℝ
Kf : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
⊢ LipschitzWith Kf fun x ↦ min a (f x)
Module: Mathlib.Topology.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
s : Set α
f : α → β
h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y
⊢ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ ↑1 * dist x y

Tactic:
simpa using

Goal state after:
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
s : Set α
f : α → β
h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y
⊢ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y
Module: Mathlib.Topology.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
f : α → ℝ
K : ℝ≥0
h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + ↑K * dist x y
⊢ LipschitzOnWith K f s

Tactic:
simpa using

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
f : α → ℝ
K : ℝ≥0
h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + ↑K * dist x y
⊢ LipschitzOnWith K f s