mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
hs : Dense s
x : α
ε : ℝ
hε : 0 < ε
⊢ (ball x ε).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
hs : Dense s
x : α
ε : ℝ
hε : 0 < ε
⊢ ∃ y ∈ s, dist x y < ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
⊢ PseudoMetricSpace.toUniformSpace = ⊥ ↔ ∃ r, 0 < r ∧ Pairwise fun x1 x2 ↦ r ≤ dist x1 x2

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
a : R
K : Ideal R
e : J * K = Ideal.span {a}
e' : I + K = ⊤
⊢ J * I ⊔ Ideal.span {a} = J

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u_3
inst✝ : PseudoMetricSpace α
r : ℝ
hpos : 0 < r
hr : Pairwise fun x1 x2 ↦ r ≤ dist x1 x2
⊢ PseudoMetricSpace.toUniformSpace.toTopologicalSpace = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: α : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : IsOrderedModule α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
⊢ sInf (a • s) = a • sInf s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
this : ∃ a K, J * K = Ideal.span {a} ∧ I + K = ⊤
⊢ ∃ a, J * I ⊔ Ideal.span {a} = J

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u_3
d : α → α → ℝ≥0
⊢ ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | ↑(d p.1 p.2) < ε} = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | ↑(d p.1 p.2) < ε}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
⊢ 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
⊢ ∀ (x : α), edist x x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
x✝¹ x✝ : α
⊢ edist x✝¹ x✝ = edist x✝ x✝¹

Tactic:
by

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: α : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : IsOrderedModule α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
⊢ sSup (a • s) = a • sSup s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
x y z : α
⊢ edist x z ≤ edist x y + edist y z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
ε : ℝ
⊢ eball x (ENNReal.ofReal ε) = ball x ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
ε : ℝ≥0
⊢ eball x ↑ε = ball x ↑ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
ε : ℝ
h : 0 ≤ ε
⊢ closedEBall x (ENNReal.ofReal ε) = closedBall x ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
ε : ℝ≥0
⊢ closedEBall x ↑ε = closedBall x ↑ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u_3
U : UniformSpace α
m : PseudoMetricSpace α
H : 𝓤 α = 𝓤 α
⊢ m.replaceUniformity H = m

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α✝
α : Type u_3
U : UniformSpace α
m : PseudoMetricSpace α
H : 𝓤 α = 𝓤 α
⊢ PseudoMetricSpace.toBornology = PseudoMetricSpace.toBornology

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: γ : Type u_3
U : TopologicalSpace γ
m : PseudoMetricSpace γ
H : U = toUniformSpace.toTopologicalSpace
⊢ m.replaceTopology H = m

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X✝ : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
X : Type u_3
e : PseudoEMetricSpace X
dist : X → X → ℝ
dist_nonneg : ∀ (x y : X), 0 ≤ dist x y
h : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)
x : X
⊢ dist x x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X✝ : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
X : Type u_3
e : PseudoEMetricSpace X
dist : X → X → ℝ
dist_nonneg : ∀ (x y : X), 0 ≤ dist x y
h : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)
x y : X
⊢ dist x y = dist y x

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
a : α
b : ℝ≥0
c : α
⊢ ↑(Pi.mulSingle a b c) = Pi.mulSingle a (↑b) c

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X✝ : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
X : Type u_3
e : PseudoEMetricSpace X
dist : X → X → ℝ
dist_nonneg : ∀ (x y : X), 0 ≤ dist x y
h : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)
x y z : X
⊢ dist x z ≤ dist x y + dist y z

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
a : α
b : ℝ≥0
c : α
⊢ ↑(Pi.single a b c) = Pi.single a (↑b) c

Tactic:
by

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: α : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Module α ℝ
inst✝ : IsOrderedModule α ℝ
a : α
ha : a ≤ 0
s : Set ℝ
⊢ sInf (a • s) = a • sSup s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X✝ : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
X : Type u_3
e : PseudoEMetricSpace X
dist : X → X → ℝ
dist_nonneg : ∀ (x y : X), 0 ≤ dist x y
h : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)
x✝¹ x✝ : X
⊢ edist x✝¹ x✝ = ENNReal.ofReal (dist x✝¹ x✝)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case refine_2
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset
p : HeightOneSpectrum R
hps : p ∈ s
⊢ 0 < J * ∏ x ∈ s \ {p}, x.asIdeal

Tactic:
rw [Ideal.zero_eq_bot, bot_lt_iff_ne_bot, ← Ideal.zero_eq_bot,
        mul_ne_zero_iff, Finset.prod_ne_zero_iff]

Goal state after:
case refine_2
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset
p : HeightOneSpectrum R
hps : p ∈ s
⊢ J ≠ 0 ∧ ∀ a ∈ s \ {p}, a.asIdeal ≠ 0
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset
⊢ ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Type u_2
s : Finset ι
f : ι → ℝ
hf : ∀ i ∈ s, 0 ≤ f i
x : ι
hxs : x ∈ s
⊢ f x = ↑(f x).toNNReal

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Type u_2
s : Finset ι
f : ι → ℝ
hf : ∀ i ∈ s, 0 ≤ f i
⊢ (∑ a ∈ s, f a).toNNReal = ∑ a ∈ s, (f a).toNNReal

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X✝ : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
X : Type u_3
e : PseudoEMetricSpace X
dist : X → X → ℝ
dist_nonneg : ∀ (x y : X), 0 ≤ dist x y
h : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)
⊢ ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε} = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α✝
α : Type u
inst✝ : PseudoEMetricSpace α
h : ∀ (x y : α), edist x y ≠ ∞
⊢ ∀ (x y : α), 0 ≤ (edist x y).toReal

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α✝
α : Type u
inst✝ : PseudoEMetricSpace α
h : ∀ (x y : α), edist x y ≠ ∞
⊢ ∀ (x y : α), edist x y = ENNReal.ofReal (edist x y).toReal

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Type u_2
s : Finset ι
f : ι → ℝ
hf : ∀ a ∈ s, 0 ≤ f a
x : ι
hxs : x ∈ s
⊢ f x = ↑(f x).toNNReal

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Type u_2
s : Finset ι
f : ι → ℝ
hf : ∀ a ∈ s, 0 ≤ f a
⊢ (∏ a ∈ s, f a).toNNReal = ∏ a ∈ s, (f a).toNNReal

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α✝
α : Type ?u.9
B : Bornology α
m : PseudoMetricSpace α
H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s
s : Set α
⊢ Bornology.IsBounded s ↔ sᶜ ∈ {s | ∃ C, ∀ x ∈ sᶜ, ∀ y ∈ sᶜ, dist x y ≤ C}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u_3
m : PseudoMetricSpace α
B : Bornology α
H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s
⊢ m.replaceBornology H = m

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α✝
α : Type u_3
B : Bornology α
m : PseudoMetricSpace α
H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s
⊢ PseudoMetricSpace.toUniformSpace = PseudoMetricSpace.toUniformSpace

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
ι' : Sort u_4
inst✝¹ : Nonempty ι
inst✝ : Nonempty ι'
f : ι → ℝ≥0
g : ι' → ℝ≥0
a : ℝ≥0
h : ∀ (i : ι) (j : ι'), a ≤ f i + g j
⊢ a ≤ (⨅ i, f i) + ⨅ j, g j

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
⊢ ∀ (x : ℝ), |x - x| = 0

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: α : Type u_3
f : α → ℝ≥0
s : Finset α
r : ℝ≥0
⊢ s.sup f / r = s.sup fun a ↦ f a / r

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x : ℝ
⊢ dist x 0 = |x|

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: s : Set ℕ
⊢ ∀ a ∈ upperBounds (Nat.cast '' s), ⌊a⌋₊ ∈ upperBounds s

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: s : Set ℕ
⊢ ∀ a ∈ upperBounds s, ↑a ∈ upperBounds (Nat.cast '' s)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: α : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Module α ℝ
inst✝ : IsOrderedModule α ℝ
a : α
ha : a ≤ 0
s : Set ℝ
⊢ sSup (a • s) = a • sInf s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x y : ℝ
⊢ x - y ≤ dist x y

Tactic:
by

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: ι : Sort u_1
r : ℝ
ha : 0 ≤ r
f : ι → ℝ
⊢ (⨅ i, f i) * r = ⨅ i, f i * r

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
f : ι → ℕ
⊢ BddAbove (range fun x ↦ ↑(f x)) ↔ BddAbove (range f)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: ι : Sort u_1
r : ℝ
ha : 0 ≤ r
f : ι → ℝ
⊢ (⨆ i, f i) * r = ⨆ i, f i * r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : HeightOneSpectrum R → R
ha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ p ∈ s, a p ∉ J * ∏ q ∈ s, q.asIdeal
⊢ J ∣ Ideal.span {∑ p ∈ s, a p}

Tactic:
by

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: ι : Sort u_1
r : ℝ
ha : r ≤ 0
f : ι → ℝ
⊢ (⨅ i, f i) * r = ⨆ i, f i * r

Tactic:
by

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: ι : Sort u_1
r : ℝ
ha : r ≤ 0
f : ι → ℝ
⊢ (⨆ i, f i) * r = ⨅ i, f i * r

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x r y : ℝ
⊢ y ∈ ball x r ↔ y ∈ Ioo (x - r) (x + r)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: r : ℝ
⊢ ball 0 r = Ioo (-r) r

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
f : ι → ℝ≥0
a : ℝ≥0
⊢ iInf f * a = ⨅ i, f i * a

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
f : ι → ℝ≥0
a : ℝ≥0
⊢ a * iInf f = ⨅ i, a * f i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x r : ℝ
⊢ closedBall x r = Icc (x - r) (x + r)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: r : ℝ
⊢ closedBall 0 r = Icc (-r) r

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
f : ι → ℝ≥0
a : ℝ≥0
⊢ a * ⨆ i, f i = ⨆ i, a * f i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x y : ℝ
⊢ Ioo x y = ball ((x + y) / 2) ((y - x) / 2)

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
f : ι → ℝ≥0
a : ℝ≥0
⊢ (⨆ i, f i) * a = ⨆ i, f i * a

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
f : ι → ℝ≥0
a : ℝ≥0
⊢ (⨆ i, f i) / a = ⨆ i, f i / a

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
a g : ℝ≥0
h : ι → ℝ≥0
H : ∀ (j : ι), g * h j ≤ a
⊢ g * iSup h ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
a : ℝ≥0
g : ι → ℝ≥0
h : ℝ≥0
H : ∀ (i : ι), g i * h ≤ a
⊢ iSup g * h ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x y : ℝ
⊢ Icc x y = closedBall ((x + y) / 2) ((y - x) / 2)

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
inst✝ : Nonempty ι
a g : ℝ≥0
h : ι → ℝ≥0
H : ∀ (j : ι), a ≤ g * h j
⊢ a ≤ g * iInf h

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
inst✝ : Nonempty ι
a : ℝ≥0
g : ι → ℝ≥0
h : ℝ≥0
H : ∀ (i : ι), a ≤ g i * h
⊢ a ≤ iInf g * h

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x r : ℝ
hr : 0 ≤ r
⊢ sphere x r = {x - r, x + r}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : HeightOneSpectrum R → R
ha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ p ∈ s, a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
e : p = ⊥
⊢ I = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
⊢ 𝓤 α = comap (fun p ↦ dist p.1 p.2) (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
ι : Type u_2
inst✝ : PseudoMetricSpace α
f : ι → α × α
p : Filter ι
⊢ Tendsto f p (𝓤 α) ↔ Tendsto (fun x ↦ dist (f x).1 (f x).2) p (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y z : α
⊢ dist (dist x y) (dist x z) ≤ dist y z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
a : α
⊢ comap (fun x ↦ dist x a) (𝓝 0) = 𝓝 a

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_4
f : ι → ℕ
⊢ ↑(⨆ i, f i) = ⨆ i, ↑(f i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
inst✝ : PseudoMetricSpace α
f : β → α
x : Filter β
a : α
⊢ Tendsto f x (𝓝 a) ↔ Tendsto (fun b ↦ dist (f b) a) x (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: case inr
ι : Sort u_4
f : ι → ℕ
hι : Nonempty ι
⊢ ↑(⨅ i, f i) = ⨅ i, ↑(f i)

Tactic:
apply eq_of_forall_le_iff

Goal state after:
case inr
ι : Sort u_4
f : ι → ℕ
hι : Nonempty ι
⊢ ∀ (c : ℝ≥0), c ≤ ↑(⨅ i, f i) ↔ c ≤ ⨅ i, ↑(f i)
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_4
f : ι → ℕ
⊢ ↑(⨅ i, f i) = ⨅ i, ↑(f i)

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: a b : ℝ≥0
⊢ a = b ↔ ↑a = ↑b

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: a b : ℝ≥0
⊢ a ≤ b ↔ ↑a ≤ ↑b

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: a b : ℝ≥0
⊢ a < b ↔ ↑a < ↑b

Tactic:
by

Goal state after:
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: a b : ℝ≥0
⊢ a ≠ b ↔ ↑a ≠ ↑b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝¹ : PseudoMetricSpace α
β : Type u_3
p : β → Prop
f : β → ℝ
l : Filter β
inst✝ : l.NeBot
hf₀ : ∀ (i : β), p i → 0 < f i
hf₁ : ∀ᶠ (i : β) in l, p i
hf : ∀ (i : ℝ), 0 < i → ∀ᶠ (x : β) in l, f x ∈ closedBall 0 i
ε : ℝ
hε : 0 < ε
i : β
hp : p i
hi : f i ∈ closedBall 0 ε
⊢ f i ≤ ε

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : HeightOneSpectrum R → R
ha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ p ∈ s, a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
⊢ p' ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝¹ : PseudoMetricSpace α
β : Type u_3
p : β → Prop
f : β → ℝ
l : Filter β
inst✝ : l.NeBot
hf₀ : ∀ (i : β), p i → 0 < f i
hf₁ : ∀ᶠ (i : β) in l, p i
hf : Tendsto f l (𝓝 0)
⊢ (𝓤 α).HasBasis p fun i ↦ {x | dist x.1 x.2 < f i}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
a : α
⊢ (∀ (i : ℝ), 0 < i → ∃ y ∈ s, y ∈ ball a i) ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
inst✝ : PseudoMetricSpace α
e : β → α
a : α
⊢ a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range e) ↔ ∀ ε > 0, ∃ k, dist a (e k) < ε

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
X✝ : Type u_2
Y : Type u_3
A : ι → Type u_4
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (A i)
X : Type u_5
t : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
⊢ (pseudoMetrizableSpaceUniformity X).toTopologicalSpace = t

Tactic:
by

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

Goal state before:
pp: X : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs✝ : IsSeparable s
u : Set X
hu : u.Countable
hs : s ⊆ closure[inst✝¹] u
this : Countable ↑u
⊢ closure[inst✝¹] u ⊆ closure[inst✝¹] (Subtype.val '' range (inclusion ⋯))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
inst✝ : PseudoMetricSpace α
e : β → α
a : α
⊢ (∀ (i : ℕ), True → ∃ y ∈ range e, y ∈ ball a (1 / (↑i + 1))) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
hs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
a : α
⊢ a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x : α
⊢ x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s ↔ ∀ r > 0, (ball x r ∩ s).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
⊢ Dense s ↔ ∀ r > 0, ⋃ c ∈ s, ball c r = univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
inst✝ : PseudoMetricSpace α
f : β → α
x : α
⊢ x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range f) ↔ ∀ r > 0, ∃ y, dist x (f y) < r

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
X✝ : Type u_2
Y : Type u_3
A : ι → Type u_4
inst✝⁶ : TopologicalSpace X✝
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → TopologicalSpace (A i)
X : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : LindelofSpace X
inst✝ : PseudoMetrizableSpace X
⊢ SecondCountableTopology X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Basic

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
f : ℕ → α
m n✝ n : ℕ
hle : m ≤ n
ihn : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1))
⊢ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1)) + dist (f n) (f (n + 1)) =
    ∑ i ∈ Finset.Ico m (n + 1), dist (f i) (f (i + 1))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Basic

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
f : ℕ → α
m n : ℕ
h : m ≤ n
⊢ dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
⊢ (∀ (i' : ℝ), 0 < i' → ∃ i, 0 < i ∧ Prod.map f f ⁻¹' {p | dist p.1 p.2 < i} ⊆ {p | dist p.1 p.2 < i'}) ↔
    ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ

Tactic:
by

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

Goal state before:
pp: X : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsSeparable s
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[inst✝¹] t

Tactic:
by

Goal state after: