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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_3
m : MetricSpace 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.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
hJ_ne_zero : J ≠ 0
⊢ {v |
      ¬↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
            ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
          0} ⊆
    {v | v.asIdeal ∣ J} ∪ {v | v.asIdeal ∣ Ideal.span {a}}

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: ι : Sort u_1
κ : Sort u_2
f : ι → ℝ≥0∞
a : ℝ≥0∞
g : κ → ℝ≥0∞
hf : ∃ i, f i ≠ ∞
hg : ∃ j, g j ≠ ∞
ha : ∀ (i : ι) (j : κ), a ≤ f i * g j
this✝ : Nonempty { a // f a ≠ ∞ }
this : Nonempty κ
⊢ ⨅ j, g j ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
γ : Type w
inst✝ : MetricSpace γ
⊢ 𝓤 PUnit.{u + 1} = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | 0 < ε}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: E : Type u_8
inst✝ : NNNorm E
x : E
r : ℝ≥0
⊢ ↑r ≤ ‖x‖ₑ ↔ r ≤ ‖x‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: E : Type u_8
inst✝ : NNNorm E
x : E
r : ℝ≥0
⊢ ‖x‖ₑ ≤ ↑r ↔ ‖x‖₊ ≤ r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: E : Type u_8
inst✝ : NNNorm E
x : E
r : ℝ≥0
⊢ ↑r < ‖x‖ₑ ↔ r < ‖x‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: E : Type u_8
inst✝ : NNNorm E
x : E
r : ℝ≥0
⊢ ‖x‖ₑ < ↑r ↔ ‖x‖₊ < r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: E : Type u_8
inst✝ : NNNorm E
x : E
⊢ ‖x‖ₑ ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: E : Type u_8
inst✝ : NNNorm E
x : E
⊢ ‖x‖ₑ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝² : Norm E
inst✝¹ : Group E
inst✝ : PseudoMetricSpace E
h₁ : ∀ (x : E), ‖x‖ = dist 1 x
h₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)
x y : E
⊢ dist x y = ‖x⁻¹ * y‖

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite,
    ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
        ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
      0

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: ι : Sort u_1
κ : Sort u_2
f : ι → ℝ≥0∞
a : ℝ≥0∞
g : κ → ℝ≥0∞
hf : ∃ i, f i ≠ ∞
hg : ∃ j, g j ≠ ∞
ha : ∀ (i : ι) (j : κ), a ≤ f i * g j
⊢ a ≤ (⨅ i, f i) * ⨅ j, g j

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝² : Norm E
inst✝¹ : Group E
inst✝ : PseudoMetricSpace E
h₁ : ∀ (x : E), ‖x‖ = dist 1 x
h₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y
x y : E
⊢ dist x y = ‖x⁻¹ * y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝ : Group E
f : GroupSeminorm E
x : E
⊢ f (x⁻¹ * x) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: ι : Sort u_1
f g : ι → ℝ≥0∞
hf : ∃ i, f i ≠ ∞
hg : ∃ j, g j ≠ ∞
h : ∀ (i j : ι), ∃ k, f k * g k ≤ f i * g j
⊢ (⨅ i, f i) * ⨅ i, g i = ⨅ i, f i * g i

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: ι : Sort u_1
R : Type u_3
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
f : ι → ℝ≥0∞
c : R
⊢ c • ⨆ i, f i = ⨆ i, c • f i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝ : Group E
f : GroupSeminorm E
x y : E
⊢ f (x⁻¹ * y) = f (y⁻¹ * x)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : FractionalIdeal R⁰ K
⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, count K v I = 0

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: R : Type u_3
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
s : Set ℝ≥0∞
c : R
⊢ c • sSup s = ⨆ a ∈ s, c • a

Tactic:
by

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

Goal state before:
pp: α : Type u_3
m m' : PseudoMetricSpace α
h : m.toDist = m'.toDist
⊢ m = m'

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: x : ℝ
hx : 0 < x
⊢ x.toNNReal ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝ : Group E
f : GroupSeminorm E
x y z : E
⊢ f (x⁻¹ * z) ≤ f (x⁻¹ * y) + f (y⁻¹ * z)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝ : Group E
f : GroupSeminorm E
x y z : E
⊢ f (x⁻¹ * z) ≤ f (x⁻¹ * y) + f (y⁻¹ * z)

Tactic:
by convert! map_mul_le_add f (x⁻¹ * y) (y⁻¹ * z) using 2; group

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: x : ℝ
hx : 0 < x
⊢ ENNReal.ofReal x⁻¹ = (ENNReal.ofReal x)⁻¹

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 x y ≤ dist z x + dist z y

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: x : ℝ
⊢ ENNReal.ofReal x⁻¹ ≤ (ENNReal.ofReal x)⁻¹

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 x y ≤ dist x z + dist y z

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: x y : ℝ
hy : 0 ≤ y
⊢ ENNReal.ofReal (x / y) ≤ ENNReal.ofReal x / ENNReal.ofReal y

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x₁ y₁ x₂ y₂ : α
⊢ dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂)

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: x y : ℝ
hy : 0 < y
⊢ ENNReal.ofReal (x / y) = ENNReal.ofReal x / ENNReal.ofReal y

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x₁ y₁ x₂ y₂ : α
⊢ dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case coe.inr
a : ℝ≥0
ha : a ≠ 0
⊢ a⁻¹ = (↑a).toNNReal⁻¹

Tactic:
toNNReal_coe

Goal state after:
case coe.inr
a : ℝ≥0
ha : a ≠ 0
⊢ a⁻¹ = a⁻¹
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ a⁻¹.toNNReal = a.toNNReal⁻¹

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
a b c d e f g h : α
⊢ dist a h ≤ dist a b + dist b c + dist c d + dist d e + dist e f + dist f g + dist g h

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
⊢ (a / b).toNNReal = a.toNNReal / b.toNNReal

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
⊢ Function.swap dist = dist

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ a⁻¹.toReal = a.toReal⁻¹

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 : α
⊢ 0 ≤ dist x y

Tactic:
by

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
⊢ (a / b).toReal = a.toReal / b.toReal

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
⊢ edist x y = ↑(nndist x y)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
⊢ nndist x y = (edist x y).toNNReal

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
c : ℝ≥0
⊢ edist x y < ↑c ↔ nndist x y < c

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
c : ℝ≥0
⊢ edist x y ≤ ↑c ↔ nndist x y ≤ c

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
r : ℝ
⊢ edist x y < ENNReal.ofReal r ↔ dist x y < r

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
r : ℝ
hr : 0 ≤ r
⊢ edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
⊢ nndist x y = (dist x y).toNNReal

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
⊢ dist x y = (edist x y).toReal

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
⊢ y ∈ ball x ε ↔ 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
x : α
ε : ℝ
h : 0 < ε
⊢ x ∈ 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
⊢ ball x ε = ∅ ↔ ε ≤ 0

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
⊢ ball x 0 = ∅

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
h : x ∈ ball y ε
⊢ ∃ ε' < ε, x ∈ ball y ε'

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
ε : ℝ
x : α
⊢ UniformSpace.ball x {p | dist p.1 p.2 < 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 y : α
ε : ℝ
⊢ y ∈ closedBall x ε ↔ 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
x y : α
ε : ℝ
⊢ y ∈ sphere x ε ↔ 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
x y : α
ε : ℝ
h : y ∈ sphere x ε
hε : ε ≠ 0
⊢ x ∉ sphere x 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✝² : PseudoMetricSpace α
x y z : α
δ ε ε₁ ε₂ : ℝ
s : Set α
inst✝¹ : Subsingleton α
inst✝ : NeZero ε
⊢ IsEmpty ↑(sphere x ε)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : PseudoMetricSpace α
x : α
ε : ℝ
inst✝ : Subsingleton α
h : 0 ≤ ε
⊢ closedBall x ε = {x}

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝¹ : PseudoMetricSpace α
x : α
ε : ℝ
inst✝ : Subsingleton α
h : 0 < ε
⊢ ball x ε = {x}

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
⊢ x ∈ 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
⊢ closedBall x ε = ∅ ↔ ε < 0

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
δ ε : ℝ
h : δ + ε ≤ dist x y
⊢ ε + δ ≤ dist y x

Tactic:
rwa [add_comm, dist_comm]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
δ ε : ℝ
h : δ + ε ≤ dist x y
⊢ ε + δ ≤ dist y x

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
⊢ sphere x ε ∪ ball x 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
⊢ closedBall x ε \ sphere 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
⊢ closedBall x ε \ ball x ε = sphere x ε

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
⊢ x ∈ ball y ε ↔ y ∈ 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 y : α
ε : ℝ
⊢ x ∈ closedBall y ε ↔ y ∈ 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 y : α
ε : ℝ
⊢ x ∈ sphere y ε ↔ y ∈ sphere 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
⊢ closedBall x ε = ⋂ δ, ⋂ (_ : δ > ε), ball x δ

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε₁ ε₂ : ℝ
h : ε₁ + dist x y ≤ ε₂
z : α
hz : z ∈ ball x ε₁
⊢ dist z x + dist x y < ε₁ + dist x y

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε₁ ε₂ : ℝ
h : ε₁ + dist x y ≤ ε₂
z : α
hz : z ∈ closedBall x ε₁
⊢ dist z x + dist x y ≤ ε₁ + dist x y

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε₁ ε₂ : ℝ
h : ε₁ + dist x y < ε₂
z : α
hz : z ∈ closedBall x ε₁
⊢ dist z x + dist x y ≤ ε₁ + dist x y

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε₁ ε₂ : ℝ
h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty
⊢ dist x y < ε₁ + ε₂

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
⊢ ⋃ n, s ∩ closedBall x ↑n = s

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε₁ ε₂ : ℝ
h : dist x y ≤ ε₂ - ε₁
z : α
zx : z ∈ ball x ε₁
⊢ z ∈ ball y ε₂

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
y : α
h : y ∈ ball x (ε / 2)
⊢ dist y x ≤ ε - ε / 2

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
h : y ∈ ball x ε
⊢ dist y x ≤ ε - (ε - dist y x)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
p : α → Prop
x : α
H : ∃ᶠ (R : ℝ) in atTop, ∀ y ∈ closedBall x R, p y
y : α
⊢ p y

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
p : α → Prop
x : α
H : ∃ᶠ (R : ℝ) in atTop, ∀ y ∈ ball x R, p y
y : α
⊢ p y

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
⊢ Bornology.IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
⊢ BoundedSpace α ↔ ∃ C, ∀ (a b : α), dist a b ≤ C

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
⊢ Bornology.IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → nndist x y ≤ C

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
⊢ BoundedSpace α ↔ ∃ C, ∀ (a b : α), nndist a b ≤ C

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
⊢ BoundedSpace α ↔ ∃ C, ∀ (a b : α), edist a b ≤ ↑C

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
⊢ (𝓤 α).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {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
inst✝ : PseudoMetricSpace α
β : Type u_3
p : β → Prop
f : β → ℝ
hf₀ : ∀ (i : β), p i → 0 < f i
hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε
⊢ (𝓤 α).HasBasis p fun i ↦ {p | dist p.1 p.2 < f i}

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
β : Type u_3
p : β → Prop
f : β → ℝ
hf₀ : ∀ (x : β), p x → 0 < f x
hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε
⊢ (𝓤 α).HasBasis p fun x ↦ {p | dist p.1 p.2 ≤ f x}

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
_ε : ℝ
ε0 : 0 < _ε
n : ℕ
hn : 1 / (↑n + 1) < _ε
⊢ 1 / ↑(n + 1) ≤ _ε

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 : Filter ι
x₀ : α
p : α × ι → Prop
⊢ (∀ᶠ (x : α × ι) in 𝓝 x₀ ×ˢ f, p x) ↔
    ∃ ε > 0, ∃ pa, (∀ᶠ (i : ι) in f, pa i) ∧ ∀ ⦃x : α⦄, dist x x₀ < ε → ∀ ⦃i : ι⦄, pa i → p (x, i)

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 : Filter ι
x₀ : α
p : ι × α → Prop
⊢ (∀ᶠ (x : ι × α) in f ×ˢ 𝓝 x₀, p x) ↔
    ∃ pa, (∀ᶠ (i : ι) in f, pa i) ∧ ∃ ε > 0, ∀ ⦃i : ι⦄, pa i → ∀ ⦃x : α⦄, dist x x₀ < ε → p (i, x)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : PseudoMetricSpace β
t : Set β
f : α → β
a : α
b : β
⊢ (∀ (ib : ℝ), 0 < ib → ∃ ia, 0 < ia ∧ ∀ x ∈ ball a ia ∩ s, f x ∈ ball b ib ∩ t) ↔
    ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : PseudoMetricSpace β
f : α → β
a : α
b : β
⊢ Tendsto f (𝓝[s] a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → dist (f x) b < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
a : α
⊢ ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, dist x a < δ → dist (f x) (f a) < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
a : α
s : Set α
⊢ ContinuousWithinAt f s a ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → dist (f x) (f a) < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
⊢ ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : TopologicalSpace β
f : β → α
b : β
⊢ ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ (x : β) in 𝓝 b, dist (f x) (f b) < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : TopologicalSpace β
f : β → α
b : β
s : Set β
⊢ ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ (x : β) in 𝓝[s] b, dist (f x) (f b) < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : TopologicalSpace β
f : β → α
s : Set β
⊢ ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ (x : β) in 𝓝[s] b, dist (f x) (f b) < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
u : β → α
a : α
⊢ (∀ (ib : ℝ), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Ici ia, u x ∈ ball a ib) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : PseudoMetricSpace α
inst✝² : Nonempty β
inst✝¹ : SemilatticeSup β
inst✝ : NoMaxOrder β
u : β → α
a : α
⊢ (∀ (ib : ℝ), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Ioi ia, u x ∈ ball a ib) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε

Tactic:
by

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

Goal state before:
pp: α : Type u_3
inst✝ : PseudoMetricSpace α
x : α
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x} ↔ ∃ ε > 0, ∀ (y : α), dist y x < ε → y = x

Tactic:
by

Goal state after: