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

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : PseudoEMetricSpace X
inst✝¹ : PseudoEMetricSpace Y
inst✝ : PseudoEMetricSpace Z
e : X ≃ᵈ Y
e' : Y ≃ᵈ Z
⊢ ratio (e.trans e') = ratio e * ratio e'

Tactic:
by

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

Goal state before:
pp: c : ℝ≥0
⊢ c • 1 = ↑c

Tactic:
by

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

Goal state before:
pp: a b c d : ℝ≥0∞
r✝ p q : ℝ≥0
A : Type u_1
inst✝¹ : Semiring A
inst✝ : Algebra ℝ≥0∞ A
r : ℝ≥0
x : A
⊢ ((algebraMap ℝ≥0∞ A).comp ofNNRealHom) r * x = x * ((algebraMap ℝ≥0∞ A).comp ofNNRealHom) r

Tactic:
by

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

Goal state before:
pp: a b c d : ℝ≥0∞
r✝ p q : ℝ≥0
A : Type u_1
inst✝¹ : Semiring A
inst✝ : Algebra ℝ≥0∞ A
r : ℝ≥0
x : A
⊢ r • x = ((algebraMap ℝ≥0∞ A).comp ofNNRealHom) r * x

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
e : X ≃ᵈ Y
⊢ ratio e.symm * ratio e = 1

Tactic:
by

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

Goal state before:
pp: R : Type u_1
r : R
s : ℝ≥0
inst✝³ : SMul R ℝ≥0
inst✝² : SMul R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ≥0∞
⊢ ↑(r • s) = r • ↑s

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝ : PseudoEMetricSpace X
e : X ≃ᵈ X
n : ℕ
⊢ ⇑(e ^ n) = (⇑e)^[n]

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : PseudoEMetricSpace X
inst✝¹ : PseudoEMetricSpace Y
inst✝ : PseudoEMetricSpace Z
e : X ≃ᵢ Y
⊢ ∀ (x y : X), edist (e.toFun x) (e.toFun y) = ↑1 * edist x y

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁵ : Semiring R
inst✝⁴ : IsDomain R
inst✝³ : Module R ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : Module.IsTorsionFree R ℝ≥0∞
inst✝ : DecidableEq R
c : R
⊢ c • ∞ = if c = 0 then 0 else ∞

Tactic:
by

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

Goal state before:
pp: x : ℝ≥0
y : ℝ≥0∞
hy : y < ∞
⊢ ↑x < ∞

Tactic:
by

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

Goal state before:
pp: x : ℝ≥0
y : ℝ≥0∞
hy : y ≠ ∞
⊢ ↑x ≠ ∞

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
e : X ≃ᵢ Y
⊢ ratio e.toDilationEquiv = 1

Tactic:
by

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

Goal state before:
pp: x : ℝ≥0
y : ℝ≥0∞
hx : x ≠ 0
⊢ x • y ≠ ∞ → y ≠ ∞

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
F : Type u_3
inst✝³ : PseudoMetricSpace X
inst✝² : PseudoMetricSpace Y
inst✝¹ : EquivLike F X Y
inst✝ : DilationEquivClass F X Y
e : F
⊢ map (⇑e) (cobounded X) = cobounded Y

Tactic:
by

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

Goal state before:
pp: x : ℝ≥0
y : ℝ≥0∞
hx : x ≠ 0
⊢ x • y < ∞ ↔ y < ∞

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: r : ℝ≥0
s : ℝ≥0∞
⊢ (r • s).toReal = r • s.toReal

Tactic:
by

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

Goal state before:
pp: a✝ b✝ c✝ d : ℝ≥0∞
r p q : ℝ≥0
a b : ℝ≥0∞
hab : a ≤ b
c : ℝ≥0
⊢ c • a ≤ c • b

Tactic:
by

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

Goal state before:
pp: a✝ b✝ c✝ d : ℝ≥0∞
r p q a b : ℝ≥0
hab : a ≤ b
c : ℝ≥0∞
⊢ a • c ≤ b • c

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
x✝¹ x✝ : PUnit.{?u.6 + 1}
⊢ ‖x✝¹ * x✝‖ ≤ ‖x✝¹‖ * ‖x✝‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
inst✝² : SeminormedAddCommGroup G
inst✝¹ : One G
inst✝ : NormOneClass G
⊢ ‖1‖ₑ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
inst✝² : SeminormedAddCommGroup G
inst✝¹ : One G
inst✝ : NormOneClass G
⊢ ‖0‖ ≠ ‖1‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝² : SeminormedAddCommGroup α
inst✝¹ : One α
inst✝ : NormOneClass α
⊢ ‖1‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝⁵ : SeminormedAddCommGroup α
inst✝⁴ : One α
inst✝³ : NormOneClass α
inst✝² : SeminormedAddCommGroup β
inst✝¹ : One β
inst✝ : NormOneClass β
⊢ ‖1‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α✝ : Type u_2
β : Type u_3
ι✝ : Type u_4
ι : Type u_5
α : ι → Type u_6
inst✝⁴ : Nonempty ι
inst✝³ : Fintype ι
inst✝² : (i : ι) → SeminormedAddCommGroup (α i)
inst✝¹ : (i : ι) → One (α i)
inst✝ : ∀ (i : ι), NormOneClass (α i)
⊢ ‖1‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: β : Type u_5
inst✝¹ : NormedRing β
inst✝ : Nontrivial β
⊢ ‖1‖ ≤ ‖1‖ * ‖1‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : NonUnitalSeminormedRing α
x y : α
⊢ ‖(AddMonoidHom.mulRight x) y‖ ≤ ‖x‖ * ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝¹ : NonUnitalSeminormedRing α
a a₁ a₂ b c : α
inst✝ : NonUnitalSeminormedRing β
x y : α × β
⊢ max (‖x.1‖ * ‖y.1‖) (‖x.2‖ * ‖y.2‖) = max (‖x.1‖ * ‖y.1‖) (‖y.2‖ * ‖x.2‖)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝¹ : NonUnitalSeminormedRing α
a a₁ a₂ b c : α
inst✝ : NonUnitalSeminormedRing β
x y : α × β
⊢ max (‖x.1‖ * ‖y.1‖) (‖y.2‖ * ‖x.2‖) ≤ max ‖x.1‖ ‖x.2‖ * max ‖y.2‖ ‖y.1‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝¹ : NonUnitalSeminormedRing α
a a₁ a₂ b c : α
inst✝ : NonUnitalSeminormedRing β
x y : α × β
⊢ max ‖x.1‖ ‖x.2‖ * max ‖y.2‖ ‖y.1‖ = max ‖x.1‖ ‖x.2‖ * max ‖y.1‖ ‖y.2‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
⊢ ‖↑0‖ ≤ ↑0 * ‖1‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
n : ℕ
⊢ ‖↑(n + 1)‖ ≤ ↑(n + 1) * ‖1‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a : α
x✝ : [a] ≠ []
⊢ ‖[a].prod‖ ≤ (map norm [a]).prod

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b : α
l : List α
x✝ : a :: b :: l ≠ []
⊢ ‖(a :: b :: l).prod‖ ≤ (map norm (a :: b :: l)).prod

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
l : List α
hl : l ≠ []
⊢ (map norm l).prod = ↑(map nnnorm l).prod

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : SeminormedRing α
inst✝ : NormOneClass α
⊢ ‖[].prod‖ ≤ (map norm []).prod

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : SeminormedRing α
inst✝ : NormOneClass α
l : List α
⊢ (map norm l).prod = ↑(map nnnorm l).prod

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: ι : Type u_4
α : Type u_5
inst✝ : NormedCommRing α
f : ι → α
val✝ : Multiset ι
l : List ι
hl : Multiset.Nodup (Quot.mk (⇑(List.isSetoid ι)) l)
hs : { val := Quot.mk (⇑(List.isSetoid ι)) l, nodup := hl }.Nonempty
⊢ List.map f l ≠ []

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: ι : Type u_4
α : Type u_5
inst✝ : NormedCommRing α
s : Finset ι
hs : s.Nonempty
f : ι → α
⊢ ‖∏ i ∈ s, f i‖ ≤ ∏ i ∈ s, ‖f i‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: ι : Type u_4
α : Type u_5
inst✝ : NormedCommRing α
s : Finset ι
hs : s.Nonempty
f : ι → α
⊢ ∏ i ∈ s, ‖f i‖ = ↑(∏ i ∈ s, ‖f i‖₊)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: ι : Type u_4
α : Type u_5
inst✝¹ : NormedCommRing α
inst✝ : NormOneClass α
s : Finset ι
f : ι → α
⊢ ‖∏ i ∈ s, f i‖ ≤ ∏ i ∈ s, ‖f i‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: ι : Type u_4
α : Type u_5
inst✝¹ : NormedCommRing α
inst✝ : NormOneClass α
s : Finset ι
f : ι → α
⊢ ∏ i ∈ s, ‖f i‖ = ↑(∏ i ∈ s, ‖f i‖₊)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
z : ℤ
⊢ ‖↑z.natAbs‖ = ‖↑z‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
z : ℤ
⊢ ‖↑z.natAbs‖₊ = ‖↑z‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
z : ℤ
⊢ ‖↑|z|‖ = ‖↑z‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
z : ℤ
⊢ ‖↑|z|‖₊ = ‖↑z‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a : α
x✝ : 0 < 1
⊢ ‖a ^ 1‖₊ ≤ ‖a‖₊ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a : α
n : ℕ
x✝ : 0 < n + 2
⊢ ‖a ^ (n + 2)‖₊ ≤ ‖a‖₊ ^ (n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : SeminormedRing α
inst✝ : NormOneClass α
a : α
n : ℕ
⊢ ‖a ^ Nat.zero‖₊ ≤ ‖a‖₊ ^ Nat.zero

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a : α
n : ℕ
h : 0 < n
⊢ ‖a ^ n‖ ≤ ‖a‖ ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : SeminormedRing α
inst✝ : NormOneClass α
a : α
n : ℕ
⊢ ‖a ^ Nat.zero‖ ≤ ‖a‖ ^ Nat.zero

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b c : α
ha : ‖a‖ ≤ 1
⊢ ‖c - a * b‖ ≤ ‖c - a‖ + ‖a * (1 - b)‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b c : α
ha : ‖a‖ ≤ 1
⊢ ‖c - a‖ + ‖a * (1 - b)‖ ≤ ‖c - a‖ + ‖a‖ * ‖1 - b‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b c : α
ha : ‖a‖ ≤ 1
⊢ ‖c - a‖ + ‖a‖ * ‖1 - b‖ ≤ ‖c - a‖ + 1 * ‖1 - b‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b c : α
ha : ‖a‖ ≤ 1
⊢ ‖c - a‖ + 1 * ‖1 - b‖ = ‖c - a‖ + ‖1 - b‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b c : α
hb : ‖b‖ ≤ 1
⊢ ‖c - a * b‖ ≤ ‖1 - a‖ + ‖c - b‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b : αˣ
⊢ ‖↑(a * b * a⁻¹ * b⁻¹) - 1‖ = ‖(↑a * ↑b - ↑b * ↑a) * ↑a⁻¹ * ↑b⁻¹‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b : αˣ
⊢ ‖↑a * ↑b - ↑b * ↑a‖ * ‖↑a⁻¹‖ * ‖↑b⁻¹‖ = ‖(↑a - 1) * (↑b - 1) - (↑b - 1) * (↑a - 1)‖ * ‖↑a⁻¹‖ * ‖↑b⁻¹‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b : αˣ
⊢ ‖(↑a - 1) * (↑b - 1) - (↑b - 1) * (↑a - 1)‖ * ‖↑a⁻¹‖ * ‖↑b⁻¹‖ ≤
    (‖(↑a - 1) * (↑b - 1)‖ + ‖(↑b - 1) * (↑a - 1)‖) * ‖↑a⁻¹‖ * ‖↑b⁻¹‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b : αˣ
⊢ (‖(↑a - 1) * (↑b - 1)‖ + ‖(↑b - 1) * (↑a - 1)‖) * ‖↑a⁻¹‖ * ‖↑b⁻¹‖ ≤
    (‖↑a - 1‖ * ‖↑b - 1‖ + ‖↑b - 1‖ * ‖↑a - 1‖) * ‖↑a⁻¹‖ * ‖↑b⁻¹‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b : αˣ
⊢ (‖↑a - 1‖ * ‖↑b - 1‖ + ‖↑b - 1‖ * ‖↑a - 1‖) * ‖↑a⁻¹‖ * ‖↑b⁻¹‖ = 2 * ‖↑a⁻¹‖ * ‖↑b⁻¹‖ * ‖↑a - 1‖ * ‖↑b - 1‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝ : SeminormedRing α
a b : αˣ
⊢ ‖↑(a * b * a⁻¹ * b⁻¹) - 1‖₊ ≤ 2 * ‖↑a⁻¹‖₊ * ‖↑b⁻¹‖₊ * ‖↑a - 1‖₊ * ‖↑b - 1‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: R : Type u_5
S : Type u_6
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
A : Subalgebra R S
f : S → ℝ
hf_pm : IsPowMul f
x : ↥A
n : ℕ
hn : 1 ≤ n
⊢ (fun x ↦ f ↑x) (x ^ n) = (fun x ↦ f ↑x) x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: x : ℝ≥0
⊢ ‖↑x‖ = ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: x : ℝ≥0
⊢ ‖↑x‖₊ = x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: x : ℝ≥0
⊢ ‖↑x‖ₑ = ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Nonempty α
inst✝² : Preorder α
inst✝¹ : IsDirectedOrder α
β : Type u_5
inst✝ : SeminormedAddCommGroup β
f : α → β
b : β
⊢ (∀ (ib : ℝ), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Set.Ici ia, f x ∈ Metric.ball b ib) ↔
    ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : α), N ≤ n → ‖f n - b‖ < ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝⁴ : Nonempty α
inst✝³ : Preorder α
inst✝² : IsDirectedOrder α
inst✝¹ : NoMaxOrder α
β : Type u_5
inst✝ : SeminormedAddCommGroup β
f : α → β
b : β
⊢ (∀ (ib : ℝ), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Set.Ioi ia, f x ∈ Metric.ball b ib) ↔
    ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : α), N < n → ‖f n - b‖ < ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝² : SeminormedAddCommGroup α
inst✝¹ : Mul α
inst✝ : NormMulClass α
a b : α
⊢ ‖a * b‖ₑ = ‖a‖ₑ * ‖b‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝² : SeminormedRing α
inst✝¹ : NormOneClass α
inst✝ : NormMulClass α
a : α
n : ℕ
⊢ ‖a ^ n‖ₑ = ‖a‖ₑ ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : NormedAddCommGroup α
inst✝² : MulOneClass α
inst✝¹ : NormMulClass α
inst✝ : Nontrivial α
⊢ ‖1‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝¹ : NormedRing α
inst✝ : NormMulClass α
a✝ b✝ : α
h : a✝ * b✝ = 0
⊢ a✝ = 0 ∨ b✝ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
R : Type u_5
inst✝ : Ring R
v : AbsoluteValue R ℝ
x : R
⊢ v (-x + x) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : SeminormedAddGroup α
inst✝² : SeminormedAddGroup β
inst✝¹ : SMulZeroClass α β
inst✝ : IsBoundedSMul α β
r : α
x : β
⊢ ‖r • x‖ ≤ ‖r‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
R : Type u_5
inst✝ : Ring R
v : AbsoluteValue R ℝ
x y : R
⊢ v (-x + y) = v (-y + x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : SeminormedAddGroup α
inst✝² : SeminormedAddGroup β
inst✝¹ : SMulZeroClass α β
inst✝ : IsBoundedSMul α β
r : α
x : β
⊢ ‖r • x‖ₑ ≤ ‖r‖ₑ * ‖x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
R : Type u_5
inst✝ : Ring R
v : AbsoluteValue R ℝ
x y z : R
⊢ v (-x + z) ≤ v (-x + y) + v (-y + z)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : SeminormedAddGroup α
inst✝² : SeminormedAddGroup β
inst✝¹ : SMulZeroClass α β
inst✝ : IsBoundedSMul α β
s : α
x y : β
⊢ dist (s • x) (s • y) ≤ ‖s‖ * dist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NonUnitalSeminormedRing α
x y₁ y₂ : α
⊢ dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NonUnitalSeminormedRing α
x₁ x₂ y : α
⊢ dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
R : Type u_5
inst✝ : Ring R
v : AbsoluteValue R ℝ
⊢ ∀ {x y : R}, v (-x + y) = 0 → x = y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NonUnitalSeminormedRing α
x : αᵐᵒᵖ
y₁ y₂ : α
⊢ dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NonUnitalSeminormedRing α
x₁ x₂ : αᵐᵒᵖ
y : α
⊢ dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : SeminormedRing α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : Module α β
h : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖
a : α
b₁ b₂ : β
⊢ dist (a • b₁) (a • b₂) ≤ dist a 0 * dist b₁ b₂

Tactic:
simpa [smul_sub, dist_eq_norm] using h a (b₁ - b₂)

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : SeminormedRing α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : Module α β
h : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖
a : α
b₁ b₂ : β
⊢ dist (a • b₁) (a • b₂) ≤ dist a 0 * dist b₁ b₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : SeminormedRing α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : Module α β
h : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖
a₁ a₂ : α
b : β
⊢ dist (a₁ • b) (a₂ • b) ≤ dist a₁ a₂ * dist b 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : SeminormedRing α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : Module α β
h : ∀ (r : α) (x : β), ‖r • x‖ₑ ≤ ‖r‖ₑ * ‖x‖ₑ
⊢ ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : SeminormedRing α
inst✝² : SeminormedAddGroup β
inst✝¹ : SMul α β
inst✝ : NormSMulClass α β
r : α
x : β
⊢ ‖r • x‖ₑ = ‖r‖ₑ * ‖x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: R : Type u_5
ι : Type u_6
ι' : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : Finite ι
inst✝³ : Finite ι'
F : Type u_8
inst✝² : FunLike F R ℝ
inst✝¹ : NonnegHomClass F R ℝ
inst✝ : MulHomClass F R ℝ
v : F
x : ι → R
y : ι' → R
⊢ ⨆ a, v (x a.1 * y a.2) = (⨆ i, v (x i)) * ⨆ j, v (y j)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β✝ : Type u_2
inst✝⁷ : SeminormedRing α
inst✝⁶ : SeminormedAddGroup β✝
inst✝⁵ : SMul α β✝
inst✝⁴ : NormSMulClass α β✝
ι : Type u_3
β : ι → Type u_4
inst✝³ : Fintype ι
inst✝² : (i : ι) → SeminormedAddGroup (β i)
inst✝¹ : (i : ι) → SMul α (β i)
inst✝ : ∀ (i : ι), NormSMulClass α (β i)
r : α
x : (i : ι) → β i
⊢ ‖r • x‖ = ‖r‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : SeminormedRing α
inst✝⁵ : SeminormedAddGroup β
inst✝⁴ : SMul α β
inst✝³ : NormSMulClass α β
γ : Type u_3
inst✝² : SeminormedAddGroup γ
inst✝¹ : SMul α γ
inst✝ : NormSMulClass α γ
r : α
x✝ : β × γ
v₁ : β
v₂ : γ
⊢ ‖r • (v₁, v₂)‖ = ‖r‖ * ‖(v₁, v₂)‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : SeminormedRing α
inst✝² : SeminormedAddCommGroup β
inst✝¹ : Module α β
inst✝ : NormSMulClass α β
s : α
x y : β
⊢ dist (s • x) (s • y) = ‖s‖ * dist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : SeminormedRing α
inst✝² : SeminormedAddCommGroup β
inst✝¹ : Module α β
inst✝ : NormSMulClass α β
s : α
x y : β
⊢ edist (s • x) (s • y) = ‖s‖₊ • edist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_5
inst✝¹ : Fintype α
ι : α → Type u
inst✝ : ∀ (a : α), Finite (ι a)
f : (a : α) → ι a → ℝ
hf₀ : ∀ (a : α) (i : ι a), 0 ≤ f a i
h : IsEmpty ((a : α) → ι a)
a : α
ha : IsEmpty (ι a)
⊢ a ∈ Finset.univ ∧ ⨆ i, f a i = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedDivisionRing α
inst✝² : SeminormedAddGroup β
inst✝¹ : MulActionWithZero α β
inst✝ : IsBoundedSMul α β
r : α
x : β
h : ¬r = 0
⊢ ‖r‖ * ‖x‖ = ‖r‖ * ‖r⁻¹ • r • x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedDivisionRing α
inst✝² : SeminormedAddGroup β
inst✝¹ : MulActionWithZero α β
inst✝ : IsBoundedSMul α β
r : α
x : β
h : ¬r = 0
⊢ ‖r‖ * ‖r⁻¹ • r • x‖ ≤ ‖r‖ * (‖r⁻¹‖ * ‖r • x‖)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_5
inst✝¹ : Fintype α
ι : α → Type u
inst✝ : ∀ (a : α), Finite (ι a)
f : (a : α) → ι a → ℝ
hf₀ : ∀ (a : α) (i : ι a), 0 ≤ f a i
h : Nonempty ((a : α) → ι a)
i : (a : α) → ι a
⊢ ∀ i_1 ∈ Finset.univ, 0 ≤ f i_1 (i i_1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedDivisionRing α
inst✝² : SeminormedAddGroup β
inst✝¹ : MulActionWithZero α β
inst✝ : IsBoundedSMul α β
r : α
x : β
h : ¬r = 0
⊢ ‖r‖ * (‖r⁻¹‖ * ‖r • x‖) = ‖r • x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedDivisionRing α
inst✝² : SeminormedAddGroup β
inst✝¹ : MulActionWithZero α β
inst✝ : IsBoundedSMul α β
r : α
x : β
⊢ ‖r • x‖ = ‖r‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Basic

Goal state before:
pp: α : Type u_5
inst✝¹ : Fintype α
ι : α → Type u
inst✝ : ∀ (a : α), Finite (ι a)
f : (a : α) → ι a → ℝ
hf₀ : ∀ (a : α) (i : ι a), 0 ≤ f a i
⊢ ⨆ i, ∏ a, f a (i a) = ∏ a, ⨆ i, f a i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedDivisionRing α
inst✝² : SeminormedAddCommGroup β
inst✝¹ : Module α β
inst✝ : NormSMulClass α β
s : α
hs : s ≠ 0
x : β
ε : ℝ
p : β
h : dist p (s • x) < ‖s‖ * ε
⊢ s • s⁻¹ • p = p

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Update

Goal state before:
pp: ι : Type u_1
π : ι → Sort u_2
x : (i : ι) → π i
inst✝ : DecidableEq ι
i : ι
y : (i_1 : ↥{i}) → π ↑i_1
⊢ updateFinset x {i} y = update x i (y ⟨i, ⋯⟩)

Tactic:
by

Goal state after: