File: Mathlib/Analysis/Normed/Order/Basic.lean

full_name: NormedOrderedAddGroup.dist_eq
code: /-- A `NormedOrderedAddGroup` is an additive group that is both a `NormedAddCommGroup` and an
`OrderedAddCommGroup`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. -/
class NormedOrderedAddGroup (α : Type*) extends OrderedAddCommGroup α, Norm α, MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
goal: dist x y = ‖x - y‖

full_name: NormedOrderedAddGroup
code: /-- A `NormedOrderedAddGroup` is an additive group that is both a `NormedAddCommGroup` and an
`OrderedAddCommGroup`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. -/
class NormedOrderedAddGroup (α : Type*) extends OrderedAddCommGroup α, Norm α, MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
goal: Type u_2

full_name: NormedOrderedGroup
code: /-- A `NormedOrderedGroup` is a group that is both a `NormedCommGroup` and an
`OrderedCommGroup`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. -/
@[to_additive]
class NormedOrderedGroup (α : Type*) extends OrderedCommGroup α, Norm α, MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
goal: Type u_2

full_name: NormedOrderedGroup.dist_eq
code: /-- A `NormedOrderedGroup` is a group that is both a `NormedCommGroup` and an
`OrderedCommGroup`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. -/
@[to_additive]
class NormedOrderedGroup (α : Type*) extends OrderedCommGroup α, Norm α, MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
goal: dist x y = ‖x / y‖

full_name: NormedLinearOrderedAddGroup.dist_eq
code: /-- A `NormedLinearOrderedAddGroup` is an additive group that is both a `NormedAddCommGroup`
and a `LinearOrderedAddCommGroup`. This class is necessary to avoid diamonds caused by both
classes carrying their own group structure. -/
class NormedLinearOrderedAddGroup (α : Type*) extends LinearOrderedAddCommGroup α, Norm α,
  MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
goal: dist x y = ‖x - y‖

full_name: NormedLinearOrderedAddGroup
code: /-- A `NormedLinearOrderedAddGroup` is an additive group that is both a `NormedAddCommGroup`
and a `LinearOrderedAddCommGroup`. This class is necessary to avoid diamonds caused by both
classes carrying their own group structure. -/
class NormedLinearOrderedAddGroup (α : Type*) extends LinearOrderedAddCommGroup α, Norm α,
  MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
goal: Type u_2

full_name: NormedLinearOrderedGroup.dist_eq
code: /-- A `NormedLinearOrderedGroup` is a group that is both a `NormedCommGroup` and a
`LinearOrderedCommGroup`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. -/
@[to_additive]
class NormedLinearOrderedGroup (α : Type*) extends LinearOrderedCommGroup α, Norm α,
  MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
goal: dist x y = ‖x / y‖

full_name: NormedLinearOrderedGroup
code: /-- A `NormedLinearOrderedGroup` is a group that is both a `NormedCommGroup` and a
`LinearOrderedCommGroup`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. -/
@[to_additive]
class NormedLinearOrderedGroup (α : Type*) extends LinearOrderedCommGroup α, Norm α,
  MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
goal: Type u_2

full_name: NormedLinearOrderedField
code: /-- A `NormedLinearOrderedField` is a field that is both a `NormedField` and a
    `LinearOrderedField`. This class is necessary to avoid diamonds. -/
class NormedLinearOrderedField (α : Type*) extends LinearOrderedField α, Norm α,
  MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ x y : α, ‖x * y‖ = ‖x‖ * ‖y‖
goal: Type u_2

full_name: NormedLinearOrderedField.norm_mul'
code: /-- A `NormedLinearOrderedField` is a field that is both a `NormedField` and a
    `LinearOrderedField`. This class is necessary to avoid diamonds. -/
class NormedLinearOrderedField (α : Type*) extends LinearOrderedField α, Norm α,
  MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ x y : α, ‖x * y‖ = ‖x‖ * ‖y‖
goal: ‖x * y‖ = ‖x‖ * ‖y‖

full_name: NormedLinearOrderedField.dist_eq
code: /-- A `NormedLinearOrderedField` is a field that is both a `NormedField` and a
    `LinearOrderedField`. This class is necessary to avoid diamonds. -/
class NormedLinearOrderedField (α : Type*) extends LinearOrderedField α, Norm α,
  MetricSpace α where
  /-- The distance function is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ x y : α, ‖x * y‖ = ‖x‖ * ‖y‖
goal: dist x y = ‖x - y‖

full_name: NormedOrderedAddGroup.toNormedAddCommGroup
code: @[to_additive]
instance (priority := 100) NormedOrderedGroup.toNormedCommGroup [NormedOrderedGroup α] :
    NormedCommGroup α 
goal: NormedAddCommGroup α

full_name: NormedOrderedGroup.toNormedCommGroup
code: @[to_additive]
instance (priority := 100) NormedOrderedGroup.toNormedCommGroup [NormedOrderedGroup α] :
    NormedCommGroup α 
goal: NormedCommGroup α

full_name: NormedLinearOrderedAddGroup.toNormedOrderedAddGroup
code: @[to_additive]
instance (priority := 100) NormedLinearOrderedGroup.toNormedOrderedGroup
    [NormedLinearOrderedGroup α] : NormedOrderedGroup α 
goal: NormedOrderedAddGroup α

full_name: NormedLinearOrderedGroup.toNormedOrderedGroup
code: @[to_additive]
instance (priority := 100) NormedLinearOrderedGroup.toNormedOrderedGroup
    [NormedLinearOrderedGroup α] : NormedOrderedGroup α 
goal: NormedOrderedGroup α

full_name: NormedLinearOrderedField.toNormedField
code: instance (priority := 100) NormedLinearOrderedField.toNormedField (α : Type*)
    [NormedLinearOrderedField α] : NormedField α 
goal: NormedField α

full_name: Rat.normedLinearOrderedField
code: instance Rat.normedLinearOrderedField : NormedLinearOrderedField ℚ 
goal: NormedLinearOrderedField ℚ

full_name: Real.normedLinearOrderedField
code: noncomputable instance Real.normedLinearOrderedField : NormedLinearOrderedField ℝ 
goal: NormedLinearOrderedField ℝ

full_name: OrderDual.normedOrderedGroup
code: @[to_additive]
instance OrderDual.normedOrderedGroup [NormedOrderedGroup α] : NormedOrderedGroup αᵒᵈ 
goal: NormedOrderedGroup αᵒᵈ

full_name: OrderDual.normedOrderedAddGroup
code: @[to_additive]
instance OrderDual.normedOrderedGroup [NormedOrderedGroup α] : NormedOrderedGroup αᵒᵈ 
goal: NormedOrderedAddGroup αᵒᵈ

full_name: OrderDual.normedLinearOrderedGroup
code: @[to_additive]
instance OrderDual.normedLinearOrderedGroup [NormedLinearOrderedGroup α] :
    NormedLinearOrderedGroup αᵒᵈ 
goal: NormedLinearOrderedGroup αᵒᵈ

full_name: OrderDual.normedLinearOrderedAddGroup
code: @[to_additive]
instance OrderDual.normedLinearOrderedGroup [NormedLinearOrderedGroup α] :
    NormedLinearOrderedGroup αᵒᵈ 
goal: NormedLinearOrderedAddGroup αᵒᵈ

full_name: Additive.normedOrderedAddGroup
code: instance Additive.normedOrderedAddGroup [NormedOrderedGroup α] :
    NormedOrderedAddGroup (Additive α) 
goal: NormedOrderedAddGroup (Additive α)

full_name: Multiplicative.normedOrderedGroup
code: instance Multiplicative.normedOrderedGroup [NormedOrderedAddGroup α] :
    NormedOrderedGroup (Multiplicative α) 
goal: NormedOrderedGroup (Multiplicative α)

full_name: Additive.normedLinearOrderedAddGroup
code: instance Additive.normedLinearOrderedAddGroup [NormedLinearOrderedGroup α] :
    NormedLinearOrderedAddGroup (Additive α) 
goal: NormedLinearOrderedAddGroup (Additive α)

full_name: Multiplicative.normedlinearOrderedGroup
code: instance Multiplicative.normedlinearOrderedGroup [NormedLinearOrderedAddGroup α] :
    NormedLinearOrderedGroup (Multiplicative α) 
goal: NormedLinearOrderedGroup (Multiplicative α)