Ordered monoid structures on Multiplicative α and Additive α.

instance instLEMultiplicative {α : Type u_1} [LE α] : LE (Multiplicative α)

Equations

• instLEMultiplicative = inst

instance instLEAdditive {α : Type u_1} [LE α] : LE (Additive α)

Equations

• instLEAdditive = inst

instance instLTMultiplicative {α : Type u_1} [LT α] : LT (Multiplicative α)

Equations

• instLTMultiplicative = inst

instance instLTAdditive {α : Type u_1} [LT α] : LT (Additive α)

Equations

• instLTAdditive = inst

instance Multiplicative.preorder {α : Type u_1} [Preorder α] : Preorder (Multiplicative α)

Equations

• Multiplicative.preorder = inst

instance Additive.preorder {α : Type u_1} [Preorder α] : Preorder (Additive α)

Equations

• Additive.preorder = inst

instance Multiplicative.partialOrder {α : Type u_1} [PartialOrder α] : PartialOrder (Multiplicative α)

Equations

• Multiplicative.partialOrder = inst

instance Additive.partialOrder {α : Type u_1} [PartialOrder α] : PartialOrder (Additive α)

Equations

• Additive.partialOrder = inst

instance Multiplicative.linearOrder {α : Type u_1} [LinearOrder α] : LinearOrder (Multiplicative α)

Equations

• Multiplicative.linearOrder = inst

instance Additive.linearOrder {α : Type u_1} [LinearOrder α] : LinearOrder (Additive α)

Equations

• Additive.linearOrder = inst

instance Multiplicative.orderBot {α : Type u_1} [LE α] [OrderBot α] : OrderBot (Multiplicative α)

Equations

• Multiplicative.orderBot = inst

instance Additive.orderBot {α : Type u_1} [LE α] [OrderBot α] : OrderBot (Additive α)

Equations

• Additive.orderBot = inst

instance Multiplicative.orderTop {α : Type u_1} [LE α] [OrderTop α] : OrderTop (Multiplicative α)

Equations

• Multiplicative.orderTop = inst

instance Additive.orderTop {α : Type u_1} [LE α] [OrderTop α] : OrderTop (Additive α)

Equations

• Additive.orderTop = inst

instance Multiplicative.boundedOrder {α : Type u_1} [LE α] [BoundedOrder α] : BoundedOrder (Multiplicative α)

Equations

• Multiplicative.boundedOrder = inst

instance Additive.boundedOrder {α : Type u_1} [LE α] [BoundedOrder α] : BoundedOrder (Additive α)

Equations

• Additive.boundedOrder = inst

instance Multiplicative.existsMulOfLe {α : Type u_1} [Add α] [LE α] [ExistsAddOfLE α] : ExistsMulOfLE (Multiplicative α)

instance Additive.existsAddOfLe {α : Type u_1} [Mul α] [LE α] [ExistsMulOfLE α] : ExistsAddOfLE (Additive α)

@[simp]

theorem Additive.ofMul_le {α : Type u_1} [Preorder α] {a b : α} : ofMul a ≤ ofMul b ↔ a ≤ b

@[simp]

theorem Additive.ofMul_lt {α : Type u_1} [Preorder α] {a b : α} : ofMul a < ofMul b ↔ a < b

@[simp]

theorem Additive.toMul_le {α : Type u_1} [Preorder α] {a b : Additive α} : toMul a ≤ toMul b ↔ a ≤ b

@[simp]

theorem Additive.toMul_lt {α : Type u_1} [Preorder α] {a b : Additive α} : toMul a < toMul b ↔ a < b

theorem Additive.toMul_mono {α : Type u_1} [Preorder α] {a b : Additive α} : a ≤ b → toMul a ≤ toMul b

Alias of the reverse direction of Additive.toMul_le.

theorem Additive.ofMul_mono {α : Type u_1} [Preorder α] {a b : α} : a ≤ b → ofMul a ≤ ofMul b

Alias of the reverse direction of Additive.ofMul_le.

theorem Additive.toMul_strictMono {α : Type u_1} [Preorder α] {a b : Additive α} : a < b → toMul a < toMul b

Alias of the reverse direction of Additive.toMul_lt.

theorem Additive.foMul_strictMono {α : Type u_1} [Preorder α] {a b : α} : a < b → ofMul a < ofMul b

Alias of the reverse direction of Additive.ofMul_lt.

@[simp]

theorem Multiplicative.ofAdd_le {α : Type u_1} [Preorder α] {a b : α} : ofAdd a ≤ ofAdd b ↔ a ≤ b

@[simp]

theorem Multiplicative.ofAdd_lt {α : Type u_1} [Preorder α] {a b : α} : ofAdd a < ofAdd b ↔ a < b

@[simp]

theorem Multiplicative.toAdd_le {α : Type u_1} [Preorder α] {a b : Multiplicative α} : toAdd a ≤ toAdd b ↔ a ≤ b

@[simp]

theorem Multiplicative.toAdd_lt {α : Type u_1} [Preorder α] {a b : Multiplicative α} : toAdd a < toAdd b ↔ a < b

theorem Multiplicative.toAdd_mono {α : Type u_1} [Preorder α] {a b : Multiplicative α} : a ≤ b → toAdd a ≤ toAdd b

Alias of the reverse direction of Multiplicative.toAdd_le.

theorem Multiplicative.ofAdd_mono {α : Type u_1} [Preorder α] {a b : α} : a ≤ b → ofAdd a ≤ ofAdd b

Alias of the reverse direction of Multiplicative.ofAdd_le.

theorem Multiplicative.toAdd_strictMono {α : Type u_1} [Preorder α] {a b : Multiplicative α} : a < b → toAdd a < toAdd b

Alias of the reverse direction of Multiplicative.toAdd_lt.

theorem Multiplicative.ofAdd_strictMono {α : Type u_1} [Preorder α] {a b : α} : a < b → ofAdd a < ofAdd b

Alias of the reverse direction of Multiplicative.ofAdd_lt.