# Ordered monoid structures on `Multiplicative α` and `Additive α`. #

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

- instLEMultiplicative = inst

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

- instLEAdditive = inst

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

- instLTMultiplicative = inst

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

- instLTAdditive = inst

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

- Multiplicative.preorder = inst

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

- Additive.preorder = inst

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

- Multiplicative.partialOrder = inst

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

- Additive.partialOrder = inst

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

- Multiplicative.linearOrder = inst

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

- Additive.linearOrder = inst

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

- Multiplicative.orderBot = inst

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

- Additive.orderBot = inst

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

- Multiplicative.orderTop = inst

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

- Additive.orderTop = inst

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

- Multiplicative.boundedOrder = inst

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

- Additive.boundedOrder = inst

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instance Multiplicative.existsMulOfLe {α : Type u_1} [Add α] [LE α] [ExistsAddOfLE α] :
ExistsMulOfLE (Multiplicative α)
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instance Additive.existsAddOfLe {α : Type u_1} [Mul α] [LE α] [ExistsMulOfLE α] :
ExistsAddOfLE (Additive α)
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@[simp]
theorem Additive.ofMul_le {α : Type u_1} [Preorder α] {a b : α} :
ofMul a ≤ ofMul b ↔ a ≤ b
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@[simp]
theorem Additive.ofMul_lt {α : Type u_1} [Preorder α] {a b : α} :
ofMul a < ofMul b ↔ a < b
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@[simp]
theorem Additive.toMul_le {α : Type u_1} [Preorder α] {a b : Additive α} :
toMul a ≤ toMul b ↔ a ≤ b
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@[simp]
theorem Additive.toMul_lt {α : Type u_1} [Preorder α] {a b : Additive α} :
toMul a < toMul b ↔ a < b
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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`.

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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`.

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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`.

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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`.

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@[simp]
theorem Multiplicative.ofAdd_le {α : Type u_1} [Preorder α] {a b : α} :
ofAdd a ≤ ofAdd b ↔ a ≤ b
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@[simp]
theorem Multiplicative.ofAdd_lt {α : Type u_1} [Preorder α] {a b : α} :
ofAdd a < ofAdd b ↔ a < b
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@[simp]
theorem Multiplicative.toAdd_le {α : Type u_1} [Preorder α] {a b : Multiplicative α} :
toAdd a ≤ toAdd b ↔ a ≤ b
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@[simp]
theorem Multiplicative.toAdd_lt {α : Type u_1} [Preorder α] {a b : Multiplicative α} :
toAdd a < toAdd b ↔ a < b
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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`.

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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`.

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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`.

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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`.