Ordered monoid structures on the order dual.

instance OrderDual.contravariantClass_add_le {α : Type u} [LE α] [Add α] [c : ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : ContravariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• ⋯ = ⋯

instance OrderDual.contravariantClass_mul_le {α : Type u} [LE α] [Mul α] [c : ContravariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : ContravariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• ⋯ = ⋯

instance OrderDual.covariantClass_add_le {α : Type u} [LE α] [Add α] [c : CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : CovariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• ⋯ = ⋯

instance OrderDual.covariantClass_mul_le {α : Type u} [LE α] [Mul α] [c : CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : CovariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• ⋯ = ⋯

instance OrderDual.contravariantClass_swap_add_le {α : Type u} [LE α] [Add α] [c : ContravariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• ⋯ = ⋯

instance OrderDual.contravariantClass_swap_mul_le {α : Type u} [LE α] [Mul α] [c : ContravariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• ⋯ = ⋯

instance OrderDual.covariantClass_swap_add_le {α : Type u} [LE α] [Add α] [c : CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• ⋯ = ⋯

instance OrderDual.covariantClass_swap_mul_le {α : Type u} [LE α] [Mul α] [c : CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• ⋯ = ⋯

instance OrderDual.contravariantClass_add_lt {α : Type u} [LT α] [Add α] [c : ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] : ContravariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• ⋯ = ⋯

instance OrderDual.contravariantClass_mul_lt {α : Type u} [LT α] [Mul α] [c : ContravariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] : ContravariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• ⋯ = ⋯

instance OrderDual.covariantClass_add_lt {α : Type u} [LT α] [Add α] [c : CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] : CovariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• ⋯ = ⋯

instance OrderDual.covariantClass_mul_lt {α : Type u} [LT α] [Mul α] [c : CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] : CovariantClass αᵒᵈ αᵒᵈ (fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• ⋯ = ⋯

instance OrderDual.contravariantClass_swap_add_lt {α : Type u} [LT α] [Add α] [c : ContravariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] : ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• ⋯ = ⋯

instance OrderDual.contravariantClass_swap_mul_lt {α : Type u} [LT α] [Mul α] [c : ContravariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] : ContravariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• ⋯ = ⋯

instance OrderDual.covariantClass_swap_add_lt {α : Type u} [LT α] [Add α] [c : CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] : CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x + x_1) fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• ⋯ = ⋯

instance OrderDual.covariantClass_swap_mul_lt {α : Type u} [LT α] [Mul α] [c : CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] : CovariantClass αᵒᵈ αᵒᵈ (Function.swap fun (x x_1 : αᵒᵈ) => x * x_1) fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• ⋯ = ⋯

instance OrderDual.orderedAddCommMonoid {α : Type u} [OrderedAddCommMonoid α] : OrderedAddCommMonoid αᵒᵈ

Equations

• OrderDual.orderedAddCommMonoid = OrderedAddCommMonoid.mk ⋯

theorem OrderDual.orderedAddCommMonoid.proof_1 {α : Type u_1} [OrderedAddCommMonoid α] : ∀ (x x_1 : αᵒᵈ), x ≤ x_1 → ∀ (c : αᵒᵈ), c + x ≤ c + x_1

instance OrderDual.orderedCommMonoid {α : Type u} [OrderedCommMonoid α] : OrderedCommMonoid αᵒᵈ

Equations

• OrderDual.orderedCommMonoid = OrderedCommMonoid.mk ⋯

instance OrderDual.OrderedCancelAddCommMonoid.to_contravariantClass {α : Type u} [OrderedCancelAddCommMonoid α] : ContravariantClass αᵒᵈ αᵒᵈ HAdd.hAdd LE.le

Equations

• ⋯ = ⋯

instance OrderDual.OrderedCancelCommMonoid.to_contravariantClass {α : Type u} [OrderedCancelCommMonoid α] : ContravariantClass αᵒᵈ αᵒᵈ HMul.hMul LE.le

Equations

• ⋯ = ⋯

theorem OrderDual.orderedAddCancelCommMonoid.proof_1 {α : Type u_1} [OrderedCancelAddCommMonoid α] : ∀ (x x_1 x_2 : α), x + x_1 ≤ x + x_2 → x_1 ≤ x_2

instance OrderDual.orderedAddCancelCommMonoid {α : Type u} [OrderedCancelAddCommMonoid α] : OrderedCancelAddCommMonoid αᵒᵈ

Equations

• OrderDual.orderedAddCancelCommMonoid = OrderedCancelAddCommMonoid.mk ⋯

instance OrderDual.orderedCancelCommMonoid {α : Type u} [OrderedCancelCommMonoid α] : OrderedCancelCommMonoid αᵒᵈ

Equations

• OrderDual.orderedCancelCommMonoid = OrderedCancelCommMonoid.mk ⋯

theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_5 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : max a b = if a ≤ b then b else a

theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_4 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : min a b = if a ≤ b then a else b

instance OrderDual.linearOrderedAddCancelCommMonoid {α : Type u} [LinearOrderedCancelAddCommMonoid α] : LinearOrderedCancelAddCommMonoid αᵒᵈ

Equations

• One or more equations did not get rendered due to their size.

theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_1 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : a ≤ b → ∀ (c : αᵒᵈ), c + a ≤ c + b

theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_6 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : compare a b = compareOfLessAndEq a b

theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_3 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : a ≤ b ∨ b ≤ a

theorem OrderDual.linearOrderedAddCancelCommMonoid.proof_2 {α : Type u_1} [LinearOrderedCancelAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) (c : αᵒᵈ) : a + b ≤ a + c → b ≤ c

instance OrderDual.linearOrderedCancelCommMonoid {α : Type u} [LinearOrderedCancelCommMonoid α] : LinearOrderedCancelCommMonoid αᵒᵈ

Equations

• One or more equations did not get rendered due to their size.

theorem OrderDual.linearOrderedAddCommMonoid.proof_4 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : max a b = if a ≤ b then b else a

theorem OrderDual.linearOrderedAddCommMonoid.proof_1 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : a ≤ b → ∀ (c : αᵒᵈ), c + a ≤ c + b

theorem OrderDual.linearOrderedAddCommMonoid.proof_5 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : compare a b = compareOfLessAndEq a b

theorem OrderDual.linearOrderedAddCommMonoid.proof_2 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : a ≤ b ∨ b ≤ a

theorem OrderDual.linearOrderedAddCommMonoid.proof_3 {α : Type u_1} [LinearOrderedAddCommMonoid α] (a : αᵒᵈ) (b : αᵒᵈ) : min a b = if a ≤ b then a else b

instance OrderDual.linearOrderedAddCommMonoid {α : Type u} [LinearOrderedAddCommMonoid α] : LinearOrderedAddCommMonoid αᵒᵈ

Equations

• One or more equations did not get rendered due to their size.

instance OrderDual.linearOrderedCommMonoid {α : Type u} [LinearOrderedCommMonoid α] : LinearOrderedCommMonoid αᵒᵈ

Equations

• One or more equations did not get rendered due to their size.