Documentation

Mathlib.Algebra.Order.Monoid.Prod

Products of ordered monoids #

instance Prod.instOrderedAddCommMonoid {α : Type u_1} {β : Type u_2} [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] :

OrderedAddCommMonoid (α × β)

Equations

Prod.instOrderedAddCommMonoid = OrderedAddCommMonoid.mk ⋯

instance Prod.instOrderedCommMonoid {α : Type u_1} {β : Type u_2} [OrderedCommMonoid α] [OrderedCommMonoid β] :

OrderedCommMonoid (α × β)

Equations

Prod.instOrderedCommMonoid = OrderedCommMonoid.mk ⋯

instance Prod.instOrderedAddCancelCommMonoid {α : Type u_1} {β : Type u_2} [OrderedCancelAddCommMonoid α] [OrderedCancelAddCommMonoid β] :

OrderedCancelAddCommMonoid (α × β)

Equations

Prod.instOrderedAddCancelCommMonoid = OrderedCancelAddCommMonoid.mk ⋯

instance Prod.instOrderedCancelCommMonoid {α : Type u_1} {β : Type u_2} [OrderedCancelCommMonoid α] [OrderedCancelCommMonoid β] :

OrderedCancelCommMonoid (α × β)

Equations

Prod.instOrderedCancelCommMonoid = OrderedCancelCommMonoid.mk ⋯

instance Prod.instExistsAddOfLE {α : Type u_1} {β : Type u_2} [LE α] [LE β] [Add α] [Add β] [ExistsAddOfLE α] [ExistsAddOfLE β] :

ExistsAddOfLE (α × β)

Equations

⋯ = ⋯

instance Prod.instExistsMulOfLE {α : Type u_1} {β : Type u_2} [LE α] [LE β] [Mul α] [Mul β] [ExistsMulOfLE α] [ExistsMulOfLE β] :

ExistsMulOfLE (α × β)

Equations

⋯ = ⋯

instance Prod.instCanonicallyOrderedAddCommMonoid {α : Type u_1} {β : Type u_2} [CanonicallyOrderedAddCommMonoid α] [CanonicallyOrderedAddCommMonoid β] :

CanonicallyOrderedAddCommMonoid (α × β)

Equations

Prod.instCanonicallyOrderedAddCommMonoid = CanonicallyOrderedAddCommMonoid.mk ⋯ ⋯

instance Prod.instCanonicallyOrderedCommMonoid {α : Type u_1} {β : Type u_2} [CanonicallyOrderedCommMonoid α] [CanonicallyOrderedCommMonoid β] :

CanonicallyOrderedCommMonoid (α × β)

Equations

Prod.instCanonicallyOrderedCommMonoid = CanonicallyOrderedCommMonoid.mk ⋯ ⋯

instance Prod.Lex.orderedAddCommMonoid {α : Type u_1} {β : Type u_2} [OrderedAddCommMonoid α] [AddLeftStrictMono α] [OrderedAddCommMonoid β] :

OrderedAddCommMonoid (Lex (α × β))

Equations

Prod.Lex.orderedAddCommMonoid = OrderedAddCommMonoid.mk ⋯

instance Prod.Lex.orderedCommMonoid {α : Type u_1} {β : Type u_2} [OrderedCommMonoid α] [MulLeftStrictMono α] [OrderedCommMonoid β] :

OrderedCommMonoid (Lex (α × β))

Equations

Prod.Lex.orderedCommMonoid = OrderedCommMonoid.mk ⋯

instance Prod.Lex.orderedAddCancelCommMonoid {α : Type u_1} {β : Type u_2} [OrderedCancelAddCommMonoid α] [OrderedCancelAddCommMonoid β] :

OrderedCancelAddCommMonoid (Lex (α × β))

Equations

Prod.Lex.orderedAddCancelCommMonoid = OrderedCancelAddCommMonoid.mk ⋯

instance Prod.Lex.orderedCancelCommMonoid {α : Type u_1} {β : Type u_2} [OrderedCancelCommMonoid α] [OrderedCancelCommMonoid β] :

OrderedCancelCommMonoid (Lex (α × β))

Equations

Prod.Lex.orderedCancelCommMonoid = OrderedCancelCommMonoid.mk ⋯

instance Prod.Lex.linearOrderedAddCancelCommMonoid {α : Type u_1} {β : Type u_2} [LinearOrderedCancelAddCommMonoid α] [LinearOrderedCancelAddCommMonoid β] :

LinearOrderedCancelAddCommMonoid (Lex (α × β))

Equations

Prod.Lex.linearOrderedAddCancelCommMonoid = LinearOrderedCancelAddCommMonoid.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯

instance Prod.Lex.linearOrderedCancelCommMonoid {α : Type u_1} {β : Type u_2} [LinearOrderedCancelCommMonoid α] [LinearOrderedCancelCommMonoid β] :

LinearOrderedCancelCommMonoid (Lex (α × β))

Equations

Prod.Lex.linearOrderedCancelCommMonoid = LinearOrderedCancelCommMonoid.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯