Covariant instances on WithZero

Adding a zero to a type with a preorder and multiplication which satisfies some axiom, gives us a new type which satisfies some variant of the axiom.

Example

If α satisfies b₁ < b₂ → a * b₁ < a * b₂ for all a, then WithZero α satisfies b₁ < b₂ → a * b₁ < a * b₂ for all a > 0, which is PosMulStrictMono (WithZero α).

Application

The type ℤₘ₀ := WithZero (Multiplicative ℤ) is used a lot in mathlib's valuation theory. These instances enable lemmas such as mul_pos to fire on ℤₘ₀.

instance instPosMulStrictMonoWithZeroOfMulLeftStrictMono {α : Type u_1} [Mul α] [Preorder α] [MulLeftStrictMono α] : PosMulStrictMono (WithZero α)

Equations

• ⋯ = ⋯

instance instMulPosStrictMonoWithZeroOfMulRightStrictMono {α : Type u_1} [Mul α] [Preorder α] [MulRightStrictMono α] : MulPosStrictMono (WithZero α)

Equations

• ⋯ = ⋯

instance instPosMulMonoWithZeroOfMulLeftMono {α : Type u_1} [Mul α] [Preorder α] [MulLeftMono α] : PosMulMono (WithZero α)

Equations

• ⋯ = ⋯

instance instMulPosMonoWithZeroOfMulRightMono {α : Type u_1} [Mul α] [Preorder α] [MulRightMono α] : MulPosMono (WithZero α)

Equations

• ⋯ = ⋯