PartialOrd Documentation

Strict partial ordering relation.

This trait extends the partial equivalence relation provided by PartialEq (==) with partial_cmp(a, b) -> Option<Ordering>, which is a trichotomy of the ordering relation when its result is Some:

• if a < b then partial_cmp(a, b) == Some(Less)
• if a > b then partial_cmp(a, b) == Some(Greater)
• if a == b then partial_cmp(a, b) == Some(Equal)

and the absence of an ordering between a and b when partial_cmp(a, b) == None. Furthermore, this trait defines <= as a < b || a == b and >= as a > b || a == b.

The comparisons must satisfy, for all a, b, and c:

• transitivity: if a < b and b < c then a < c
• duality: if a < b then b > a

The lt (<), le (<=), gt (>), and ge (>=) methods are implemented in terms of partial_cmp according to these rules. The default implementations can be overridden for performance reasons, but manual implementations must satisfy the rules above.

From these rules it follows that PartialOrd must be implemented symmetrically and transitively: if T: PartialOrd<U> and U: PartialOrd<V> then U: PartialOrd<T> and T: PartialOrd<V>.

The following corollaries follow from transitivity of <, duality, and from the definition of < et al. in terms of
the same partial_cmp:

• irreflexivity of <: !(a < a)
• transitivity of >: if a > b and b > c then a > c
• asymmetry of <: if a < b then !(b < a)
• antisymmetry of <: if a < b then !(a > b)

Stronger ordering relations can be expressed by using the Eq and Ord traits, where the PartialOrd methods provide:

• a non-strict partial ordering relation if T: PartialOrd + Eq
• a total ordering relation if T: Ord