Trait for types that form a total order.

Implementations must be consistent with the PartialOrd implementation, and ensure max, min, and clamp are consistent with cmp:

• partial_cmp(a, b) == Some(cmp(a, b)).
• max(a, b) == max_by(a, b, cmp) (ensured by the default implementation).
• min(a, b) == min_by(a, b, cmp) (ensured by the default implementation).
• For a.clamp(min, max), see the method docs (ensured by the default implementation).

Violating these requirements is a logic error. The behavior resulting from a logic error is not specified, but users of the trait must ensure that such logic errors do not result in undefined behavior. This means that unsafe code must not rely on the correctness of these methods.

Corollaries

From the above and the requirements of PartialOrd, it follows that for all a, b and c:

• exactly one of a < b, a == b or a > b is true; and
• < is transitive: a < b and b < c implies a < c. The same must hold for both == and >.

Mathematically speaking, the < operator defines a strict weak order. In cases where == conforms to mathematical equality, it also defines a strict total order.

Derivable

This trait can be used with #[derive].

When derived on structs, it will produce a lexicographic ordering based on the top-to-bottom declaration order of the struct's members.

When derived on enums, variants are ordered primarily by their discriminants. Secondarily, they are ordered by their fields. By default, the discriminant is smallest for variants at the top, and largest for variants at the bottom. Here's an example:

#[derive(PartialEq, Eq, PartialOrd, Ord)]
enum E {
    Top,
    Bottom,
}

assert!(E::Top < E::Bottom);

However, manually setting the discriminants can override this default behavior:

#[derive(PartialEq, Eq, PartialOrd, Ord)]
enum E {
    Top = 2,
    Bottom = 1,
}

assert!(E::Bottom < E::Top);

Lexicographical comparison

Lexicographical comparison is an operation with the following properties:

• Two sequences are compared element by element.
• The first mismatching element defines which sequence is lexicographically less or greater than the other.
• If one sequence is a prefix of another, the shorter sequence is lexicographically less than the other.
• If two sequences have equivalent elements and are of the same length, then the sequences are lexicographically equal.
• An empty sequence is lexicographically less than any non-empty sequence.
• Two empty sequences are lexicographically equal.

How can I implement Ord?

Ord requires that the type also be PartialOrd, PartialEq, and Eq.

Because Ord implies a stronger ordering relationship than PartialOrd, and both Ord and PartialOrd must agree, you must choose how to implement Ord first. You can choose to derive it, or implement it manually. If you derive it, you should derive all four traits. If you implement it manually, you should manually implement all four traits, based on the implementation of Ord.

Here's an example where you want to define the Character comparison by health and experience only, disregarding the field mana:

use std::cmp::Ordering;

struct Character {
    health: u32,
    experience: u32,
    mana: f32,
}

impl Ord for Character {
    fn cmp(&self, other: &Self) -> Ordering {
        self.experience
            .cmp(&other.experience)
            .then(self.health.cmp(&other.health))
    }
}

impl PartialOrd for Character {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}

impl PartialEq for Character {
    fn eq(&self, other: &Self) -> bool {
        self.health == other.health && self.experience == other.experience
    }
}

impl Eq for Character {}

If all you need is to slice::sort a type by a field value, it can be simpler to use slice::sort_by_key.

Examples of incorrect Ord implementations

use std::cmp::Ordering;

#[derive(Debug)]
struct Character {
    health: f32,
}

impl Ord for Character {
    fn cmp(&self, other: &Self) -> std::cmp::Ordering {
        if self.health < other.health {
            Ordering::Less
        } else if self.health > other.health {
            Ordering::Greater
        } else {
            Ordering::Equal
        }
    }
}

impl PartialOrd for Character {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}

impl PartialEq for Character {
    fn eq(&self, other: &Self) -> bool {
        self.health == other.health
    }
}

impl Eq for Character {}

let a = Character { health: 4.5 };
let b = Character { health: f32::NAN };

// Mistake: floating-point values do not form a total order and using the built-in comparison
// operands to implement `Ord` irregardless of that reality does not change it. Use
// `f32::total_cmp` if you need a total order for floating-point values.

// Reflexivity requirement of `Ord` is not given.
assert!(a == a);
assert!(b != b);

// Antisymmetry requirement of `Ord` is not given. Only one of a < c and c < a is allowed to be
// true, not both or neither.
assert_eq!((a < b) as u8 + (b < a) as u8, 0);

use std::cmp::Ordering;

#[derive(Debug)]
struct Character {
    health: u32,
    experience: u32,
}

impl PartialOrd for Character {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}

impl Ord for Character {
    fn cmp(&self, other: &Self) -> std::cmp::Ordering {
        if self.health < 50 {
            self.health.cmp(&other.health)
        } else {
            self.experience.cmp(&other.experience)
        }
    }
}

// For performance reasons implementing `PartialEq` this way is not the idiomatic way, but it
// ensures consistent behavior between `PartialEq`, `PartialOrd` and `Ord` in this example.
impl PartialEq for Character {
    fn eq(&self, other: &Self) -> bool {
        self.cmp(other) == Ordering::Equal
    }
}

impl Eq for Character {}

let a = Character {
    health: 3,
    experience: 5,
};
let b = Character {
    health: 10,
    experience: 77,
};
let c = Character {
    health: 143,
    experience: 2,
};

// Mistake: The implementation of `Ord` compares different fields depending on the value of
// `self.health`, the resulting order is not total.

// Transitivity requirement of `Ord` is not given. If a is smaller than b and b is smaller than
// c, by transitive property a must also be smaller than c.
assert!(a < b && b < c && c < a);

// Antisymmetry requirement of `Ord` is not given. Only one of a < c and c < a is allowed to be
// true, not both or neither.
assert_eq!((a < c) as u8 + (c < a) as u8, 2);

The documentation of PartialOrd contains further examples, for example it's wrong for PartialOrd and PartialEq to disagree.