# LessThanComparable

### Description

A type is LessThanComparable if it is ordered: it must be possible to compare two objects of that type using `operator<`, and `operator<` must be a strict weak ordering relation.

### Refinement of

### Associated types

### Notation

`X` | A type that is a model of LessThanComparable
`x`, `y`, `z` | Object of type `X`

### Definitions

Consider the relation `!(x < y) && !(y < x)`. If this relation is transitive (that is, if `!(x < y) && !(y < x) && !(y < z) && !(z < y)` implies `!(x < z) && !(z < x)`), then it satisfies the mathematical definition of an equivalence relation. In this case, `operator<` is a strict weak ordering.

If `operator<` is a strict weak ordering, and if each equivalence class has only a single element, then `operator<` is a total ordering.

### Valid expressions

Name | Expression | Type requirements | Return type
--- | --- | --- | ---
Less | `x < y` |  | Convertible to `bool`

### Expression semantics

Name | Expression | Precondition | Semantics | Postcondition
--- | --- | --- | --- | ---
Less | `x < y` | `x` and `y` are in the domain of `<` |

### Complexity guarantees

### Invariants

Irreflexivity | `x < x` must be false.
Antisymmetry | `x < y` implies !(y < x) [2]
Transitivity | `x < y` and `y < z` implies `x < z` [3]

### Models

- int

### Notes

[1] Only `operator<` is fundamental; the other inequality operators are essentially syntactic sugar.

[2] Antisymmetry is a theorem, not an axiom: it follows from irreflexivity and transitivity.

[3] Because of irreflexivity and transitivity, `operator<` always satisfies the definition of a partial ordering. The definition of a strict weak ordering is stricter, and the definition of a total ordering is stricter still.

### See also

EqualityComparable, StrictWeakOrdering

Copyright © 2000 | Jeremy Siek, Univ.of Notre Dame (jsiek@lsc.nd.edu)