# LessThan Comparable

 Category: utilities Component type: concept

### 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 partial ordering.

### 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
Greater x > y   Convertible to bool
Less or equal x <= y   Convertible to bool
Greater or equal x >= y   Convertible to bool

### Expression semantics

Name Expression Precondition Semantics Postcondition
Less x < y x and y are in the domain of <
Greater x > y x and y are in the domain of < Equivalent to y < x [1]
Less or equal x <= y x and y are in the domain of < Equivalent to !(y < x) [1]
Greater or equal x >= y x and y are in the domain of < Equivalent to !(x < y) [1]

### Invariants

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

• 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.