# 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.

### 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`
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] |

### 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

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