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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
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