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LessThanComparable
[Concepts]

Detailed 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:
NameExpressionType requirementsReturn 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:
NameExpressionPreconditionSemanticsPostcondition
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]
Type(s) Modeling this Concept:
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:

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