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
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X
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A type that is a model of LessThanComparable
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x, y, z
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Object of type X
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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
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Name
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Expression
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Type requirements
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Return type
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Less
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x < y
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Convertible to bool
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Expression semantics
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Name
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Expression
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Precondition
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Semantics
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Postcondition
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Less
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x < y
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x and y are in the domain of <
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Complexity guarantees
Invariants
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Irreflexivity
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x < x must be false.
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Antisymmetry
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x < y implies !(y < x) [2]
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Transitivity
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x < y and y < z implies x < z [3]
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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