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+LessThanComparable
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+LessThanComparable
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+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
+
+
+|
+X
+ |
+
+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
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+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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