
Category: utilities  Component type: concept 
X  A type that is a model of LessThanComparable 
x, y, z  Object of type X 
If operator< is a strict weak ordering, and if each equivalence class has only a single element, then operator< is a total ordering.
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 
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] 
Irreflexivity  x < x must be false. 
Antisymmetry  x < y implies !(y < x) [2] 
Transitivity  x < y and y < z implies x < z [3] 
[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.
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