
Category: functors  Component type: concept 
First argument type  The type of the Strict Weak Ordering's first argument. 
Second argument type  The type of the Strict Weak Ordering's second argument. The first argument type and second argument type must be the same. 
Result type  The type returned when the Strict Weak Ordering is called. The result type must be convertible to bool. 
F  A type that is a model of Strict Weak Ordering 
X  The type of Strict Weak Ordering's arguments. 
f  Object of type F 
x, y, z  Object of type X 
Name  Expression  Precondition  Semantics  Postcondition 

Function call  f(x, y)  The ordered pair (x,y) is in the domain of f  Returns true if x precedes y, and false otherwise  The result is either true or false 
Irreflexivity  f(x, x) must be false. 
Antisymmetry  f(x, y) implies !f(y, x) 
Transitivity  f(x, y) and f(y, z) imply f(x, z). 
Transitivity of equivalence  Equivalence (as defined above) is transitive: if x is equivalent to y and y is equivalent to z, then x is equivalent to z. (This implies that equivalence does in fact satisfy the mathematical definition of an equivalence relation.) [1] 
[1] The first three axioms, irreflexivity, antisymmetry, and transitivity, are the definition of a partial ordering; transitivity of equivalence is required by the definition of a strict weak ordering. A total ordering is one that satisfies an even stronger condition: equivalence must be the same as equality.
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