ord -package:sbv -package:genvalidity-hspec package:LambdaHack

The Ord class is used for totally ordered datatypes. Instances of Ord can be derived for any user-defined datatype whose constituent types are in Ord. The declared order of the constructors in the data declaration determines the ordering in derived Ord instances. The Ordering datatype allows a single comparison to determine the precise ordering of two objects. The Haskell Report defines no laws for Ord. However, <= is customarily expected to implement a non-strict partial order and have the following properties:
  • Transitivity if x <= y && y <= z = True, then x <= z = True
  • Reflexivity x <= x = True
  • Antisymmetry if x <= y && y <= x = True, then x == y = True
Note that the following operator interactions are expected to hold:
  1. x >= y = y <= x
  2. x < y = x <= y && x /= y
  3. x > y = y < x
  4. x < y = compare x y == LT
  5. x > y = compare x y == GT
  6. x == y = compare x y == EQ
  7. min x y == if x <= y then x else y = True
  8. max x y == if x >= y then x else y = True
Note that (7.) and (8.) do not require min and max to return either of their arguments. The result is merely required to equal one of the arguments in terms of (==). Minimal complete definition: either compare or <=. Using compare can be more efficient for complex types.
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