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43 lines
1.4 KiB
Idris
43 lines
1.4 KiB
Idris
||| An order is a particular kind of binary relation. The order
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||| relation is intended to proceed in some direction, though not
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||| necessarily with a unique path.
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||| Orders are often defined simply as bundles of binary relation
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||| properties.
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||| A prominent example of an order relation is LTE over Nat.
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module Control.Order
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import Control.Relation
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||| A preorder is reflexive and transitive.
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public export
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interface (Reflexive ty rel, Transitive ty rel) => Preorder ty rel where
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||| A partial order is an antisymmetrics preorder.
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public export
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interface (Preorder ty rel, Antisymmetric ty rel) => PartialOrder ty rel where
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||| A relation is connex if for any two distinct x and y, either x ~ y or y ~ x.
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||| This can also be stated as a trichotomy: x ~ y or x = y or y ~ x.
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public export
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interface Connex ty rel where
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connex : {x, y : ty} -> Not (x = y) -> Either (rel x y) (rel y x)
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||| A relation is strongly connex if for any two x and y, either x ~ y or y ~ x.
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public export
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interface StronglyConnex ty rel where
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order : (x, y : ty) -> Either (rel x y) (rel y x)
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||| A linear order is a connex partial order.
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public export
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interface (PartialOrder ty rel, Connex ty rel) => LinearOrder ty rel where
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----------------------------------------
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||| Every equivalence relation is a preorder.
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public export
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[EP] Equivalence ty rel => Preorder ty rel where
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