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41 lines
1.3 KiB
Idris
41 lines
1.3 KiB
Idris
||| Implementation of ordering relations for `Fin`ite numbers
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module Data.Fin.Order
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import Data.Fin
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import Data.Fun
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import Data.Rel
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import Data.Nat
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import Data.Nat.Order
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import Decidable.Decidable
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import Decidable.Order
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using (k : Nat)
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public export
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data FinLTE : Fin k -> Fin k -> Type where
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FromNatPrf : {m, n : Fin k} -> LTE (finToNat m) (finToNat n) -> FinLTE m n
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public export
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implementation Preorder (Fin k) FinLTE where
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transitive m n o (FromNatPrf p1) (FromNatPrf p2) =
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FromNatPrf (LTEIsTransitive (finToNat m) (finToNat n) (finToNat o) p1 p2)
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reflexive n = FromNatPrf (LTEIsReflexive (finToNat n))
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public export
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implementation Poset (Fin k) FinLTE where
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antisymmetric m n (FromNatPrf p1) (FromNatPrf p2) =
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finToNatInjective m n (LTEIsAntisymmetric (finToNat m) (finToNat n) p1 p2)
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public export
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implementation Decidable 2 [Fin k, Fin k] FinLTE where
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decide m n with (decideLTE (finToNat m) (finToNat n))
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decide m n | Yes prf = Yes (FromNatPrf prf)
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decide m n | No disprf = No (\ (FromNatPrf prf) => disprf prf)
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public export
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implementation Ordered (Fin k) FinLTE where
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order m n =
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either (Left . FromNatPrf)
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(Right . FromNatPrf)
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(order (finToNat m) (finToNat n))
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