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55 lines
1.5 KiB
Idris
55 lines
1.5 KiB
Idris
||| Implementation of ordering relations for `Fin`ite numbers
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module Data.Fin.Order
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import Control.Relation
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import Control.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 Decidable.Decidable
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%default total
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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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Transitive (Fin k) FinLTE where
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transitive (FromNatPrf xy) (FromNatPrf yz) =
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FromNatPrf $ transitive xy yz
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public export
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Reflexive (Fin k) FinLTE where
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reflexive = FromNatPrf $ reflexive
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public export
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Preorder (Fin k) FinLTE where
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public export
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Antisymmetric (Fin k) FinLTE where
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antisymmetric {x} {y} (FromNatPrf xy) (FromNatPrf yx) =
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finToNatInjective x y $
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antisymmetric xy yx
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public export
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PartialOrder (Fin k) FinLTE where
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public export
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Connex (Fin k) FinLTE where
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connex {x = FZ} _ = Left $ FromNatPrf LTEZero
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connex {y = FZ} _ = Right $ FromNatPrf LTEZero
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connex {x = FS k} {y = FS j} prf =
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case connex {rel = FinLTE} $ prf . (cong FS) of
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Left (FromNatPrf p) => Left $ FromNatPrf $ LTESucc p
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Right (FromNatPrf p) => Right $ FromNatPrf $ LTESucc p
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public export
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Decidable 2 [Fin k, Fin k] FinLTE where
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decide m n with (isLTE (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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