45 lines
1 KiB
Idris
45 lines
1 KiB
Idris
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module Quox.FinExtra
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import public Data.Fin
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import Quox.Decidable
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public export
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data LT : Rel (Fin n) where
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LTZ : FZ `LT` FS i
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LTS : i `LT` j -> FS i `LT` FS j
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%builtin Natural FinExtra.LT
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%name FinExtra.LT lt
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public export %inline
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GT : Rel (Fin n)
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GT = flip LT
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export
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Transitive (Fin n) LT where
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transitive LTZ (LTS _) = LTZ
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transitive (LTS p) (LTS q) = LTS $ transitive p q
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export Uninhabited (i `FinExtra.LT` i) where uninhabited (LTS p) = uninhabited p
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export Uninhabited (FS i `LT` FZ) where uninhabited _ impossible
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public export
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data Compare : Rel (Fin n) where
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IsLT : (lt : i `LT` j) -> Compare i j
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IsEQ : Compare i i
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IsGT : (gt : i `GT` j) -> Compare i j
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%name Compare cmp
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export
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compareS : Compare i j -> Compare (FS i) (FS j)
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compareS (IsLT lt) = IsLT (LTS lt)
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compareS IsEQ = IsEQ
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compareS (IsGT gt) = IsGT (LTS gt)
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export
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compareP : (i, j : Fin n) -> Compare i j
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compareP FZ FZ = IsEQ
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compareP FZ (FS j) = IsLT LTZ
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compareP (FS i) FZ = IsGT LTZ
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compareP (FS i) (FS j) = compareS $ compareP i j
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