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https://github.com/idris-lang/Idris2.git
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85 lines
2.8 KiB
Idris
85 lines
2.8 KiB
Idris
||| Implementation of ordering relations for `Nat`ural numbers
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module Data.Nat.Order
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import Data.Nat
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import Data.Fun
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import Data.Rel
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import Decidable.Decidable
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import Decidable.Equality
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import Decidable.Order
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public export
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total LTEIsTransitive : (m : Nat) -> (n : Nat) -> (o : Nat) ->
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LTE m n -> LTE n o ->
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LTE m o
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LTEIsTransitive Z n o LTEZero nlteo = LTEZero
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LTEIsTransitive (S m) (S n) (S o) (LTESucc mlten) (LTESucc nlteo) = LTESucc (LTEIsTransitive m n o mlten nlteo)
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public export
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total LTEIsReflexive : (n : Nat) -> LTE n n
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LTEIsReflexive Z = LTEZero
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LTEIsReflexive (S n) = LTESucc (LTEIsReflexive n)
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public export
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implementation Preorder Nat LTE where
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transitive = LTEIsTransitive
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reflexive = LTEIsReflexive
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public export
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total LTEIsAntisymmetric : (m : Nat) -> (n : Nat) ->
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LTE m n -> LTE n m -> m = n
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LTEIsAntisymmetric Z Z LTEZero LTEZero = Refl
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LTEIsAntisymmetric (S n) (S m) (LTESucc mLTEn) (LTESucc nLTEm) with (LTEIsAntisymmetric n m mLTEn nLTEm)
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LTEIsAntisymmetric (S n) (S n) (LTESucc mLTEn) (LTESucc nLTEm) | Refl = Refl
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public export
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implementation Poset Nat LTE where
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antisymmetric = LTEIsAntisymmetric
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public export
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total zeroNeverGreater : {n : Nat} -> LTE (S n) Z -> Void
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zeroNeverGreater {n} LTEZero impossible
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zeroNeverGreater {n} (LTESucc _) impossible
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public export
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total zeroAlwaysSmaller : {n : Nat} -> LTE Z n
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zeroAlwaysSmaller = LTEZero
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public export
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total ltesuccinjective : {0 n : Nat} -> {0 m : Nat} -> (LTE n m -> Void) -> LTE (S n) (S m) -> Void
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ltesuccinjective {n} {m} disprf (LTESucc nLTEm) = void (disprf nLTEm)
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ltesuccinjective {n} {m} disprf LTEZero impossible
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public export
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total decideLTE : (n : Nat) -> (m : Nat) -> Dec (LTE n m)
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decideLTE Z y = Yes LTEZero
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decideLTE (S x) Z = No zeroNeverGreater
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decideLTE (S x) (S y) with (decEq (S x) (S y))
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decideLTE (S x) (S y) | Yes eq = rewrite eq in Yes (reflexive (S y))
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decideLTE (S x) (S y) | No _ with (decideLTE x y)
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decideLTE (S x) (S y) | No _ | Yes nLTEm = Yes (LTESucc nLTEm)
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decideLTE (S x) (S y) | No _ | No nGTm = No (ltesuccinjective nGTm)
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public export
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implementation Decidable 2 [Nat,Nat] LTE where
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decide = decideLTE
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public export
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total
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lte : (m : Nat) -> (n : Nat) -> Dec (LTE m n)
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lte m n = decide {ts = [Nat,Nat]} {p = LTE} m n
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public export
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total
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shift : (m : Nat) -> (n : Nat) -> LTE m n -> LTE (S m) (S n)
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shift m n mLTEn = LTESucc mLTEn
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public export
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implementation Ordered Nat LTE where
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order Z n = Left LTEZero
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order m Z = Right LTEZero
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order (S k) (S j) with (order {to=LTE} k j)
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order (S k) (S j) | Left prf = Left (shift k j prf)
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order (S k) (S j) | Right prf = Right (shift j k prf)
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