Idris2/libs/base/Data/Nat/Order.idr

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||| Implementation of ordering relations for `Nat`ural numbers
module Data.Nat.Order
import Data.Nat
import Data.Fun
import Data.Rel
import Decidable.Decidable
%default total
public export
2021-07-13 17:32:01 +03:00
zeroNeverGreater : Not (LTE (S n) Z)
zeroNeverGreater LTEZero impossible
zeroNeverGreater (LTESucc _) impossible
public export
zeroAlwaysSmaller : {n : Nat} -> LTE Z n
zeroAlwaysSmaller = LTEZero
public export
ltesuccinjective : {0 n, m : Nat} -> Not (LTE n m) -> Not (LTE (S n) (S m))
ltesuccinjective disprf (LTESucc nLTEm) = void (disprf nLTEm)
ltesuccinjective disprf LTEZero impossible
-- Remove any time after release of 0.6.0
||| Deprecated. Use `Nat.isLTE`.
public export
decideLTE : (n : Nat) -> (m : Nat) -> Dec (LTE n m)
decideLTE = isLTE
public export
Decidable 2 [Nat,Nat] LTE where
decide = decideLTE
public export
Decidable 2 [Nat,Nat] LT where
decide m = decideLTE (S m)
public export
lte : (m : Nat) -> (n : Nat) -> Dec (LTE m n)
lte m n = decide {ts = [Nat,Nat]} {p = LTE} m n
public export
shift : (m : Nat) -> (n : Nat) -> LTE m n -> LTE (S m) (S n)
shift m n mLTEn = LTESucc mLTEn