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68 lines
2.0 KiB
Idris
68 lines
2.0 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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%default total
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public export
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zeroNeverGreater : Not (LTE (S n) Z)
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zeroNeverGreater LTEZero impossible
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zeroNeverGreater (LTESucc _) impossible
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public export
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zeroAlwaysSmaller : {n : Nat} -> LTE Z n
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zeroAlwaysSmaller = LTEZero
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public export
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ltesuccinjective : {0 n, m : Nat} -> Not (LTE n m) -> Not (LTE (S n) (S m))
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ltesuccinjective disprf (LTESucc nLTEm) = void (disprf nLTEm)
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ltesuccinjective disprf LTEZero impossible
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||| If a predicate is decidable then we can decide whether it holds on
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||| a bounded domain.
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public export
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decideLTBounded :
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{0 p : Nat -> Type} ->
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((n : Nat) -> Dec (p n)) ->
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(n : Nat) -> Dec ((k : Nat) -> k `LT` n -> p k)
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decideLTBounded pdec 0 = Yes (\ k, bnd => absurd bnd)
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decideLTBounded pdec (S n) with (pdec 0)
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_ | No np0 = No (\ prf => np0 (prf 0 (LTESucc LTEZero)))
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_ | Yes p0 with (decideLTBounded (\ n => pdec (S n)) n)
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_ | No nprf = No (\ prf => nprf (\ k, bnd => prf (S k) (LTESucc bnd)))
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_ | Yes prf = Yes $ \ k, bnd =>
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case view bnd of
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LTZero => p0
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(LTSucc bnd) => prf _ bnd
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||| If a predicate is decidable then we can decide whether it holds on
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||| a bounded domain.
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public export
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decideLTEBounded :
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{0 p : Nat -> Type} ->
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((n : Nat) -> Dec (p n)) ->
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(n : Nat) -> Dec ((k : Nat) -> k `LTE` n -> p k)
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decideLTEBounded pdec n with (decideLTBounded pdec (S n))
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_ | Yes prf = Yes (\ k, bnd => prf k (LTESucc bnd))
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_ | No nprf = No (\ prf => nprf (\ k, bnd => prf k (fromLteSucc bnd)))
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public export
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Decidable 2 [Nat,Nat] LTE where
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decide = isLTE
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public export
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Decidable 2 [Nat,Nat] LT where
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decide m = isLTE (S m)
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public export
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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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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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