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70 lines
2.1 KiB
Idris
70 lines
2.1 KiB
Idris
module Data.SnocList.Elem
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import Data.SnocList
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import Decidable.Equality
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||| A proof that some element is found in a list.
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public export
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data Elem : a -> SnocList a -> Type where
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||| A proof that the element is at the head of the list
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Here : Elem x (sx :< x)
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||| A proof that the element is in the tail of the list
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There : Elem x sx -> Elem x (sx :< y)
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export
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Uninhabited (Here = There e) where
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uninhabited Refl impossible
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export
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Uninhabited (There e = Here) where
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uninhabited Refl impossible
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export
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Uninhabited (Elem {a} x [<]) where
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uninhabited Here impossible
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uninhabited (There p) impossible
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export
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thereInjective : {0 e1, e2 : Elem x sx} -> There e1 = There e2 -> e1 = e2
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thereInjective Refl = Refl
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export
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DecEq (x `Elem` sx) where
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decEq Here Here = Yes Refl
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decEq Here (There _) = No absurd
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decEq (There _) Here = No absurd
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decEq (There y) (There z) with (decEq y z)
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decEq (There y) (There y) | Yes Refl = Yes Refl
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decEq (There y) (There z) | No neq = No $ neq . thereInjective
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||| Remove the element at the given position.
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public export
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dropElem : (sx : SnocList a) -> (p : Elem x sx) -> SnocList a
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dropElem (sy :< _) Here = sy
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dropElem (sy :< y) (There p) = (dropElem sy p) :< y
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||| Erase the indices, returning the numeric position of the element
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public export
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elemToNat : Elem x sx -> Nat
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elemToNat Here = Z
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elemToNat (There p) = S (elemToNat p)
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||| Find the element with a proof at a given position (in reverse), if it is valid
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public export
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indexElem : Nat -> (sx : SnocList a) -> Maybe (x ** Elem x sx)
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indexElem _ [<] = Nothing
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indexElem Z (_ :< y) = Just (y ** Here)
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indexElem (S k) (sy :< _) = (\(y ** p) => (y ** There p)) `map` (indexElem k sy)
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||| Lift the membership proof to a mapped list
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export
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elemMap : (0 f : a -> b) -> Elem x sx -> Elem (f x) (map f sx)
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elemMap f Here = Here
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elemMap f (There el) = There $ elemMap f el
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||| An item not in the head and not in the tail is not in the list at all.
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export
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neitherHereNorThere : Not (x = y) -> Not (Elem x sx) -> Not (Elem x (sx :< y))
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neitherHereNorThere xny _ Here = xny Refl
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neitherHereNorThere _ xnxs (There xxs) = xnxs xxs
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