base.Data.Vect.Elem.idr Maven / Gradle / Ivy
The newest version!
module Data.Vect.Elem
import Data.Singleton
import Data.Vect
import Decidable.Equality
%default total
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-- Vector membership proof
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||| A proof that some element is found in a vector
public export
data Elem : a -> Vect k a -> Type where
Here : Elem x (x::xs)
There : (later : Elem x xs) -> Elem x (y::xs)
export
Uninhabited (Here = There e) where
uninhabited Refl impossible
export
Uninhabited (There e = Here) where
uninhabited Refl impossible
export
Injective (There {x} {y} {xs}) where
injective Refl = Refl
export
DecEq (Elem x xs) where
decEq Here Here = Yes Refl
decEq (There this) (There that) = decEqCong $ decEq this that
decEq Here (There later) = No absurd
decEq (There later) Here = No absurd
export
Uninhabited (Elem x []) where
uninhabited Here impossible
export
Uninhabited (x = z) => Uninhabited (Elem z xs) => Uninhabited (Elem z $ x::xs) where
uninhabited Here @{xz} = uninhabited Refl @{xz}
uninhabited (There y) = uninhabited y
||| An item not in the head and not in the tail is not in the Vect at all
export
neitherHereNorThere : Not (x = y) -> Not (Elem x xs) -> Not (Elem x (y :: xs))
neitherHereNorThere xneqy xninxs Here = xneqy Refl
neitherHereNorThere xneqy xninxs (There xinxs) = xninxs xinxs
||| A decision procedure for Elem
public export
isElem : DecEq a => (x : a) -> (xs : Vect n a) -> Dec (Elem x xs)
isElem x [] = No uninhabited
isElem x (y::xs) with (decEq x y)
isElem x (x::xs) | (Yes Refl) = Yes Here
isElem x (y::xs) | (No xneqy) with (isElem x xs)
isElem x (y::xs) | (No xneqy) | (Yes xinxs) = Yes (There xinxs)
isElem x (y::xs) | (No xneqy) | (No xninxs) = No (neitherHereNorThere xneqy xninxs)
||| Get the element at the given position.
public export
get : (xs : Vect n a) -> (p : Elem x xs) -> a
get (x :: _) Here = x
get (_ :: xs) (There p) = get xs p
||| Get the element at the given position, with proof that it is the desired element.
public export
lookup : (xs : Vect n a) -> (p : Elem x xs) -> Singleton x
lookup (x :: _) Here = Val x
lookup (_ :: xs) (There p) = lookup xs p
public export
replaceElem : (xs : Vect k t) -> Elem x xs -> (y : t) -> (ys : Vect k t ** Elem y ys)
replaceElem (x::xs) Here y = (y :: xs ** Here)
replaceElem (x::xs) (There xinxs) y with (replaceElem xs xinxs y)
replaceElem (x::xs) (There xinxs) y | (ys ** yinys) = (x :: ys ** There yinys)
public export
replaceByElem : (xs : Vect k t) -> Elem x xs -> t -> Vect k t
replaceByElem (x::xs) Here y = y :: xs
replaceByElem (x::xs) (There xinxs) y = x :: replaceByElem xs xinxs y
public export
mapElem : {0 xs : Vect k t} -> {0 f : t -> u} ->
(1 _ : Elem x xs) -> Elem (f x) (map f xs)
mapElem Here = Here
mapElem (There e) = There (mapElem e)
||| Remove the element at the given position.
|||
||| @xs The vector to be removed from
||| @p A proof that the element to be removed is in the vector
public export
dropElem : {k : _} -> (xs : Vect (S k) t) -> Elem x xs -> Vect k t
dropElem (x::ys) Here = ys
dropElem {k = S k} (x::ys) (There later) = x :: dropElem ys later
||| Erase the indices, returning the bounded numeric position of the element
public export
elemToFin : {0 xs : Vect n a} -> Elem x xs -> Fin n
elemToFin Here = FZ
elemToFin (There p) = FS (elemToFin p)
||| Find the element with a proof at a given bounded position
public export
indexElem : (1 _ : Fin n) -> (xs : Vect n a) -> (x ** Elem x xs)
indexElem FZ (y::_) = (y ** Here)
indexElem (FS n) (_::ys) = let (x ** p) = indexElem n ys in
(x ** There p)