mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-18 08:42:11 +03:00
85 lines
2.7 KiB
Idris
85 lines
2.7 KiB
Idris
module Data.List.Quantifiers
|
|
|
|
import Data.List
|
|
import Data.List.Elem
|
|
|
|
||| A proof that some element of a list satisfies some property
|
|
|||
|
|
||| @ p the property to be satsified
|
|
public export
|
|
data Any : (0 p : a -> Type) -> List a -> Type where
|
|
||| A proof that the satisfying element is the first one in the `List`
|
|
Here : {0 xs : List a} -> p x -> Any p (x :: xs)
|
|
||| A proof that the satsifying element is in the tail of the `List`
|
|
There : {0 xs : List a} -> Any p xs -> Any p (x :: xs)
|
|
|
|
export
|
|
Uninhabited (Any p Nil) where
|
|
uninhabited (Here _) impossible
|
|
uninhabited (There _) impossible
|
|
|
|
||| Eliminator for `Any`
|
|
export
|
|
anyElim : (Any p xs -> b) -> (p x -> b) -> Any p (x :: xs) -> b
|
|
anyElim _ f (Here p) = f p
|
|
anyElim f _ (There p) = f p
|
|
|
|
||| Given a decision procedure for a property, determine if an element of a
|
|
||| list satisfies it.
|
|
|||
|
|
||| @ p the property to be satisfied
|
|
||| @ dec the decision procedure
|
|
||| @ xs the list to examine
|
|
export
|
|
any : (dec : (x : a) -> Dec (p x)) -> (xs : List a) -> Dec (Any p xs)
|
|
any _ Nil = No uninhabited
|
|
any p (x::xs) with (p x)
|
|
any p (x::xs) | Yes prf = Yes (Here prf)
|
|
any p (x::xs) | No ctra =
|
|
case any p xs of
|
|
Yes prf' => Yes (There prf')
|
|
No ctra' => No (anyElim ctra' ctra)
|
|
|
|
||| A proof that all elements of a list satisfy a property. It is a list of
|
|
||| proofs, corresponding element-wise to the `List`.
|
|
public export
|
|
data All : (0 p : a -> Type) -> List a -> Type where
|
|
Nil : All p Nil
|
|
(::) : {0 xs : List a} -> p x -> All p xs -> All p (x :: xs)
|
|
|
|
||| If there does not exist an element that satifies the property, then it is
|
|
||| the case that all elements do not satisfy it.
|
|
export
|
|
negAnyAll : {xs : List a} -> Not (Any p xs) -> All (Not . p) xs
|
|
negAnyAll {xs=Nil} _ = Nil
|
|
negAnyAll {xs=x::xs} f = (f . Here) :: negAnyAll (f . There)
|
|
|
|
||| If there exists an element that doesn't satify the property, then it is
|
|
||| the case that not all elements satisfy it.
|
|
export
|
|
negAllAny : Any (Not . p) xs -> Not (All p xs)
|
|
negAllAny (Here ctra) (p::_) = ctra p
|
|
negAllAny (There np) (_::ps) = negAllAny np ps
|
|
|
|
||| Given a decision procedure for a property, decide whether all elements of
|
|
||| a list satisfy it.
|
|
|||
|
|
||| @ p the property
|
|
||| @ dec the decision procedure
|
|
||| @ xs the list to examine
|
|
export
|
|
all : (dec : (x : a) -> Dec (p x)) -> (xs : List a) -> Dec (All p xs)
|
|
all _ Nil = Yes Nil
|
|
all d (x::xs) with (d x)
|
|
all d (x::xs) | No ctra = No $ \(p::_) => ctra p
|
|
all d (x::xs) | Yes prf =
|
|
case all d xs of
|
|
Yes prf' => Yes (prf :: prf')
|
|
No ctra' => No $ \(_::ps) => ctra' ps
|
|
|
|
||| Given a proof of membership for some element, extract the property proof for it
|
|
export
|
|
indexAll : Elem x xs -> All p xs -> p x
|
|
indexAll Here (p::_ ) = p
|
|
indexAll (There e) ( _::ps) = indexAll e ps
|