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https://github.com/idris-lang/Idris2.git
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21c6f4fb79
* [ breaking ] remove parsing of dangling binders It used to be the case that ``` ID : Type -> Type ID a = a test : ID (a : Type) -> a -> a test = \ a, x => x ``` and ``` head : List $ a -> Maybe a head [] = Nothing head (x :: _) = Just x ``` were accepted but these are now rejected because: * `ID (a : Type) -> a -> a` is parsed as `(ID (a : Type)) -> a -> a` * `List $ a -> Maybe a` is parsed as `List (a -> Maybe a)` Similarly if you want to use a lambda / rewrite / let expression as part of the last argument of an application, the use of `$` or parens is now mandatory. This should hopefully allow us to make progress on #1703
108 lines
4.4 KiB
Idris
108 lines
4.4 KiB
Idris
module Data.List.Equalities
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import Data.List
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%default total
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||| A list constructued using snoc cannot be empty.
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export
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snocNonEmpty : {x : a} -> {xs : List a} -> Not (xs ++ [x] = [])
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snocNonEmpty {xs = []} prf = uninhabited prf
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snocNonEmpty {xs = y :: ys} prf = uninhabited prf
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||| Proof that snoc'ed list is not empty in terms of `NonEmpty`.
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export %hint
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SnocNonEmpty : (xs : List a) -> (x : a) -> NonEmpty (xs `snoc` x)
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SnocNonEmpty [] _ = IsNonEmpty
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SnocNonEmpty (_::_) _ = IsNonEmpty
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||| Two lists are equal, if their heads are equal and their tails are equal.
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export
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consCong2 : {x : a} -> {xs : List a} -> {y : b} -> {ys : List b} ->
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x = y -> xs = ys -> x :: xs = y :: ys
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consCong2 Refl Refl = Refl
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||| Equal non-empty lists should result in equal components after destructuring 'snoc'.
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export
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snocInjective : {x : a} -> {xs : List a} -> {y : a} -> {ys : List a} ->
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(xs `snoc` x) = (ys `snoc` y) -> (xs = ys, x = y)
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snocInjective {xs = []} {ys = []} Refl = (Refl, Refl)
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snocInjective {xs = []} {ys = z :: zs} prf =
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let nilIsSnoc = snd $ consInjective {xs = []} {ys = zs ++ [y]} prf
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in void $ snocNonEmpty (sym nilIsSnoc)
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snocInjective {xs = z :: xs} {ys = []} prf =
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let snocIsNil = snd $ consInjective {x = z} {xs = xs ++ [x]} {ys = []} prf
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in void $ snocNonEmpty snocIsNil
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snocInjective {xs = w :: xs} {ys = z :: ys} prf =
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let (wEqualsZ, xsSnocXEqualsYsSnocY) = consInjective prf
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(xsEqualsYS, xEqualsY) = snocInjective xsSnocXEqualsYsSnocY
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in (consCong2 wEqualsZ xsEqualsYS, xEqualsY)
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||| Appending pairwise equal lists gives equal lists
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export
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appendCong2 : {x1 : List a} -> {x2 : List a} ->
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{y1 : List b} -> {y2 : List b} ->
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x1 = y1 -> x2 = y2 -> x1 ++ x2 = y1 ++ y2
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appendCong2 {x1=[]} {y1=(_ :: _)} Refl _ impossible
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appendCong2 {x1=(_ :: _)} {y1=[]} Refl _ impossible
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appendCong2 {x1=[]} {y1=[]} _ eq2 = eq2
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appendCong2 {x1=(_ :: _)} {y1=(_ :: _)} eq1 eq2 =
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let (hdEqual, tlEqual) = consInjective eq1
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in consCong2 hdEqual (appendCong2 tlEqual eq2)
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||| List.map is distributive over appending.
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export
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mapDistributesOverAppend
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: (f : a -> b)
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-> (xs : List a)
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-> (ys : List a)
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-> map f (xs ++ ys) = map f xs ++ map f ys
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mapDistributesOverAppend _ [] _ = Refl
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mapDistributesOverAppend f (x :: xs) ys =
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cong (f x ::) $ mapDistributesOverAppend f xs ys
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||| List.length is distributive over appending.
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export
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lengthDistributesOverAppend
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: (xs, ys : List a)
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-> length (xs ++ ys) = length xs + length ys
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lengthDistributesOverAppend [] ys = Refl
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lengthDistributesOverAppend (x :: xs) ys =
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cong S $ lengthDistributesOverAppend xs ys
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||| Length of a snoc'd list is the same as Succ of length list.
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export
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lengthSnoc : (x : _) -> (xs : List a) -> length (snoc xs x) = S (length xs)
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lengthSnoc x [] = Refl
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lengthSnoc x (_ :: xs) = cong S (lengthSnoc x xs)
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||| Appending the same list at left is injective.
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export
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appendSameLeftInjective : (xs, ys, zs : List a) -> zs ++ xs = zs ++ ys -> xs = ys
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appendSameLeftInjective xs ys [] = id
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appendSameLeftInjective xs ys (_::zs) = appendSameLeftInjective xs ys zs . snd . consInjective
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||| Appending the same list at right is injective.
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export
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appendSameRightInjective : (xs, ys, zs : List a) -> xs ++ zs = ys ++ zs -> xs = ys
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appendSameRightInjective xs ys [] = rewrite appendNilRightNeutral xs in
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rewrite appendNilRightNeutral ys in
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id
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appendSameRightInjective xs ys (z::zs) = rewrite appendAssociative xs [z] zs in
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rewrite appendAssociative ys [z] zs in
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fst . snocInjective . appendSameRightInjective (xs ++ [z]) (ys ++ [z]) zs
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||| List cannot be equal to itself prepended with some non-empty list.
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export
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appendNonEmptyLeftNotEq : (zs, xs : List a) -> NonEmpty xs => Not (zs = xs ++ zs)
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appendNonEmptyLeftNotEq [] (_::_) Refl impossible
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appendNonEmptyLeftNotEq (z::zs) (_::xs) prf
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= appendNonEmptyLeftNotEq zs (xs ++ [z]) @{SnocNonEmpty xs z}
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$ rewrite sym $ appendAssociative xs [z] zs in snd $ consInjective prf
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||| List cannot be equal to itself appended with some non-empty list.
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export
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appendNonEmptyRightNotEq : (zs, xs : List a) -> NonEmpty xs => Not (zs = zs ++ xs)
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appendNonEmptyRightNotEq [] (_::_) Refl impossible
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appendNonEmptyRightNotEq (_::zs) (x::xs) prf = appendNonEmptyRightNotEq zs (x::xs) $ snd $ consInjective prf
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