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78 lines
2.8 KiB
Idris
78 lines
2.8 KiB
Idris
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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} -> xs ++ [x] = [] -> Void
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snocNonEmpty {xs = []} prf = uninhabited prf
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snocNonEmpty {xs = y :: ys} prf = uninhabited prf
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||| (::) is injective
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export
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consInjective : {x : a} -> {xs : List a} -> {y : b} -> {ys : List b} ->
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(x :: xs) = (y :: ys) -> (x = y, xs = ys)
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consInjective Refl = (Refl, Refl)
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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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snocCong2 : {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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snocCong2 {xs = []} {ys = []} Refl = (Refl, Refl)
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snocCong2 {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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snocCong2 {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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snocCong2 {xs = w :: xs} {ys = z :: ys} prf =
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let (wEqualsZ, xsSnocXEqualsYsSnocY) = consInjective prf
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(xsEqualsYS, xEqualsY) = snocCong2 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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