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20 lines
948 B
Idris
20 lines
948 B
Idris
In : (x : a) -> (l : List a) -> Type
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In x [] = Void
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In x (x' :: xs) = (x' = x) `Either` In x xs
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appendEmpty : (xs : List a) -> xs ++ [] = xs
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appendEmpty [] = Refl
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appendEmpty (x :: xs) = rewrite appendEmpty xs in Refl
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in_app_iff : {l : List b} -> {l' : List b} -> In a (l++l') -> (In a l `Either` In a l')
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in_app_iff {l = []} { l' = []} x = Left x
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in_app_iff {l = []} { l' = (y :: xs)} x = Right x
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in_app_iff {l = (y::xs)} { l' = []} x = rewrite sym (appendEmpty xs) in Left x
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in_app_iff {l = (y::xs)} {l' = (z::ys)} (Left prf) = Left (Left prf)
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in_app_iff {l = (y::xs)} {l' = (z::ys)} (Right prf) =
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let induc : Either (In a xs) (Either (z = a) (In a ys))= in_app_iff {l = xs} {l' = z :: ys} prf in
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case induc of
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(Left l) => Left $ Right l
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(Right r) => Right r
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