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import Data.Vect
import Data.Fin
%default total
fins : Vect n (Fin n)
fins {n = Z} = []
fins {n = S n} = FZ :: map FS fins
Permutation : Nat -> Type
Permutation n = {a : Type} -> Vect n a -> Vect n a
permutations : {n : Nat} -> Vect (fact n) (Permutation n)
permutations {n = Z} = [id]
permutations {n = S n} =
rewrite multCommutative (S n) (fact n) in
concat . map inserts $ permutations {n = n}
where
inserts : Permutation k -> Vect (S k) (Permutation (S k))
inserts pi = map (\i => \(x :: xs) => insertAt i x . pi $ xs) fins
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