Agda-2.3.2: A dependently typed functional programming language and proof assistant

Safe Haskell | Safe-Infered

Agda.Utils.Permutation

Synopsis

- data Permutation = Perm {
  - permRange :: Int
  - permPicks :: [Int]
  }
- permute :: Permutation -> [a] -> [a]
- idP :: Int -> Permutation
- takeP :: Int -> Permutation -> Permutation
- liftP :: Int -> Permutation -> Permutation
- composeP :: Permutation -> Permutation -> Permutation
- invertP :: Permutation -> Permutation
- compactP :: Permutation -> Permutation
- reverseP :: Permutation -> Permutation
- expandP :: Int -> Int -> Permutation -> Permutation
- topoSort :: (a -> a -> Bool) -> [a] -> Maybe Permutation

# Documentation

data Permutation

    permute [1,2,0] [x0,x1,x2] = [x1,x2,x0]

Agda typing would be: `Perm : {m : Nat}(n : Nat) -> Vec (Fin n) m -> Permutation` `m` is the `size` of the permutation.

Constructors

Perm |
Fields permRange :: Int permPicks :: [Int]

Instances

Eq Permutation |
Show Permutation |
Typeable Permutation |
Sized Permutation |
Abstract Permutation |
Apply Permutation |
DropArgs Permutation |
EmbPrj Permutation |

permute :: Permutation -> [a] -> [a]

`permute [1,2,0] [x0,x1,x2] = [x1,x2,x0]` More precisely, `permute indices list = sublist`, generates `sublist` from `list` by picking the elements of list as indicated by `indices`. `permute [1,3,0] [x0,x1,x2,x3] = [x1,x3,x0]`

Agda typing: `permute (Perm {m} n is) : Vec A m -> Vec A n`

idP :: Int -> Permutation

takeP :: Int -> Permutation -> Permutation

liftP :: Int -> Permutation -> Permutation

`liftP k` takes a `Perm {m} n` to a `Perm {m+k} (n+k)`. Analogous to `liftS`, but Permutations operate on de Bruijn LEVELS, not indices.

composeP :: Permutation -> Permutation -> Permutation

    permute (compose p1 p2) == permute p1 . permute p2

invertP :: Permutation -> Permutation

compactP :: Permutation -> Permutation

Turn a possible non-surjective permutation into a surjective permutation.

reverseP :: Permutation -> Permutation

    permute (reverseP p) xs ==
        reverse $ permute p $ reverse xs

Example: `permute (reverseP (Perm 4 [1,3,0])) [x0,x1,x2,x3] == permute (Perm 4 $ map (3-) [0,3,1]) [x0,x1,x2,x3] == permute (Perm 4 [3,0,2]) [x0,x1,x2,x3] == [x3,x0,x2] == reverse [x2,x0,x3] == reverse $ permute (Perm 4 [1,3,0]) [x3,x2,x1,x0] == reverse $ permute (Perm 4 [1,3,0]) $ reverse [x0,x1,x2,x3]`

expandP :: Int -> Int -> Permutation -> Permutation

`expandP i n π` in the domain of `π` replace the ith element by n elements.

topoSort :: (a -> a -> Bool) -> [a] -> Maybe Permutation

Stable topologic sort. The first argument decides whether its first argument is an immediate parent to its second argument.