import Control.SetFunctions

-- The nodes of a graph
data Node = A | B | C | D | E
 deriving (Eq,Show)

-- The edges of a graph
next :: Node -> Node
next A = B
next A = C
next B = A
next B = C
next B = D
next C = D
next C = E
next D = E
next E = C

succ :: Node -> Node
succ A = B ? C
succ B = A ? C ? D
succ C = D ? E
succ D = E
succ E = C

-- Compute a path from a given node to a target node.
pathFromTo :: Node -> Node -> [Node]
pathFromTo n1 n2 = extendPath n2 [n1]

-- Extends a given path (in reverse order) to a target node.
extendPath :: Node -> [Node] -> [Node]
extendPath target rpath@(n:_) =
  if target == n then reverse rpath
                 else extendPath target ((succ n) `consNoCycle` rpath)

-- Constrained constructor to construct acyclic lists only:
consNoCycle :: Node -> [Node] -> [Node]
consNoCycle x xs | all (/= x) xs
                 = x : xs

shortestPath :: Node -> Node -> [Node]
shortestPath n1 n2 =  minValueBy shorter ((set2 pathFromTo) n1 n2)
 where
  shorter p1 p2 = compare (length p1) (length p2)