map length (tiers :: [[ Xs Word4 ]])  =  [0,1,1,2,4,8,16,32,64,128,256,512,...]

length (list :: [ Xs Word4 ])  =  Infinity

allUnique (list :: [ Xs Word4 ])  =  True

ratioRepetitions (list :: [ Xs Word4 ])  =  0 % 1

tiers :: [Xs Word4]  =
  [ []
  , [Xs []]
  , [Xs [0]]
  , [ Xs [0,0]
    , Xs [1]
    ]
  , [ Xs [0,0,0]
    , Xs [0,1]
    , Xs [1,0]
    , Xs [2]
    ]
  , [ Xs [0,0,0,0]
    , Xs [0,0,1]
    , Xs [0,1,0]
    , Xs [0,2]
    , Xs [1,0,0]
    , Xs [1,1]
    , Xs [2,0]
    , Xs [15]
    ]
  , [ Xs [0,0,0,0,0]
    , Xs [0,0,0,1]
    , Xs [0,0,1,0]
    , Xs [0,0,2]
    , Xs [0,1,0,0]
    , Xs [0,1,1]
    , Xs [0,2,0]
    , Xs [0,15]
    , Xs [1,0,0,0]
    , Xs [1,0,1]
    , Xs [1,1,0]
    , Xs [1,2]
    , Xs [2,0,0]
    , Xs [2,1]
    , Xs [15,0]
    , Xs [3]
    ]
  , [ Xs [0,0,0,0,0,0]
    , Xs [0,0,0,0,1]
    , Xs [0,0,0,1,0]
    , Xs [0,0,0,2]
    , Xs [0,0,1,0,0]
    , Xs [0,0,1,1]
    , Xs [0,0,2,0]
    , Xs [0,0,15]
    , Xs [0,1,0,0,0]
    , Xs [0,1,0,1]
    , Xs [0,1,1,0]
    , Xs [0,1,2]
    , Xs [0,2,0,0]
    , Xs [0,2,1]
    , Xs [0,15,0]
    , Xs [0,3]
    , Xs [1,0,0,0,0]
    , Xs [1,0,0,1]
    , Xs [1,0,1,0]
    , Xs [1,0,2]
    , Xs [1,1,0,0]
    , Xs [1,1,1]
    , Xs [1,2,0]
    , Xs [1,15]
    , Xs [2,0,0,0]
    , Xs [2,0,1]
    , Xs [2,1,0]
    , Xs [2,2]
    , Xs [15,0,0]
    , Xs [15,1]
    , Xs [3,0]
    , Xs [4]
    ]
  , [ Xs [0,0,0,0,0,0,0]
    , Xs [0,0,0,0,0,1]
    , Xs [0,0,0,0,1,0]
    , Xs [0,0,0,0,2]
    , Xs [0,0,0,1,0,0]
    , Xs [0,0,0,1,1]
    , Xs [0,0,0,2,0]
    , Xs [0,0,0,15]
    , Xs [0,0,1,0,0,0]
    , Xs [0,0,1,0,1]
    , Xs [0,0,1,1,0]
    , Xs [0,0,1,2]
    , Xs [0,0,2,0,0]
    , Xs [0,0,2,1]
    , Xs [0,0,15,0]
    , Xs [0,0,3]
    , Xs [0,1,0,0,0,0]
    , Xs [0,1,0,0,1]
    , Xs [0,1,0,1,0]
    , Xs [0,1,0,2]
    , Xs [0,1,1,0,0]
    , Xs [0,1,1,1]
    , Xs [0,1,2,0]
    , Xs [0,1,15]
    , Xs [0,2,0,0,0]
    , Xs [0,2,0,1]
    , Xs [0,2,1,0]
    , Xs [0,2,2]
    , Xs [0,15,0,0]
    , Xs [0,15,1]
    , Xs [0,3,0]
    , Xs [0,4]
    , Xs [1,0,0,0,0,0]
    , Xs [1,0,0,0,1]
    , Xs [1,0,0,1,0]
    , Xs [1,0,0,2]
    , Xs [1,0,1,0,0]
    , Xs [1,0,1,1]
    , Xs [1,0,2,0]
    , Xs [1,0,15]
    , Xs [1,1,0,0,0]
    , Xs [1,1,0,1]
    , Xs [1,1,1,0]
    , Xs [1,1,2]
    , Xs [1,2,0,0]
    , Xs [1,2,1]
    , Xs [1,15,0]
    , Xs [1,3]
    , Xs [2,0,0,0,0]
    , Xs [2,0,0,1]
    , Xs [2,0,1,0]
    , Xs [2,0,2]
    , Xs [2,1,0,0]
    , Xs [2,1,1]
    , Xs [2,2,0]
    , Xs [2,15]
    , Xs [15,0,0,0]
    , Xs [15,0,1]
    , Xs [15,1,0]
    , Xs [15,2]
    , Xs [3,0,0]
    , Xs [3,1]
    , Xs [4,0]
    , Xs [14]
    ]
  , [ Xs [0,0,0,0,0,0,0,0]
    , Xs [0,0,0,0,0,0,1]
    , Xs [0,0,0,0,0,1,0]
    , Xs [0,0,0,0,0,2]
    , Xs [0,0,0,0,1,0,0]
    , Xs [0,0,0,0,1,1]
    , Xs [0,0,0,0,2,0]
    , Xs [0,0,0,0,15]
    , Xs [0,0,0,1,0,0,0]
    , Xs [0,0,0,1,0,1]
    , Xs [0,0,0,1,1,0]
    , Xs [0,0,0,1,2]
    , Xs [0,0,0,2,0,0]
    , Xs [0,0,0,2,1]
    , Xs [0,0,0,15,0]
    , Xs [0,0,0,3]
    , Xs [0,0,1,0,0,0,0]
    , Xs [0,0,1,0,0,1]
    , Xs [0,0,1,0,1,0]
    , Xs [0,0,1,0,2]
    , Xs [0,0,1,1,0,0]
    , Xs [0,0,1,1,1]
    , Xs [0,0,1,2,0]
    , Xs [0,0,1,15]
    , Xs [0,0,2,0,0,0]
    , Xs [0,0,2,0,1]
    , Xs [0,0,2,1,0]
    , Xs [0,0,2,2]
    , Xs [0,0,15,0,0]
    , Xs [0,0,15,1]
    , Xs [0,0,3,0]
    , Xs [0,0,4]
    , Xs [0,1,0,0,0,0,0]
    , Xs [0,1,0,0,0,1]
    , Xs [0,1,0,0,1,0]
    , Xs [0,1,0,0,2]
    , Xs [0,1,0,1,0,0]
    , Xs [0,1,0,1,1]
    , Xs [0,1,0,2,0]
    , Xs [0,1,0,15]
    , Xs [0,1,1,0,0,0]
    , Xs [0,1,1,0,1]
    , Xs [0,1,1,1,0]
    , Xs [0,1,1,2]
    , Xs [0,1,2,0,0]
    , Xs [0,1,2,1]
    , Xs [0,1,15,0]
    , Xs [0,1,3]
    , Xs [0,2,0,0,0,0]
    , Xs [0,2,0,0,1]
    , Xs [0,2,0,1,0]
    , Xs [0,2,0,2]
    , Xs [0,2,1,0,0]
    , Xs [0,2,1,1]
    , Xs [0,2,2,0]
    , Xs [0,2,15]
    , Xs [0,15,0,0,0]
    , Xs [0,15,0,1]
    , Xs [0,15,1,0]
    , Xs [0,15,2]
    , Xs [0,3,0,0]
    , Xs [0,3,1]
    , Xs [0,4,0]
    , Xs [0,14]
    , Xs [1,0,0,0,0,0,0]
    , Xs [1,0,0,0,0,1]
    , Xs [1,0,0,0,1,0]
    , Xs [1,0,0,0,2]
    , Xs [1,0,0,1,0,0]
    , Xs [1,0,0,1,1]
    , Xs [1,0,0,2,0]
    , Xs [1,0,0,15]
    , Xs [1,0,1,0,0,0]
    , Xs [1,0,1,0,1]
    , Xs [1,0,1,1,0]
    , Xs [1,0,1,2]
    , Xs [1,0,2,0,0]
    , Xs [1,0,2,1]
    , Xs [1,0,15,0]
    , Xs [1,0,3]
    , Xs [1,1,0,0,0,0]
    , Xs [1,1,0,0,1]
    , Xs [1,1,0,1,0]
    , Xs [1,1,0,2]
    , Xs [1,1,1,0,0]
    , Xs [1,1,1,1]
    , Xs [1,1,2,0]
    , Xs [1,1,15]
    , Xs [1,2,0,0,0]
    , Xs [1,2,0,1]
    , Xs [1,2,1,0]
    , Xs [1,2,2]
    , Xs [1,15,0,0]
    , Xs [1,15,1]
    , Xs [1,3,0]
    , Xs [1,4]
    , Xs [2,0,0,0,0,0]
    , Xs [2,0,0,0,1]
    , Xs [2,0,0,1,0]
    , Xs [2,0,0,2]
    , Xs [2,0,1,0,0]
    , Xs [2,0,1,1]
    , Xs [2,0,2,0]
    , Xs [2,0,15]
    , Xs [2,1,0,0,0]
    , Xs [2,1,0,1]
    , Xs [2,1,1,0]
    , Xs [2,1,2]
    , Xs [2,2,0,0]
    , Xs [2,2,1]
    , Xs [2,15,0]
    , Xs [2,3]
    , Xs [15,0,0,0,0]
    , Xs [15,0,0,1]
    , Xs [15,0,1,0]
    , Xs [15,0,2]
    , Xs [15,1,0,0]
    , Xs [15,1,1]
    , Xs [15,2,0]
    , Xs [15,15]
    , Xs [3,0,0,0]
    , Xs [3,0,1]
    , Xs [3,1,0]
    , Xs [3,2]
    , Xs [4,0,0]
    , Xs [4,1]
    , Xs [14,0]
    , Xs [5]
    ]
  , [ Xs [0,0,0,0,0,0,0,0,0]
    , Xs [0,0,0,0,0,0,0,1]
    , Xs [0,0,0,0,0,0,1,0]
    , Xs [0,0,0,0,0,0,2]
    , Xs [0,0,0,0,0,1,0,0]
    , Xs [0,0,0,0,0,1,1]
    , Xs [0,0,0,0,0,2,0]
    , Xs [0,0,0,0,0,15]
    , Xs [0,0,0,0,1,0,0,0]
    , Xs [0,0,0,0,1,0,1]
    , Xs [0,0,0,0,1,1,0]
    , Xs [0,0,0,0,1,2]
    , Xs [0,0,0,0,2,0,0]
    , Xs [0,0,0,0,2,1]
    , Xs [0,0,0,0,15,0]
    , Xs [0,0,0,0,3]
    , Xs [0,0,0,1,0,0,0,0]
    , Xs [0,0,0,1,0,0,1]
    , Xs [0,0,0,1,0,1,0]
    , Xs [0,0,0,1,0,2]
    , Xs [0,0,0,1,1,0,0]
    , Xs [0,0,0,1,1,1]
    , Xs [0,0,0,1,2,0]
    , Xs [0,0,0,1,15]
    , Xs [0,0,0,2,0,0,0]
    , Xs [0,0,0,2,0,1]
    , Xs [0,0,0,2,1,0]
    , Xs [0,0,0,2,2]
    , Xs [0,0,0,15,0,0]
    , Xs [0,0,0,15,1]
    , Xs [0,0,0,3,0]
    , Xs [0,0,0,4]
    , Xs [0,0,1,0,0,0,0,0]
    , Xs [0,0,1,0,0,0,1]
    , Xs [0,0,1,0,0,1,0]
    , Xs [0,0,1,0,0,2]
    , Xs [0,0,1,0,1,0,0]
    , Xs [0,0,1,0,1,1]
    , Xs [0,0,1,0,2,0]
    , Xs [0,0,1,0,15]
    , Xs [0,0,1,1,0,0,0]
    , Xs [0,0,1,1,0,1]
    , Xs [0,0,1,1,1,0]
    , Xs [0,0,1,1,2]
    , Xs [0,0,1,2,0,0]
    , Xs [0,0,1,2,1]
    , Xs [0,0,1,15,0]
    , Xs [0,0,1,3]
    , Xs [0,0,2,0,0,0,0]
    , Xs [0,0,2,0,0,1]
    , Xs [0,0,2,0,1,0]
    , Xs [0,0,2,0,2]
    , Xs [0,0,2,1,0,0]
    , Xs [0,0,2,1,1]
    , Xs [0,0,2,2,0]
    , Xs [0,0,2,15]
    , Xs [0,0,15,0,0,0]
    , Xs [0,0,15,0,1]
    , Xs [0,0,15,1,0]
    , Xs [0,0,15,2]
    , Xs [0,0,3,0,0]
    , Xs [0,0,3,1]
    , Xs [0,0,4,0]
    , Xs [0,0,14]
    , Xs [0,1,0,0,0,0,0,0]
    , Xs [0,1,0,0,0,0,1]
    , Xs [0,1,0,0,0,1,0]
    , Xs [0,1,0,0,0,2]
    , Xs [0,1,0,0,1,0,0]
    , Xs [0,1,0,0,1,1]
    , Xs [0,1,0,0,2,0]
    , Xs [0,1,0,0,15]
    , Xs [0,1,0,1,0,0,0]
    , Xs [0,1,0,1,0,1]
    , Xs [0,1,0,1,1,0]
    , Xs [0,1,0,1,2]
    , Xs [0,1,0,2,0,0]
    , Xs [0,1,0,2,1]
    , Xs [0,1,0,15,0]
    , Xs [0,1,0,3]
    , Xs [0,1,1,0,0,0,0]
    , Xs [0,1,1,0,0,1]
    , Xs [0,1,1,0,1,0]
    , Xs [0,1,1,0,2]
    , Xs [0,1,1,1,0,0]
    , Xs [0,1,1,1,1]
    , Xs [0,1,1,2,0]
    , Xs [0,1,1,15]
    , Xs [0,1,2,0,0,0]
    , Xs [0,1,2,0,1]
    , Xs [0,1,2,1,0]
    , Xs [0,1,2,2]
    , Xs [0,1,15,0,0]
    , Xs [0,1,15,1]
    , Xs [0,1,3,0]
    , Xs [0,1,4]
    , Xs [0,2,0,0,0,0,0]
    , Xs [0,2,0,0,0,1]
    , Xs [0,2,0,0,1,0]
    , Xs [0,2,0,0,2]
    , Xs [0,2,0,1,0,0]
    , Xs [0,2,0,1,1]
    , Xs [0,2,0,2,0]
    , Xs [0,2,0,15]
    , Xs [0,2,1,0,0,0]
    , Xs [0,2,1,0,1]
    , Xs [0,2,1,1,0]
    , Xs [0,2,1,2]
    , Xs [0,2,2,0,0]
    , Xs [0,2,2,1]
    , Xs [0,2,15,0]
    , Xs [0,2,3]
    , Xs [0,15,0,0,0,0]
    , Xs [0,15,0,0,1]
    , Xs [0,15,0,1,0]
    , Xs [0,15,0,2]
    , Xs [0,15,1,0,0]
    , Xs [0,15,1,1]
    , Xs [0,15,2,0]
    , Xs [0,15,15]
    , Xs [0,3,0,0,0]
    , Xs [0,3,0,1]
    , Xs [0,3,1,0]
    , Xs [0,3,2]
    , Xs [0,4,0,0]
    , Xs [0,4,1]
    , Xs [0,14,0]
    , Xs [0,5]
    , Xs [1,0,0,0,0,0,0,0]
    , Xs [1,0,0,0,0,0,1]
    , Xs [1,0,0,0,0,1,0]
    , Xs [1,0,0,0,0,2]
    , Xs [1,0,0,0,1,0,0]
    , Xs [1,0,0,0,1,1]
    , Xs [1,0,0,0,2,0]
    , Xs [1,0,0,0,15]
    , Xs [1,0,0,1,0,0,0]
    , Xs [1,0,0,1,0,1]
    , Xs [1,0,0,1,1,0]
    , Xs [1,0,0,1,2]
    , Xs [1,0,0,2,0,0]
    , Xs [1,0,0,2,1]
    , Xs [1,0,0,15,0]
    , Xs [1,0,0,3]
    , Xs [1,0,1,0,0,0,0]
    , Xs [1,0,1,0,0,1]
    , Xs [1,0,1,0,1,0]
    , Xs [1,0,1,0,2]
    , Xs [1,0,1,1,0,0]
    , Xs [1,0,1,1,1]
    , Xs [1,0,1,2,0]
    , Xs [1,0,1,15]
    , Xs [1,0,2,0,0,0]
    , Xs [1,0,2,0,1]
    , Xs [1,0,2,1,0]
    , Xs [1,0,2,2]
    , Xs [1,0,15,0,0]
    , Xs [1,0,15,1]
    , Xs [1,0,3,0]
    , Xs [1,0,4]
    , Xs [1,1,0,0,0,0,0]
    , Xs [1,1,0,0,0,1]
    , Xs [1,1,0,0,1,0]
    , Xs [1,1,0,0,2]
    , Xs [1,1,0,1,0,0]
    , Xs [1,1,0,1,1]
    , Xs [1,1,0,2,0]
    , Xs [1,1,0,15]
    , Xs [1,1,1,0,0,0]
    , Xs [1,1,1,0,1]
    , Xs [1,1,1,1,0]
    , Xs [1,1,1,2]
    , Xs [1,1,2,0,0]
    , Xs [1,1,2,1]
    , Xs [1,1,15,0]
    , Xs [1,1,3]
    , Xs [1,2,0,0,0,0]
    , Xs [1,2,0,0,1]
    , Xs [1,2,0,1,0]
    , Xs [1,2,0,2]
    , Xs [1,2,1,0,0]
    , Xs [1,2,1,1]
    , Xs [1,2,2,0]
    , Xs [1,2,15]
    , Xs [1,15,0,0,0]
    , Xs [1,15,0,1]
    , Xs [1,15,1,0]
    , Xs [1,15,2]
    , Xs [1,3,0,0]
    , Xs [1,3,1]
    , Xs [1,4,0]
    , Xs [1,14]
    , Xs [2,0,0,0,0,0,0]
    , Xs [2,0,0,0,0,1]
    , Xs [2,0,0,0,1,0]
    , Xs [2,0,0,0,2]
    , Xs [2,0,0,1,0,0]
    , Xs [2,0,0,1,1]
    , Xs [2,0,0,2,0]
    , Xs [2,0,0,15]
    , Xs [2,0,1,0,0,0]
    , Xs [2,0,1,0,1]
    , Xs [2,0,1,1,0]
    , Xs [2,0,1,2]
    , Xs [2,0,2,0,0]
    , Xs [2,0,2,1]
    , Xs [2,0,15,0]
    , Xs [2,0,3]
    , Xs [2,1,0,0,0,0]
    , Xs [2,1,0,0,1]
    , Xs [2,1,0,1,0]
    , Xs [2,1,0,2]
    , Xs [2,1,1,0,0]
    , Xs [2,1,1,1]
    , Xs [2,1,2,0]
    , Xs [2,1,15]
    , Xs [2,2,0,0,0]
    , Xs [2,2,0,1]
    , Xs [2,2,1,0]
    , Xs [2,2,2]
    , Xs [2,15,0,0]
    , Xs [2,15,1]
    , Xs [2,3,0]
    , Xs [2,4]
    , Xs [15,0,0,0,0,0]
    , Xs [15,0,0,0,1]
    , Xs [15,0,0,1,0]
    , Xs [15,0,0,2]
    , Xs [15,0,1,0,0]
    , Xs [15,0,1,1]
    , Xs [15,0,2,0]
    , Xs [15,0,15]
    , Xs [15,1,0,0,0]
    , Xs [15,1,0,1]
    , Xs [15,1,1,0]
    , Xs [15,1,2]
    , Xs [15,2,0,0]
    , Xs [15,2,1]
    , Xs [15,15,0]
    , Xs [15,3]
    , Xs [3,0,0,0,0]
    , Xs [3,0,0,1]
    , Xs [3,0,1,0]
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    , Xs [2,1,2,0,0]
    , Xs [2,1,2,1]
    , Xs [2,1,15,0]
    , Xs [2,1,3]
    , Xs [2,2,0,0,0,0]
    , Xs [2,2,0,0,1]
    , Xs [2,2,0,1,0]
    , Xs [2,2,0,2]
    , Xs [2,2,1,0,0]
    , Xs [2,2,1,1]
    , Xs [2,2,2,0]
    , Xs [2,2,15]
    , Xs [2,15,0,0,0]
    , Xs [2,15,0,1]
    , Xs [2,15,1,0]
    , Xs [2,15,2]
    , Xs [2,3,0,0]
    , Xs [2,3,1]
    , Xs [2,4,0]
    , Xs [2,14]
    , Xs [15,0,0,0,0,0,0]
    , Xs [15,0,0,0,0,1]
    , Xs [15,0,0,0,1,0]
    , Xs [15,0,0,0,2]
    , Xs [15,0,0,1,0,0]
    , Xs [15,0,0,1,1]
    , Xs [15,0,0,2,0]
    , Xs [15,0,0,15]
    , Xs [15,0,1,0,0,0]
    , Xs [15,0,1,0,1]
    , Xs [15,0,1,1,0]
    , Xs [15,0,1,2]
    , Xs [15,0,2,0,0]
    , Xs [15,0,2,1]
    , Xs [15,0,15,0]
    , Xs [15,0,3]
    , Xs [15,1,0,0,0,0]
    , Xs [15,1,0,0,1]
    , Xs [15,1,0,1,0]
    , Xs [15,1,0,2]
    , Xs [15,1,1,0,0]
    , Xs [15,1,1,1]
    , Xs [15,1,2,0]
    , Xs [15,1,15]
    , Xs [15,2,0,0,0]
    , Xs [15,2,0,1]
    , Xs [15,2,1,0]
    , Xs [15,2,2]
    , Xs [15,15,0,0]
    , Xs [15,15,1]
    , Xs [15,3,0]
    , Xs [15,4]
    , Xs [3,0,0,0,0,0]
    , Xs [3,0,0,0,1]
    , Xs [3,0,0,1,0]
    , Xs [3,0,0,2]
    , Xs [3,0,1,0,0]
    , Xs [3,0,1,1]
    , Xs [3,0,2,0]
    , Xs [3,0,15]
    , Xs [3,1,0,0,0]
    , Xs [3,1,0,1]
    , Xs [3,1,1,0]
    , Xs [3,1,2]
    , Xs [3,2,0,0]
    , Xs [3,2,1]
    , Xs [3,15,0]
    , Xs [3,3]
    , Xs [4,0,0,0,0]
    , Xs [4,0,0,1]
    , Xs [4,0,1,0]
    , Xs [4,0,2]
    , Xs [4,1,0,0]
    , Xs [4,1,1]
    , Xs [4,2,0]
    , Xs [4,15]
    , Xs [14,0,0,0]
    , Xs [14,0,1]
    , Xs [14,1,0]
    , Xs [14,2]
    , Xs [5,0,0]
    , Xs [5,1]
    , Xs [6,0]
    , Xs [13]
    ]
  , ...
  ]
