map length (tiers :: [[ Xs Nat7 ]])  =  [0,1,1,2,4,8,16,32,64,127,253,504,...]

length (list :: [ Xs Nat7 ])  =  Infinity

allUnique (list :: [ Xs Nat7 ])  =  True

ratioRepetitions (list :: [ Xs Nat7 ])  =  0 % 1

tiers :: [Xs Nat7]  =
  [ []
  , [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 [6]
    ]
  , [ 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 [0,6]
    , Xs [1,0,0,0]
    , Xs [1,0,1]
    , Xs [1,1,0]
    , Xs [1,2]
    , Xs [2,0,0]
    , Xs [2,1]
    , Xs [6,0]
    , 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 [0,0,6]
    , 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 [0,6,0]
    , 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 [1,6]
    , Xs [2,0,0,0]
    , Xs [2,0,1]
    , Xs [2,1,0]
    , Xs [2,2]
    , Xs [6,0,0]
    , Xs [6,1]
    , 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 [0,0,0,6]
    , 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 [0,0,6,0]
    , 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 [0,1,6]
    , Xs [0,2,0,0,0]
    , Xs [0,2,0,1]
    , Xs [0,2,1,0]
    , Xs [0,2,2]
    , Xs [0,6,0,0]
    , Xs [0,6,1]
    , 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 [1,0,6]
    , 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 [1,6,0]
    , 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 [2,6]
    , Xs [6,0,0,0]
    , Xs [6,0,1]
    , Xs [6,1,0]
    , Xs [6,2]
    , Xs [3,0,0]
    , Xs [3,1]
    , Xs [4,0]
    , Xs [5]
    ]
  , [ 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 [0,0,0,0,6]
    , 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 [0,0,0,6,0]
    , 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 [0,0,1,6]
    , 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 [0,0,6,0,0]
    , Xs [0,0,6,1]
    , 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 [0,1,0,6]
    , 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 [0,1,6,0]
    , 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 [0,2,6]
    , Xs [0,6,0,0,0]
    , Xs [0,6,0,1]
    , Xs [0,6,1,0]
    , Xs [0,6,2]
    , Xs [0,3,0,0]
    , Xs [0,3,1]
    , Xs [0,4,0]
    , Xs [0,5]
    , 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 [1,0,0,6]
    , 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 [1,0,6,0]
    , 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 [1,1,6]
    , Xs [1,2,0,0,0]
    , Xs [1,2,0,1]
    , Xs [1,2,1,0]
    , Xs [1,2,2]
    , Xs [1,6,0,0]
    , Xs [1,6,1]
    , 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 [2,0,6]
    , 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 [2,6,0]
    , Xs [2,3]
    , Xs [6,0,0,0,0]
    , Xs [6,0,0,1]
    , Xs [6,0,1,0]
    , Xs [6,0,2]
    , Xs [6,1,0,0]
    , Xs [6,1,1]
    , Xs [6,2,0]
    , Xs [6,6]
    , Xs [3,0,0,0]
    , Xs [3,0,1]
    , Xs [3,1,0]
    , Xs [3,2]
    , Xs [4,0,0]
    , Xs [4,1]
    , Xs [5,0]
    ]
  , [ 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 [0,0,0,0,0,6]
    , 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 [0,0,0,0,6,0]
    , 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 [0,0,0,1,6]
    , 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 [0,0,0,6,0,0]
    , Xs [0,0,0,6,1]
    , 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 [0,0,1,0,6]
    , 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 [0,0,1,6,0]
    , 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 [0,0,2,6]
    , Xs [0,0,6,0,0,0]
    , Xs [0,0,6,0,1]
    , Xs [0,0,6,1,0]
    , Xs [0,0,6,2]
    , Xs [0,0,3,0,0]
    , Xs [0,0,3,1]
    , Xs [0,0,4,0]
    , Xs [0,0,5]
    , 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 [0,1,0,0,6]
    , 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 [0,1,0,6,0]
    , 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 [0,1,1,6]
    , 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 [0,1,6,0,0]
    , Xs [0,1,6,1]
    , 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 [0,2,0,6]
    , 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 [0,2,6,0]
    , Xs [0,2,3]
    , Xs [0,6,0,0,0,0]
    , Xs [0,6,0,0,1]
    , Xs [0,6,0,1,0]
    , Xs [0,6,0,2]
    , Xs [0,6,1,0,0]
    , Xs [0,6,1,1]
    , Xs [0,6,2,0]
    , Xs [0,6,6]
    , 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 [0,5,0]
    , 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 [1,0,0,0,6]
    , 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 [1,0,0,6,0]
    , 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 [1,0,1,6]
    , 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 [1,0,6,0,0]
    , Xs [1,0,6,1]
    , 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 [1,1,0,6]
    , 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 [1,1,6,0]
    , 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 [1,2,6]
    , Xs [1,6,0,0,0]
    , Xs [1,6,0,1]
    , Xs [1,6,1,0]
    , Xs [1,6,2]
    , Xs [1,3,0,0]
    , Xs [1,3,1]
    , Xs [1,4,0]
    , Xs [1,5]
    , 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 [2,0,0,6]
    , 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 [2,0,6,0]
    , 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 [2,1,6]
    , Xs [2,2,0,0,0]
    , Xs [2,2,0,1]
    , Xs [2,2,1,0]
    , Xs [2,2,2]
    , Xs [2,6,0,0]
    , Xs [2,6,1]
    , Xs [2,3,0]
    , Xs [2,4]
    , Xs [6,0,0,0,0,0]
    , Xs [6,0,0,0,1]
    , Xs [6,0,0,1,0]
    , Xs [6,0,0,2]
    , Xs [6,0,1,0,0]
    , Xs [6,0,1,1]
    , Xs [6,0,2,0]
    , Xs [6,0,6]
    , Xs [6,1,0,0,0]
    , Xs [6,1,0,1]
    , Xs [6,1,1,0]
    , Xs [6,1,2]
    , Xs [6,2,0,0]
    , Xs [6,2,1]
    , Xs [6,6,0]
    , Xs [6,3]
    , Xs [3,0,0,0,0]
    , Xs [3,0,0,1]
    , Xs [3,0,1,0]
    , Xs [3,0,2]
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    , Xs [1,6,0,2]
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    , Xs [1,6,6]
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    , Xs [1,3,1,0]
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    , 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 [2,2,6]
    , Xs [2,6,0,0,0]
    , Xs [2,6,0,1]
    , Xs [2,6,1,0]
    , Xs [2,6,2]
    , Xs [2,3,0,0]
    , Xs [2,3,1]
    , Xs [2,4,0]
    , Xs [2,5]
    , Xs [6,0,0,0,0,0,0]
    , Xs [6,0,0,0,0,1]
    , Xs [6,0,0,0,1,0]
    , Xs [6,0,0,0,2]
    , Xs [6,0,0,1,0,0]
    , Xs [6,0,0,1,1]
    , Xs [6,0,0,2,0]
    , Xs [6,0,0,6]
    , Xs [6,0,1,0,0,0]
    , Xs [6,0,1,0,1]
    , Xs [6,0,1,1,0]
    , Xs [6,0,1,2]
    , Xs [6,0,2,0,0]
    , Xs [6,0,2,1]
    , Xs [6,0,6,0]
    , Xs [6,0,3]
    , Xs [6,1,0,0,0,0]
    , Xs [6,1,0,0,1]
    , Xs [6,1,0,1,0]
    , Xs [6,1,0,2]
    , Xs [6,1,1,0,0]
    , Xs [6,1,1,1]
    , Xs [6,1,2,0]
    , Xs [6,1,6]
    , Xs [6,2,0,0,0]
    , Xs [6,2,0,1]
    , Xs [6,2,1,0]
    , Xs [6,2,2]
    , Xs [6,6,0,0]
    , Xs [6,6,1]
    , Xs [6,3,0]
    , Xs [6,4]
    , 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 [3,0,6]
    , 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 [3,6,0]
    , 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 [4,6]
    , Xs [5,0,0,0]
    , Xs [5,0,1]
    , Xs [5,1,0]
    , Xs [5,2]
    ]
  , ...
  ]
