MAYBE

Problem:
 a__from(X) -> cons(mark(X),from(s(X)))
 a__sel(0(),cons(X,Y)) -> mark(X)
 a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
 mark(from(X)) -> a__from(mark(X))
 mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(s(X)) -> s(mark(X))
 mark(0()) -> 0()
 a__from(X) -> from(X)
 a__sel(X1,X2) -> sel(X1,X2)

Proof:
 DP Processor:
  DPs:
   a__from#(X) -> mark#(X)
   a__sel#(0(),cons(X,Y)) -> mark#(X)
   a__sel#(s(X),cons(Y,Z)) -> mark#(Z)
   a__sel#(s(X),cons(Y,Z)) -> mark#(X)
   a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z))
   mark#(from(X)) -> mark#(X)
   mark#(from(X)) -> a__from#(mark(X))
   mark#(sel(X1,X2)) -> mark#(X2)
   mark#(sel(X1,X2)) -> mark#(X1)
   mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
   mark#(cons(X1,X2)) -> mark#(X1)
   mark#(s(X)) -> mark#(X)
  TRS:
   a__from(X) -> cons(mark(X),from(s(X)))
   a__sel(0(),cons(X,Y)) -> mark(X)
   a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
   mark(from(X)) -> a__from(mark(X))
   mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
   mark(cons(X1,X2)) -> cons(mark(X1),X2)
   mark(s(X)) -> s(mark(X))
   mark(0()) -> 0()
   a__from(X) -> from(X)
   a__sel(X1,X2) -> sel(X1,X2)
  TDG Processor:
   DPs:
    a__from#(X) -> mark#(X)
    a__sel#(0(),cons(X,Y)) -> mark#(X)
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z)
    a__sel#(s(X),cons(Y,Z)) -> mark#(X)
    a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z))
    mark#(from(X)) -> mark#(X)
    mark#(from(X)) -> a__from#(mark(X))
    mark#(sel(X1,X2)) -> mark#(X2)
    mark#(sel(X1,X2)) -> mark#(X1)
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X)
   TRS:
    a__from(X) -> cons(mark(X),from(s(X)))
    a__sel(0(),cons(X,Y)) -> mark(X)
    a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
    mark(from(X)) -> a__from(mark(X))
    mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
    mark(cons(X1,X2)) -> cons(mark(X1),X2)
    mark(s(X)) -> s(mark(X))
    mark(0()) -> 0()
    a__from(X) -> from(X)
    a__sel(X1,X2) -> sel(X1,X2)
   graph:
    a__sel#(0(),cons(X,Y)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    a__sel#(0(),cons(X,Y)) -> mark#(X) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    a__sel#(0(),cons(X,Y)) -> mark#(X) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    a__sel#(0(),cons(X,Y)) -> mark#(X) ->
    mark#(sel(X1,X2)) -> mark#(X1)
    a__sel#(0(),cons(X,Y)) -> mark#(X) ->
    mark#(sel(X1,X2)) -> mark#(X2)
    a__sel#(0(),cons(X,Y)) -> mark#(X) ->
    mark#(from(X)) -> a__from#(mark(X))
    a__sel#(0(),cons(X,Y)) -> mark#(X) ->
    mark#(from(X)) -> mark#(X)
    a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z)) ->
    a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z))
    a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z)) ->
    a__sel#(s(X),cons(Y,Z)) -> mark#(X)
    a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z)) ->
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z)
    a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z)) ->
    a__sel#(0(),cons(X,Y)) -> mark#(X)
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z) ->
    mark#(s(X)) -> mark#(X)
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z) ->
    mark#(sel(X1,X2)) -> mark#(X1)
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z) ->
    mark#(sel(X1,X2)) -> mark#(X2)
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z) ->
    mark#(from(X)) -> a__from#(mark(X))
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z) ->
    mark#(from(X)) -> mark#(X)
    a__sel#(s(X),cons(Y,Z)) -> mark#(X) ->
    mark#(s(X)) -> mark#(X)
    a__sel#(s(X),cons(Y,Z)) -> mark#(X) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    a__sel#(s(X),cons(Y,Z)) -> mark#(X) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    a__sel#(s(X),cons(Y,Z)) -> mark#(X) ->
    mark#(sel(X1,X2)) -> mark#(X1)
    a__sel#(s(X),cons(Y,Z)) -> mark#(X) ->
    mark#(sel(X1,X2)) -> mark#(X2)
    a__sel#(s(X),cons(Y,Z)) -> mark#(X) ->
    mark#(from(X)) -> a__from#(mark(X))
    a__sel#(s(X),cons(Y,Z)) -> mark#(X) ->
    mark#(from(X)) -> mark#(X)
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) ->
    a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z))
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) ->
    a__sel#(s(X),cons(Y,Z)) -> mark#(X)
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) ->
    a__sel#(s(X),cons(Y,Z)) -> mark#(Z)
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) ->
    a__sel#(0(),cons(X,Y)) -> mark#(X)
    mark#(sel(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X)
    mark#(sel(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(sel(X1,X2)) -> mark#(X2) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    mark#(sel(X1,X2)) -> mark#(X2) -> mark#(sel(X1,X2)) -> mark#(X1)
    mark#(sel(X1,X2)) -> mark#(X2) -> mark#(sel(X1,X2)) -> mark#(X2)
    mark#(sel(X1,X2)) -> mark#(X2) ->
    mark#(from(X)) -> a__from#(mark(X))
    mark#(sel(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X)
    mark#(sel(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(sel(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(sel(X1,X2)) -> mark#(X1) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    mark#(sel(X1,X2)) -> mark#(X1) -> mark#(sel(X1,X2)) -> mark#(X1)
    mark#(sel(X1,X2)) -> mark#(X1) -> mark#(sel(X1,X2)) -> mark#(X2)
    mark#(sel(X1,X2)) -> mark#(X1) ->
    mark#(from(X)) -> a__from#(mark(X))
    mark#(sel(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(sel(X1,X2)) -> mark#(X1)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(sel(X1,X2)) -> mark#(X2)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(from(X)) -> a__from#(mark(X))
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X)
    mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(from(X)) -> mark#(X) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    mark#(from(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1)
    mark#(from(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2)
    mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X))
    mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X)
    mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    mark#(s(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2)
    mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X))
    mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X)
    a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    a__from#(X) -> mark#(X) ->
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    a__from#(X) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1)
    a__from#(X) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2)
    a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X))
    a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [a__sel#](x0, x1) = x0 + x1,
     
     [mark#](x0) = x0,
     
     [a__from#](x0) = x0,
     
     [sel](x0, x1) = 7x0 + 6x1 + 0,
     
     [a__sel](x0, x1) = 7x0 + 6x1 + 0,
     
     [0] = 0,
     
     [cons](x0, x1) = x0 + x1 + 0,
     
     [from](x0) = x0 + 0,
     
     [s](x0) = x0 + 0,
     
     [mark](x0) = x0,
     
     [a__from](x0) = x0 + 0
    orientation:
     a__from#(X) = X >= X = mark#(X)
     
     a__sel#(0(),cons(X,Y)) = X + Y + 0 >= X = mark#(X)
     
     a__sel#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Z = mark#(Z)
     
     a__sel#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X = mark#(X)
     
     a__sel#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + Z = a__sel#(mark(X),mark(Z))
     
     mark#(from(X)) = X + 0 >= X = mark#(X)
     
     mark#(from(X)) = X + 0 >= X = a__from#(mark(X))
     
     mark#(sel(X1,X2)) = 7X1 + 6X2 + 0 >= X2 = mark#(X2)
     
     mark#(sel(X1,X2)) = 7X1 + 6X2 + 0 >= X1 = mark#(X1)
     
     mark#(sel(X1,X2)) = 7X1 + 6X2 + 0 >= X1 + X2 = a__sel#(mark(X1),mark(X2))
     
     mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1)
     
     mark#(s(X)) = X + 0 >= X = mark#(X)
     
     a__from(X) = X + 0 >= X + 0 = cons(mark(X),from(s(X)))
     
     a__sel(0(),cons(X,Y)) = 6X + 6Y + 7 >= X = mark(X)
     
     a__sel(s(X),cons(Y,Z)) = 7X + 6Y + 6Z + 7 >= 7X + 6Z + 0 = a__sel(mark(X),mark(Z))
     
     mark(from(X)) = X + 0 >= X + 0 = a__from(mark(X))
     
     mark(sel(X1,X2)) = 7X1 + 6X2 + 0 >= 7X1 + 6X2 + 0 = a__sel(mark(X1),mark(X2))
     
     mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2)
     
     mark(s(X)) = X + 0 >= X + 0 = s(mark(X))
     
     mark(0()) = 0 >= 0 = 0()
     
     a__from(X) = X + 0 >= X + 0 = from(X)
     
     a__sel(X1,X2) = 7X1 + 6X2 + 0 >= 7X1 + 6X2 + 0 = sel(X1,X2)
    problem:
     DPs:
      a__from#(X) -> mark#(X)
      a__sel#(0(),cons(X,Y)) -> mark#(X)
      a__sel#(s(X),cons(Y,Z)) -> mark#(Z)
      a__sel#(s(X),cons(Y,Z)) -> mark#(X)
      a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z))
      mark#(from(X)) -> mark#(X)
      mark#(from(X)) -> a__from#(mark(X))
      mark#(cons(X1,X2)) -> mark#(X1)
      mark#(s(X)) -> mark#(X)
     TRS:
      a__from(X) -> cons(mark(X),from(s(X)))
      a__sel(0(),cons(X,Y)) -> mark(X)
      a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
      mark(from(X)) -> a__from(mark(X))
      mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
      mark(cons(X1,X2)) -> cons(mark(X1),X2)
      mark(s(X)) -> s(mark(X))
      mark(0()) -> 0()
      a__from(X) -> from(X)
      a__sel(X1,X2) -> sel(X1,X2)
    SCC Processor:
     #sccs: 2
     #rules: 6
     #arcs: 72/81
     DPs:
      a__sel#(s(X),cons(Y,Z)) -> a__sel#(mark(X),mark(Z))
     TRS:
      a__from(X) -> cons(mark(X),from(s(X)))
      a__sel(0(),cons(X,Y)) -> mark(X)
      a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
      mark(from(X)) -> a__from(mark(X))
      mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
      mark(cons(X1,X2)) -> cons(mark(X1),X2)
      mark(s(X)) -> s(mark(X))
      mark(0()) -> 0()
      a__from(X) -> from(X)
      a__sel(X1,X2) -> sel(X1,X2)
     Open
     
     DPs:
      mark#(from(X)) -> mark#(X)
      mark#(from(X)) -> a__from#(mark(X))
      a__from#(X) -> mark#(X)
      mark#(cons(X1,X2)) -> mark#(X1)
      mark#(s(X)) -> mark#(X)
     TRS:
      a__from(X) -> cons(mark(X),from(s(X)))
      a__sel(0(),cons(X,Y)) -> mark(X)
      a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
      mark(from(X)) -> a__from(mark(X))
      mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
      mark(cons(X1,X2)) -> cons(mark(X1),X2)
      mark(s(X)) -> s(mark(X))
      mark(0()) -> 0()
      a__from(X) -> from(X)
      a__sel(X1,X2) -> sel(X1,X2)
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [mark#](x0) = 3x0,
       
       [a__from#](x0) = 4x0,
       
       [sel](x0, x1) = x1,
       
       [a__sel](x0, x1) = x1,
       
       [0] = 4,
       
       [cons](x0, x1) = 1x0 + x1,
       
       [from](x0) = 1x0,
       
       [s](x0) = x0,
       
       [mark](x0) = x0,
       
       [a__from](x0) = 1x0
      orientation:
       mark#(from(X)) = 4X >= 3X = mark#(X)
       
       mark#(from(X)) = 4X >= 4X = a__from#(mark(X))
       
       a__from#(X) = 4X >= 3X = mark#(X)
       
       mark#(cons(X1,X2)) = 4X1 + 3X2 >= 3X1 = mark#(X1)
       
       mark#(s(X)) = 3X >= 3X = mark#(X)
       
       a__from(X) = 1X >= 1X = cons(mark(X),from(s(X)))
       
       a__sel(0(),cons(X,Y)) = 1X + Y >= X = mark(X)
       
       a__sel(s(X),cons(Y,Z)) = 1Y + Z >= Z = a__sel(mark(X),mark(Z))
       
       mark(from(X)) = 1X >= 1X = a__from(mark(X))
       
       mark(sel(X1,X2)) = X2 >= X2 = a__sel(mark(X1),mark(X2))
       
       mark(cons(X1,X2)) = 1X1 + X2 >= 1X1 + X2 = cons(mark(X1),X2)
       
       mark(s(X)) = X >= X = s(mark(X))
       
       mark(0()) = 4 >= 4 = 0()
       
       a__from(X) = 1X >= 1X = from(X)
       
       a__sel(X1,X2) = X2 >= X2 = sel(X1,X2)
      problem:
       DPs:
        mark#(from(X)) -> a__from#(mark(X))
        mark#(s(X)) -> mark#(X)
       TRS:
        a__from(X) -> cons(mark(X),from(s(X)))
        a__sel(0(),cons(X,Y)) -> mark(X)
        a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
        mark(from(X)) -> a__from(mark(X))
        mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
        mark(cons(X1,X2)) -> cons(mark(X1),X2)
        mark(s(X)) -> s(mark(X))
        mark(0()) -> 0()
        a__from(X) -> from(X)
        a__sel(X1,X2) -> sel(X1,X2)
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 17/4
       DPs:
        mark#(s(X)) -> mark#(X)
       TRS:
        a__from(X) -> cons(mark(X),from(s(X)))
        a__sel(0(),cons(X,Y)) -> mark(X)
        a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
        mark(from(X)) -> a__from(mark(X))
        mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
        mark(cons(X1,X2)) -> cons(mark(X1),X2)
        mark(s(X)) -> s(mark(X))
        mark(0()) -> 0()
        a__from(X) -> from(X)
        a__sel(X1,X2) -> sel(X1,X2)
       Subterm Criterion Processor:
        simple projection:
         pi(mark#) = 0
        problem:
         DPs:
          
         TRS:
          a__from(X) -> cons(mark(X),from(s(X)))
          a__sel(0(),cons(X,Y)) -> mark(X)
          a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z))
          mark(from(X)) -> a__from(mark(X))
          mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
          mark(cons(X1,X2)) -> cons(mark(X1),X2)
          mark(s(X)) -> s(mark(X))
          mark(0()) -> 0()
          a__from(X) -> from(X)
          a__sel(X1,X2) -> sel(X1,X2)
        Qed