/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem:
 a__first(0(),X) -> nil()
 a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
 a__from(X) -> cons(mark(X),from(s(X)))
 mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
 mark(from(X)) -> a__from(mark(X))
 mark(0()) -> 0()
 mark(nil()) -> nil()
 mark(s(X)) -> s(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 a__first(X1,X2) -> first(X1,X2)
 a__from(X) -> from(X)

Proof:
 DP Processor:
  DPs:
   a__first#(s(X),cons(Y,Z)) -> mark#(Y)
   a__from#(X) -> mark#(X)
   mark#(first(X1,X2)) -> mark#(X2)
   mark#(first(X1,X2)) -> mark#(X1)
   mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
   mark#(from(X)) -> mark#(X)
   mark#(from(X)) -> a__from#(mark(X))
   mark#(s(X)) -> mark#(X)
   mark#(cons(X1,X2)) -> mark#(X1)
  TRS:
   a__first(0(),X) -> nil()
   a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
   a__from(X) -> cons(mark(X),from(s(X)))
   mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(0()) -> 0()
   mark(nil()) -> nil()
   mark(s(X)) -> s(mark(X))
   mark(cons(X1,X2)) -> cons(mark(X1),X2)
   a__first(X1,X2) -> first(X1,X2)
   a__from(X) -> from(X)
  TDG Processor:
   DPs:
    a__first#(s(X),cons(Y,Z)) -> mark#(Y)
    a__from#(X) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X2)
    mark#(first(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(from(X)) -> mark#(X)
    mark#(from(X)) -> a__from#(mark(X))
    mark#(s(X)) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1)
   TRS:
    a__first(0(),X) -> nil()
    a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
    a__from(X) -> cons(mark(X),from(s(X)))
    mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
    mark(from(X)) -> a__from(mark(X))
    mark(0()) -> 0()
    mark(nil()) -> nil()
    mark(s(X)) -> s(mark(X))
    mark(cons(X1,X2)) -> cons(mark(X1),X2)
    a__first(X1,X2) -> first(X1,X2)
    a__from(X) -> from(X)
   graph:
    a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X))
    a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X)
    a__from#(X) -> mark#(X) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    a__from#(X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1)
    a__from#(X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2)
    mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X)
    mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    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)) -> mark#(X) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(from(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1)
    mark#(from(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(from(X)) -> a__from#(mark(X))
    mark#(first(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(first(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(first(X1,X2)) -> mark#(X2)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(from(X)) -> a__from#(mark(X))
    mark#(first(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> mark#(X2)
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) ->
    a__first#(s(X),cons(Y,Z)) -> mark#(Y)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    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#(cons(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> mark#(X1)
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2)
    mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X))
    mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(s(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) ->
    mark#(first(X1,X2)) -> mark#(X2)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(s(X)) -> mark#(X)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(from(X)) -> a__from#(mark(X))
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(from(X)) -> mark#(X)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(first(X1,X2)) -> mark#(X1)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(first(X1,X2)) -> mark#(X2)
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     a__first(0(),X) -> nil()
     a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
     a__from(X) -> cons(mark(X),from(s(X)))
     mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
     mark(from(X)) -> a__from(mark(X))
     mark(0()) -> 0()
     mark(nil()) -> nil()
     mark(s(X)) -> s(mark(X))
     mark(cons(X1,X2)) -> cons(mark(X1),X2)
     a__first(X1,X2) -> first(X1,X2)
     a__from(X) -> from(X)
    interpretation:
     [a__from#](x0) = x0 + 4,
     
     [mark#](x0) = x0,
     
     [a__first#](x0, x1) = 7x0 + x1 + 0,
     
     [from](x0) = x0 + 4,
     
     [a__from](x0) = x0 + 4,
     
     [first](x0, x1) = 7x0 + x1 + 7,
     
     [mark](x0) = x0 + 0,
     
     [cons](x0, x1) = x0 + x1 + 4,
     
     [s](x0) = x0 + 0,
     
     [nil] = 2,
     
     [a__first](x0, x1) = 7x0 + x1 + 7,
     
     [0] = 0
    orientation:
     a__first#(s(X),cons(Y,Z)) = 7X + Y + Z + 7 >= Y = mark#(Y)
     
     a__from#(X) = X + 4 >= X = mark#(X)
     
     mark#(first(X1,X2)) = 7X1 + X2 + 7 >= X2 = mark#(X2)
     
     mark#(first(X1,X2)) = 7X1 + X2 + 7 >= X1 = mark#(X1)
     
     mark#(first(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = a__first#(mark(X1),mark(X2))
     
     mark#(from(X)) = X + 4 >= X = mark#(X)
     
     mark#(from(X)) = X + 4 >= X + 4 = a__from#(mark(X))
     
     mark#(s(X)) = X + 0 >= X = mark#(X)
     
     mark#(cons(X1,X2)) = X1 + X2 + 4 >= X1 = mark#(X1)
     
     a__first(0(),X) = X + 7 >= 2 = nil()
     
     a__first(s(X),cons(Y,Z)) = 7X + Y + Z + 7 >= 7X + Y + Z + 7 = cons(mark(Y),first(X,Z))
     
     a__from(X) = X + 4 >= X + 4 = cons(mark(X),from(s(X)))
     
     mark(first(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = a__first(mark(X1),mark(X2))
     
     mark(from(X)) = X + 4 >= X + 4 = a__from(mark(X))
     
     mark(0()) = 0 >= 0 = 0()
     
     mark(nil()) = 2 >= 2 = nil()
     
     mark(s(X)) = X + 0 >= X + 0 = s(mark(X))
     
     mark(cons(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = cons(mark(X1),X2)
     
     a__first(X1,X2) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = first(X1,X2)
     
     a__from(X) = X + 4 >= X + 4 = from(X)
    problem:
     DPs:
      a__first#(s(X),cons(Y,Z)) -> mark#(Y)
      a__from#(X) -> mark#(X)
      mark#(first(X1,X2)) -> mark#(X2)
      mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
      mark#(from(X)) -> mark#(X)
      mark#(from(X)) -> a__from#(mark(X))
      mark#(s(X)) -> mark#(X)
      mark#(cons(X1,X2)) -> mark#(X1)
     TRS:
      a__first(0(),X) -> nil()
      a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
      a__from(X) -> cons(mark(X),from(s(X)))
      mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
      mark(from(X)) -> a__from(mark(X))
      mark(0()) -> 0()
      mark(nil()) -> nil()
      mark(s(X)) -> s(mark(X))
      mark(cons(X1,X2)) -> cons(mark(X1),X2)
      a__first(X1,X2) -> first(X1,X2)
      a__from(X) -> from(X)
    Restore Modifier:
     DPs:
      a__first#(s(X),cons(Y,Z)) -> mark#(Y)
      a__from#(X) -> mark#(X)
      mark#(first(X1,X2)) -> mark#(X2)
      mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
      mark#(from(X)) -> mark#(X)
      mark#(from(X)) -> a__from#(mark(X))
      mark#(s(X)) -> mark#(X)
      mark#(cons(X1,X2)) -> mark#(X1)
     TRS:
      a__first(0(),X) -> nil()
      a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
      a__from(X) -> cons(mark(X),from(s(X)))
      mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
      mark(from(X)) -> a__from(mark(X))
      mark(0()) -> 0()
      mark(nil()) -> nil()
      mark(s(X)) -> s(mark(X))
      mark(cons(X1,X2)) -> cons(mark(X1),X2)
      a__first(X1,X2) -> first(X1,X2)
      a__from(X) -> from(X)
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       a__first(0(),X) -> nil()
       a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
       a__from(X) -> cons(mark(X),from(s(X)))
       mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
       mark(from(X)) -> a__from(mark(X))
       mark(0()) -> 0()
       mark(nil()) -> nil()
       mark(s(X)) -> s(mark(X))
       mark(cons(X1,X2)) -> cons(mark(X1),X2)
       a__first(X1,X2) -> first(X1,X2)
       a__from(X) -> from(X)
      interpretation:
       [a__from#](x0) = 1x0 + -16,
       
       [mark#](x0) = x0,
       
       [a__first#](x0, x1) = 1x0 + 1x1 + -16,
       
       [from](x0) = 1x0 + 7,
       
       [a__from](x0) = 1x0 + 7,
       
       [first](x0, x1) = 2x0 + 2x1 + 0,
       
       [mark](x0) = x0 + -9,
       
       [cons](x0, x1) = 1x0 + -1,
       
       [s](x0) = 1x0 + -5,
       
       [nil] = 7,
       
       [a__first](x0, x1) = 2x0 + 2x1 + 0,
       
       [0] = 5
      orientation:
       a__first#(s(X),cons(Y,Z)) = 2X + 2Y + 0 >= Y = mark#(Y)
       
       a__from#(X) = 1X + -16 >= X = mark#(X)
       
       mark#(first(X1,X2)) = 2X1 + 2X2 + 0 >= X2 = mark#(X2)
       
       mark#(first(X1,X2)) = 2X1 + 2X2 + 0 >= 1X1 + 1X2 + -8 = a__first#(mark(X1),mark(X2))
       
       mark#(from(X)) = 1X + 7 >= X = mark#(X)
       
       mark#(from(X)) = 1X + 7 >= 1X + -8 = a__from#(mark(X))
       
       mark#(s(X)) = 1X + -5 >= X = mark#(X)
       
       mark#(cons(X1,X2)) = 1X1 + -1 >= X1 = mark#(X1)
       
       a__first(0(),X) = 2X + 7 >= 7 = nil()
       
       a__first(s(X),cons(Y,Z)) = 3X + 3Y + 1 >= 1Y + -1 = cons(mark(Y),first(X,Z))
       
       a__from(X) = 1X + 7 >= 1X + -1 = cons(mark(X),from(s(X)))
       
       mark(first(X1,X2)) = 2X1 + 2X2 + 0 >= 2X1 + 2X2 + 0 = a__first(mark(X1),mark(X2))
       
       mark(from(X)) = 1X + 7 >= 1X + 7 = a__from(mark(X))
       
       mark(0()) = 5 >= 5 = 0()
       
       mark(nil()) = 7 >= 7 = nil()
       
       mark(s(X)) = 1X + -5 >= 1X + -5 = s(mark(X))
       
       mark(cons(X1,X2)) = 1X1 + -1 >= 1X1 + -1 = cons(mark(X1),X2)
       
       a__first(X1,X2) = 2X1 + 2X2 + 0 >= 2X1 + 2X2 + 0 = first(X1,X2)
       
       a__from(X) = 1X + 7 >= 1X + 7 = from(X)
      problem:
       DPs:
        mark#(from(X)) -> a__from#(mark(X))
       TRS:
        a__first(0(),X) -> nil()
        a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
        a__from(X) -> cons(mark(X),from(s(X)))
        mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
        mark(from(X)) -> a__from(mark(X))
        mark(0()) -> 0()
        mark(nil()) -> nil()
        mark(s(X)) -> s(mark(X))
        mark(cons(X1,X2)) -> cons(mark(X1),X2)
        a__first(X1,X2) -> first(X1,X2)
        a__from(X) -> from(X)
      Restore Modifier:
       DPs:
        mark#(from(X)) -> a__from#(mark(X))
       TRS:
        a__first(0(),X) -> nil()
        a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
        a__from(X) -> cons(mark(X),from(s(X)))
        mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
        mark(from(X)) -> a__from(mark(X))
        mark(0()) -> 0()
        mark(nil()) -> nil()
        mark(s(X)) -> s(mark(X))
        mark(cons(X1,X2)) -> cons(mark(X1),X2)
        a__first(X1,X2) -> first(X1,X2)
        a__from(X) -> from(X)
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 51/1