/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem:
 a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
 a(a(x)) -> f(b(),a(f(a(x),b())))
 f(a(x),b()) -> f(b(),a(x))

Proof:
 DP Processor:
  DPs:
   a#(a(f(b(),a(x)))) -> a#(a(x))
   a#(a(f(b(),a(x)))) -> a#(a(a(x)))
   a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
   a#(a(x)) -> f#(a(x),b())
   a#(a(x)) -> a#(f(a(x),b()))
   a#(a(x)) -> f#(b(),a(f(a(x),b())))
   f#(a(x),b()) -> f#(b(),a(x))
  TRS:
   a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
   a(a(x)) -> f(b(),a(f(a(x),b())))
   f(a(x),b()) -> f(b(),a(x))
  TDG Processor:
   DPs:
    a#(a(f(b(),a(x)))) -> a#(a(x))
    a#(a(f(b(),a(x)))) -> a#(a(a(x)))
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
    a#(a(x)) -> f#(a(x),b())
    a#(a(x)) -> a#(f(a(x),b()))
    a#(a(x)) -> f#(b(),a(f(a(x),b())))
    f#(a(x),b()) -> f#(b(),a(x))
   TRS:
    a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
    a(a(x)) -> f(b(),a(f(a(x),b())))
    f(a(x),b()) -> f(b(),a(x))
   graph:
    f#(a(x),b()) -> f#(b(),a(x)) ->
    f#(a(x),b()) -> f#(b(),a(x))
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b()) ->
    f#(a(x),b()) -> f#(b(),a(x))
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(x)) -> f#(b(),a(f(a(x),b())))
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(x)) -> a#(f(a(x),b()))
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) -> a#(a(x)) -> f#(a(x),b())
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(f(b(),a(x)))) -> a#(a(a(x)))
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(f(b(),a(x)))) -> a#(a(x))
    a#(a(f(b(),a(x)))) -> a#(a(x)) ->
    a#(a(x)) -> f#(b(),a(f(a(x),b())))
    a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(x)) -> a#(f(a(x),b()))
    a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(x)) -> f#(a(x),b())
    a#(a(f(b(),a(x)))) -> a#(a(x)) ->
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
    a#(a(f(b(),a(x)))) -> a#(a(x)) ->
    a#(a(f(b(),a(x)))) -> a#(a(a(x)))
    a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(f(b(),a(x)))) -> a#(a(x))
    a#(a(x)) -> f#(a(x),b()) -> f#(a(x),b()) -> f#(b(),a(x))
    a#(a(x)) -> f#(b(),a(f(a(x),b()))) -> f#(a(x),b()) -> f#(b(),a(x))
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(x)) -> f#(b(),a(f(a(x),b())))
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(x)) -> a#(f(a(x),b()))
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(x)) -> f#(a(x),b())
    a#(a(x)) -> a#(f(a(x),b())) ->
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(f(b(),a(x)))) -> a#(a(a(x)))
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(f(b(),a(x)))) -> a#(a(x))
   SCC Processor:
    #sccs: 2
    #rules: 4
    #arcs: 22/49
    DPs:
     a#(a(f(b(),a(x)))) -> a#(a(a(x)))
     a#(a(f(b(),a(x)))) -> a#(a(x))
     a#(a(x)) -> a#(f(a(x),b()))
    TRS:
     a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
     a(a(x)) -> f(b(),a(f(a(x),b())))
     f(a(x),b()) -> f(b(),a(x))
    EDG Processor:
     DPs:
      a#(a(f(b(),a(x)))) -> a#(a(a(x)))
      a#(a(f(b(),a(x)))) -> a#(a(x))
      a#(a(x)) -> a#(f(a(x),b()))
     TRS:
      a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
      a(a(x)) -> f(b(),a(f(a(x),b())))
      f(a(x),b()) -> f(b(),a(x))
     graph:
      a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
      a#(a(f(b(),a(x)))) -> a#(a(x))
      a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
      a#(a(f(b(),a(x)))) -> a#(a(a(x)))
      a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
      a#(a(x)) -> a#(f(a(x),b()))
      a#(a(f(b(),a(x)))) -> a#(a(x)) ->
      a#(a(f(b(),a(x)))) -> a#(a(x))
      a#(a(f(b(),a(x)))) -> a#(a(x)) ->
      a#(a(f(b(),a(x)))) -> a#(a(a(x)))
      a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(x)) -> a#(f(a(x),b()))
     SCC Processor:
      #sccs: 1
      #rules: 2
      #arcs: 6/9
      DPs:
       a#(a(f(b(),a(x)))) -> a#(a(a(x)))
       a#(a(f(b(),a(x)))) -> a#(a(x))
      TRS:
       a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
       a(a(x)) -> f(b(),a(f(a(x),b())))
       f(a(x),b()) -> f(b(),a(x))
      Semantic Labeling Processor:
       dimension: 1
       usable rules:
        
       interpretation:
        [f](x0, x1) = x0 + x1,
        
        [a](x0) = x0 + 1,
        
        [b] = 0
        labeled:
         a#
        usable (for model):
         a#
         a
         f
         b
        argument filtering:
         pi(b) = []
         pi(a) = []
         pi(f) = 0
         pi(a#) = []
       precedence:
        a# ~ f ~ a ~ b
       problem:
        DPs:
         a#(a(f(b(),a(x)))) -> a#(a(a(x)))
        TRS:
         a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
         a(a(x)) -> f(b(),a(f(a(x),b())))
         f(a(x),b()) -> f(b(),a(x))
       Restore Modifier:
        DPs:
         a#(a(f(b(),a(x)))) -> a#(a(a(x)))
        TRS:
         a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
         a(a(x)) -> f(b(),a(f(a(x),b())))
         f(a(x),b()) -> f(b(),a(x))
        Semantic Labeling Processor:
         dimension: 2
         usable rules:
          
         interpretation:
                        [0 0]     [0 0]     [0]
          [f](x0, x1) = [0 1]x0 + [0 1]x1 + [1],
          
                    [0 1]     [1]
          [a](x0) = [1 0]x0 + [0],
          
                [0]
          [b] = [0]
          labeled:
           a#
          usable (for model):
           a#
           a
           f
           b
          argument filtering:
           pi(b) = []
           pi(a) = 0
           pi(f) = []
           pi(a#) = []
         precedence:
          a# ~ f ~ a ~ b
         problem:
          DPs:
           
          TRS:
           a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
           a(a(x)) -> f(b(),a(f(a(x),b())))
           f(a(x),b()) -> f(b(),a(x))
         Qed
    
    DPs:
     f#(a(x),b()) -> f#(b(),a(x))
    TRS:
     a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
     a(a(x)) -> f(b(),a(f(a(x),b())))
     f(a(x),b()) -> f(b(),a(x))
    EDG Processor:
     DPs:
      f#(a(x),b()) -> f#(b(),a(x))
     TRS:
      a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
      a(a(x)) -> f(b(),a(f(a(x),b())))
      f(a(x),b()) -> f(b(),a(x))
     graph:
      
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 0/1