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


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YES

Problem:
 f(s(x)) -> f(id_inc(c(x,x)))
 f(c(s(x),y)) -> g(c(x,y))
 g(c(s(x),y)) -> g(c(y,x))
 g(c(x,s(y))) -> g(c(y,x))
 g(c(x,x)) -> f(x)
 id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
 id_inc(s(x)) -> s(id_inc(x))
 id_inc(0()) -> 0()
 id_inc(0()) -> s(0())

Proof:
 DP Processor:
  DPs:
   f#(s(x)) -> id_inc#(c(x,x))
   f#(s(x)) -> f#(id_inc(c(x,x)))
   f#(c(s(x),y)) -> g#(c(x,y))
   g#(c(s(x),y)) -> g#(c(y,x))
   g#(c(x,s(y))) -> g#(c(y,x))
   g#(c(x,x)) -> f#(x)
   id_inc#(c(x,y)) -> id_inc#(y)
   id_inc#(c(x,y)) -> id_inc#(x)
   id_inc#(s(x)) -> id_inc#(x)
  TRS:
   f(s(x)) -> f(id_inc(c(x,x)))
   f(c(s(x),y)) -> g(c(x,y))
   g(c(s(x),y)) -> g(c(y,x))
   g(c(x,s(y))) -> g(c(y,x))
   g(c(x,x)) -> f(x)
   id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
   id_inc(s(x)) -> s(id_inc(x))
   id_inc(0()) -> 0()
   id_inc(0()) -> s(0())
  TDG Processor:
   DPs:
    f#(s(x)) -> id_inc#(c(x,x))
    f#(s(x)) -> f#(id_inc(c(x,x)))
    f#(c(s(x),y)) -> g#(c(x,y))
    g#(c(s(x),y)) -> g#(c(y,x))
    g#(c(x,s(y))) -> g#(c(y,x))
    g#(c(x,x)) -> f#(x)
    id_inc#(c(x,y)) -> id_inc#(y)
    id_inc#(c(x,y)) -> id_inc#(x)
    id_inc#(s(x)) -> id_inc#(x)
   TRS:
    f(s(x)) -> f(id_inc(c(x,x)))
    f(c(s(x),y)) -> g(c(x,y))
    g(c(s(x),y)) -> g(c(y,x))
    g(c(x,s(y))) -> g(c(y,x))
    g(c(x,x)) -> f(x)
    id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
    id_inc(s(x)) -> s(id_inc(x))
    id_inc(0()) -> 0()
    id_inc(0()) -> s(0())
   graph:
    g#(c(s(x),y)) -> g#(c(y,x)) -> g#(c(x,x)) -> f#(x)
    g#(c(s(x),y)) -> g#(c(y,x)) -> g#(c(x,s(y))) -> g#(c(y,x))
    g#(c(s(x),y)) -> g#(c(y,x)) -> g#(c(s(x),y)) -> g#(c(y,x))
    g#(c(x,s(y))) -> g#(c(y,x)) -> g#(c(x,x)) -> f#(x)
    g#(c(x,s(y))) -> g#(c(y,x)) -> g#(c(x,s(y))) -> g#(c(y,x))
    g#(c(x,s(y))) -> g#(c(y,x)) -> g#(c(s(x),y)) -> g#(c(y,x))
    g#(c(x,x)) -> f#(x) -> f#(c(s(x),y)) -> g#(c(x,y))
    g#(c(x,x)) -> f#(x) -> f#(s(x)) -> f#(id_inc(c(x,x)))
    g#(c(x,x)) -> f#(x) -> f#(s(x)) -> id_inc#(c(x,x))
    id_inc#(c(x,y)) -> id_inc#(y) -> id_inc#(s(x)) -> id_inc#(x)
    id_inc#(c(x,y)) -> id_inc#(y) -> id_inc#(c(x,y)) -> id_inc#(x)
    id_inc#(c(x,y)) -> id_inc#(y) -> id_inc#(c(x,y)) -> id_inc#(y)
    id_inc#(c(x,y)) -> id_inc#(x) -> id_inc#(s(x)) -> id_inc#(x)
    id_inc#(c(x,y)) -> id_inc#(x) -> id_inc#(c(x,y)) -> id_inc#(x)
    id_inc#(c(x,y)) -> id_inc#(x) -> id_inc#(c(x,y)) -> id_inc#(y)
    id_inc#(s(x)) -> id_inc#(x) -> id_inc#(s(x)) -> id_inc#(x)
    id_inc#(s(x)) -> id_inc#(x) -> id_inc#(c(x,y)) -> id_inc#(x)
    id_inc#(s(x)) -> id_inc#(x) -> id_inc#(c(x,y)) -> id_inc#(y)
    f#(c(s(x),y)) -> g#(c(x,y)) -> g#(c(x,x)) -> f#(x)
    f#(c(s(x),y)) -> g#(c(x,y)) -> g#(c(x,s(y))) -> g#(c(y,x))
    f#(c(s(x),y)) -> g#(c(x,y)) -> g#(c(s(x),y)) -> g#(c(y,x))
    f#(s(x)) -> id_inc#(c(x,x)) -> id_inc#(s(x)) -> id_inc#(x)
    f#(s(x)) -> id_inc#(c(x,x)) -> id_inc#(c(x,y)) -> id_inc#(x)
    f#(s(x)) -> id_inc#(c(x,x)) -> id_inc#(c(x,y)) -> id_inc#(y)
    f#(s(x)) -> f#(id_inc(c(x,x))) -> f#(c(s(x),y)) -> g#(c(x,y))
    f#(s(x)) -> f#(id_inc(c(x,x))) -> f#(s(x)) -> f#(id_inc(c(x,x)))
    f#(s(x)) -> f#(id_inc(c(x,x))) -> f#(s(x)) -> id_inc#(c(x,x))
   SCC Processor:
    #sccs: 2
    #rules: 8
    #arcs: 27/81
    DPs:
     g#(c(s(x),y)) -> g#(c(y,x))
     g#(c(x,s(y))) -> g#(c(y,x))
     g#(c(x,x)) -> f#(x)
     f#(s(x)) -> f#(id_inc(c(x,x)))
     f#(c(s(x),y)) -> g#(c(x,y))
    TRS:
     f(s(x)) -> f(id_inc(c(x,x)))
     f(c(s(x),y)) -> g(c(x,y))
     g(c(s(x),y)) -> g(c(y,x))
     g(c(x,s(y))) -> g(c(y,x))
     g(c(x,x)) -> f(x)
     id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
     id_inc(s(x)) -> s(id_inc(x))
     id_inc(0()) -> 0()
     id_inc(0()) -> s(0())
    Usable Rule Processor:
     DPs:
      g#(c(s(x),y)) -> g#(c(y,x))
      g#(c(x,s(y))) -> g#(c(y,x))
      g#(c(x,x)) -> f#(x)
      f#(s(x)) -> f#(id_inc(c(x,x)))
      f#(c(s(x),y)) -> g#(c(x,y))
     TRS:
      id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
      id_inc(s(x)) -> s(id_inc(x))
      id_inc(0()) -> 0()
      id_inc(0()) -> s(0())
     Matrix Interpretation Processor: dim=1
      
      usable rules:
       id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
       id_inc(s(x)) -> s(id_inc(x))
       id_inc(0()) -> 0()
       id_inc(0()) -> s(0())
      interpretation:
       [g#](x0) = x0 + 3,
       
       [f#](x0) = x0 + 2,
       
       [0] = 1,
       
       [id_inc](x0) = 2x0 + 2,
       
       [c](x0, x1) = 1/2x0 + 1/2x1,
       
       [s](x0) = 2x0 + 2
      orientation:
       g#(c(s(x),y)) = x + 1/2y + 4 >= 1/2x + 1/2y + 3 = g#(c(y,x))
       
       g#(c(x,s(y))) = 1/2x + y + 4 >= 1/2x + 1/2y + 3 = g#(c(y,x))
       
       g#(c(x,x)) = x + 3 >= x + 2 = f#(x)
       
       f#(s(x)) = 2x + 4 >= 2x + 4 = f#(id_inc(c(x,x)))
       
       f#(c(s(x),y)) = x + 1/2y + 3 >= 1/2x + 1/2y + 3 = g#(c(x,y))
       
       id_inc(c(x,y)) = x + y + 2 >= x + y + 2 = c(id_inc(x),id_inc(y))
       
       id_inc(s(x)) = 4x + 6 >= 4x + 6 = s(id_inc(x))
       
       id_inc(0()) = 4 >= 1 = 0()
       
       id_inc(0()) = 4 >= 4 = s(0())
      problem:
       DPs:
        f#(s(x)) -> f#(id_inc(c(x,x)))
        f#(c(s(x),y)) -> g#(c(x,y))
       TRS:
        id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
        id_inc(s(x)) -> s(id_inc(x))
        id_inc(0()) -> 0()
        id_inc(0()) -> s(0())
      Restore Modifier:
       DPs:
        f#(s(x)) -> f#(id_inc(c(x,x)))
        f#(c(s(x),y)) -> g#(c(x,y))
       TRS:
        f(s(x)) -> f(id_inc(c(x,x)))
        f(c(s(x),y)) -> g(c(x,y))
        g(c(s(x),y)) -> g(c(y,x))
        g(c(x,s(y))) -> g(c(y,x))
        g(c(x,x)) -> f(x)
        id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
        id_inc(s(x)) -> s(id_inc(x))
        id_inc(0()) -> 0()
        id_inc(0()) -> s(0())
       SCC Processor:
        #sccs: 1
        #rules: 1
        #arcs: 13/4
        DPs:
         f#(s(x)) -> f#(id_inc(c(x,x)))
        TRS:
         f(s(x)) -> f(id_inc(c(x,x)))
         f(c(s(x),y)) -> g(c(x,y))
         g(c(s(x),y)) -> g(c(y,x))
         g(c(x,s(y))) -> g(c(y,x))
         g(c(x,x)) -> f(x)
         id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
         id_inc(s(x)) -> s(id_inc(x))
         id_inc(0()) -> 0()
         id_inc(0()) -> s(0())
        Usable Rule Processor:
         DPs:
          f#(s(x)) -> f#(id_inc(c(x,x)))
         TRS:
          id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
          id_inc(s(x)) -> s(id_inc(x))
          id_inc(0()) -> 0()
          id_inc(0()) -> s(0())
         Arctic Interpretation Processor:
          dimension: 1
          usable rules:
           id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
           id_inc(s(x)) -> s(id_inc(x))
           id_inc(0()) -> 0()
           id_inc(0()) -> s(0())
          interpretation:
           [f#](x0) = x0 + 0,
           
           [0] = 5,
           
           [id_inc](x0) = x0 + 0,
           
           [c](x0, x1) = 0,
           
           [s](x0) = 4
          orientation:
           f#(s(x)) = 4 >= 0 = f#(id_inc(c(x,x)))
           
           id_inc(c(x,y)) = 0 >= 0 = c(id_inc(x),id_inc(y))
           
           id_inc(s(x)) = 4 >= 4 = s(id_inc(x))
           
           id_inc(0()) = 5 >= 5 = 0()
           
           id_inc(0()) = 5 >= 4 = s(0())
          problem:
           DPs:
            
           TRS:
            id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
            id_inc(s(x)) -> s(id_inc(x))
            id_inc(0()) -> 0()
            id_inc(0()) -> s(0())
          Qed
    
    DPs:
     id_inc#(c(x,y)) -> id_inc#(y)
     id_inc#(c(x,y)) -> id_inc#(x)
     id_inc#(s(x)) -> id_inc#(x)
    TRS:
     f(s(x)) -> f(id_inc(c(x,x)))
     f(c(s(x),y)) -> g(c(x,y))
     g(c(s(x),y)) -> g(c(y,x))
     g(c(x,s(y))) -> g(c(y,x))
     g(c(x,x)) -> f(x)
     id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
     id_inc(s(x)) -> s(id_inc(x))
     id_inc(0()) -> 0()
     id_inc(0()) -> s(0())
    Subterm Criterion Processor:
     simple projection:
      pi(id_inc#) = 0
     problem:
      DPs:
       
      TRS:
       f(s(x)) -> f(id_inc(c(x,x)))
       f(c(s(x),y)) -> g(c(x,y))
       g(c(s(x),y)) -> g(c(y,x))
       g(c(x,s(y))) -> g(c(y,x))
       g(c(x,x)) -> f(x)
       id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
       id_inc(s(x)) -> s(id_inc(x))
       id_inc(0()) -> 0()
       id_inc(0()) -> s(0())
     Qed