/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(0()) -> 0()
 f(s(0())) -> s(0())
 f(s(s(x))) -> p(h(g(x)))
 g(0()) -> pair(s(0()),s(0()))
 g(s(x)) -> h(g(x))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y
 +(x,0()) -> x
 +(x,s(y)) -> s(+(x,y))
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))

Proof:
 DP Processor:
  DPs:
   f#(s(s(x))) -> g#(x)
   f#(s(s(x))) -> h#(g(x))
   f#(s(s(x))) -> p#(h(g(x)))
   g#(s(x)) -> g#(x)
   g#(s(x)) -> h#(g(x))
   h#(x) -> q#(x)
   h#(x) -> p#(x)
   h#(x) -> +#(p(x),q(x))
   +#(x,s(y)) -> +#(x,y)
   f#(s(s(x))) -> q#(g(x))
   f#(s(s(x))) -> p#(g(x))
   f#(s(s(x))) -> +#(p(g(x)),q(g(x)))
   g#(s(x)) -> q#(g(x))
   g#(s(x)) -> p#(g(x))
   g#(s(x)) -> +#(p(g(x)),q(g(x)))
  TRS:
   f(0()) -> 0()
   f(s(0())) -> s(0())
   f(s(s(x))) -> p(h(g(x)))
   g(0()) -> pair(s(0()),s(0()))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x),q(x)),p(x))
   p(pair(x,y)) -> x
   q(pair(x,y)) -> y
   +(x,0()) -> x
   +(x,s(y)) -> s(+(x,y))
   f(s(s(x))) -> +(p(g(x)),q(g(x)))
   g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
  TDG Processor:
   DPs:
    f#(s(s(x))) -> g#(x)
    f#(s(s(x))) -> h#(g(x))
    f#(s(s(x))) -> p#(h(g(x)))
    g#(s(x)) -> g#(x)
    g#(s(x)) -> h#(g(x))
    h#(x) -> q#(x)
    h#(x) -> p#(x)
    h#(x) -> +#(p(x),q(x))
    +#(x,s(y)) -> +#(x,y)
    f#(s(s(x))) -> q#(g(x))
    f#(s(s(x))) -> p#(g(x))
    f#(s(s(x))) -> +#(p(g(x)),q(g(x)))
    g#(s(x)) -> q#(g(x))
    g#(s(x)) -> p#(g(x))
    g#(s(x)) -> +#(p(g(x)),q(g(x)))
   TRS:
    f(0()) -> 0()
    f(s(0())) -> s(0())
    f(s(s(x))) -> p(h(g(x)))
    g(0()) -> pair(s(0()),s(0()))
    g(s(x)) -> h(g(x))
    h(x) -> pair(+(p(x),q(x)),p(x))
    p(pair(x,y)) -> x
    q(pair(x,y)) -> y
    +(x,0()) -> x
    +(x,s(y)) -> s(+(x,y))
    f(s(s(x))) -> +(p(g(x)),q(g(x)))
    g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
   graph:
    +#(x,s(y)) -> +#(x,y) -> +#(x,s(y)) -> +#(x,y)
    h#(x) -> +#(p(x),q(x)) -> +#(x,s(y)) -> +#(x,y)
    g#(s(x)) -> +#(p(g(x)),q(g(x))) -> +#(x,s(y)) -> +#(x,y)
    g#(s(x)) -> h#(g(x)) -> h#(x) -> +#(p(x),q(x))
    g#(s(x)) -> h#(g(x)) -> h#(x) -> p#(x)
    g#(s(x)) -> h#(g(x)) -> h#(x) -> q#(x)
    g#(s(x)) -> g#(x) -> g#(s(x)) -> +#(p(g(x)),q(g(x)))
    g#(s(x)) -> g#(x) -> g#(s(x)) -> p#(g(x))
    g#(s(x)) -> g#(x) -> g#(s(x)) -> q#(g(x))
    g#(s(x)) -> g#(x) -> g#(s(x)) -> h#(g(x))
    g#(s(x)) -> g#(x) -> g#(s(x)) -> g#(x)
    f#(s(s(x))) -> +#(p(g(x)),q(g(x))) -> +#(x,s(y)) -> +#(x,y)
    f#(s(s(x))) -> h#(g(x)) -> h#(x) -> +#(p(x),q(x))
    f#(s(s(x))) -> h#(g(x)) -> h#(x) -> p#(x)
    f#(s(s(x))) -> h#(g(x)) -> h#(x) -> q#(x)
    f#(s(s(x))) -> g#(x) -> g#(s(x)) -> +#(p(g(x)),q(g(x)))
    f#(s(s(x))) -> g#(x) -> g#(s(x)) -> p#(g(x))
    f#(s(s(x))) -> g#(x) -> g#(s(x)) -> q#(g(x))
    f#(s(s(x))) -> g#(x) -> g#(s(x)) -> h#(g(x))
    f#(s(s(x))) -> g#(x) -> g#(s(x)) -> g#(x)
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 20/225
    DPs:
     g#(s(x)) -> g#(x)
    TRS:
     f(0()) -> 0()
     f(s(0())) -> s(0())
     f(s(s(x))) -> p(h(g(x)))
     g(0()) -> pair(s(0()),s(0()))
     g(s(x)) -> h(g(x))
     h(x) -> pair(+(p(x),q(x)),p(x))
     p(pair(x,y)) -> x
     q(pair(x,y)) -> y
     +(x,0()) -> x
     +(x,s(y)) -> s(+(x,y))
     f(s(s(x))) -> +(p(g(x)),q(g(x)))
     g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
    Subterm Criterion Processor:
     simple projection:
      pi(g#) = 0
     problem:
      DPs:
       
      TRS:
       f(0()) -> 0()
       f(s(0())) -> s(0())
       f(s(s(x))) -> p(h(g(x)))
       g(0()) -> pair(s(0()),s(0()))
       g(s(x)) -> h(g(x))
       h(x) -> pair(+(p(x),q(x)),p(x))
       p(pair(x,y)) -> x
       q(pair(x,y)) -> y
       +(x,0()) -> x
       +(x,s(y)) -> s(+(x,y))
       f(s(s(x))) -> +(p(g(x)),q(g(x)))
       g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
     Qed
    
    DPs:
     +#(x,s(y)) -> +#(x,y)
    TRS:
     f(0()) -> 0()
     f(s(0())) -> s(0())
     f(s(s(x))) -> p(h(g(x)))
     g(0()) -> pair(s(0()),s(0()))
     g(s(x)) -> h(g(x))
     h(x) -> pair(+(p(x),q(x)),p(x))
     p(pair(x,y)) -> x
     q(pair(x,y)) -> y
     +(x,0()) -> x
     +(x,s(y)) -> s(+(x,y))
     f(s(s(x))) -> +(p(g(x)),q(g(x)))
     g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
    Subterm Criterion Processor:
     simple projection:
      pi(+#) = 1
     problem:
      DPs:
       
      TRS:
       f(0()) -> 0()
       f(s(0())) -> s(0())
       f(s(s(x))) -> p(h(g(x)))
       g(0()) -> pair(s(0()),s(0()))
       g(s(x)) -> h(g(x))
       h(x) -> pair(+(p(x),q(x)),p(x))
       p(pair(x,y)) -> x
       q(pair(x,y)) -> y
       +(x,0()) -> x
       +(x,s(y)) -> s(+(x,y))
       f(s(s(x))) -> +(p(g(x)),q(g(x)))
       g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
     Qed