/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:
 p(0()) -> 0()
 p(s(x)) -> x
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(x,0()) -> x
 minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
 if(true(),x,y) -> x
 if(false(),x,y) -> y

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(x,s(y)) -> p#(s(y))
   minus#(x,s(y)) -> minus#(x,p(s(y)))
   minus#(x,s(y)) -> p#(minus(x,p(s(y))))
   minus#(x,s(y)) -> le#(x,s(y))
   minus#(x,s(y)) -> if#(le(x,s(y)),0(),p(minus(x,p(s(y)))))
  TRS:
   p(0()) -> 0()
   p(s(x)) -> x
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   minus(x,0()) -> x
   minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
   if(true(),x,y) -> x
   if(false(),x,y) -> y
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(x,s(y)) -> p#(s(y))
    minus#(x,s(y)) -> minus#(x,p(s(y)))
    minus#(x,s(y)) -> p#(minus(x,p(s(y))))
    minus#(x,s(y)) -> le#(x,s(y))
    minus#(x,s(y)) -> if#(le(x,s(y)),0(),p(minus(x,p(s(y)))))
   TRS:
    p(0()) -> 0()
    p(s(x)) -> x
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    minus(x,0()) -> x
    minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
    if(true(),x,y) -> x
    if(false(),x,y) -> y
   graph:
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> if#(le(x,s(y)),0(),p(minus(x,p(s(y)))))
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> le#(x,s(y))
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> p#(minus(x,p(s(y))))
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> minus#(x,p(s(y)))
    minus#(x,s(y)) -> minus#(x,p(s(y))) -> minus#(x,s(y)) -> p#(s(y))
    minus#(x,s(y)) -> le#(x,s(y)) -> le#(s(x),s(y)) -> le#(x,y)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 7/36
    DPs:
     minus#(x,s(y)) -> minus#(x,p(s(y)))
    TRS:
     p(0()) -> 0()
     p(s(x)) -> x
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,0()) -> x
     minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
     if(true(),x,y) -> x
     if(false(),x,y) -> y
    Usable Rule Processor:
     DPs:
      minus#(x,s(y)) -> minus#(x,p(s(y)))
     TRS:
      p(s(x)) -> x
     Semantic Labeling Processor:
      dimension: 2
      usable rules:
       
      interpretation:
                 [1 1]     [1]
       [s](x0) = [1 1]x0 + [0],
       
                 [0 1]     [0]
       [p](x0) = [0 1]x0 + [1]
       labeled:
        minus#
       usable (for model):
        minus#
        s
        p
       argument filtering:
        pi(p) = []
        pi(s) = 0
        pi(minus#) = []
      precedence:
       minus# ~ s ~ p
      problem:
       DPs:
        
       TRS:
        p(s(x)) -> x
      Qed
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     p(0()) -> 0()
     p(s(x)) -> x
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,0()) -> x
     minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
     if(true(),x,y) -> x
     if(false(),x,y) -> y
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 0
     problem:
      DPs:
       
      TRS:
       p(0()) -> 0()
       p(s(x)) -> x
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,0()) -> x
       minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
       if(true(),x,y) -> x
       if(false(),x,y) -> y
     Qed