/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:
 cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
 cond2(true(),x,y) -> cond2(gr(y,0()),x,p(y))
 cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
 gr(0(),x) -> false()
 gr(s(x),0()) -> true()
 gr(s(x),s(y)) -> gr(x,y)
 p(0()) -> 0()
 p(s(x)) -> x

Proof:
 DP Processor:
  DPs:
   cond1#(true(),x,y) -> gr#(y,0())
   cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
   cond2#(true(),x,y) -> p#(y)
   cond2#(true(),x,y) -> gr#(y,0())
   cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
   cond2#(false(),x,y) -> p#(x)
   cond2#(false(),x,y) -> gr#(x,0())
   cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
   gr#(s(x),s(y)) -> gr#(x,y)
  TRS:
   cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
   cond2(true(),x,y) -> cond2(gr(y,0()),x,p(y))
   cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
   gr(0(),x) -> false()
   gr(s(x),0()) -> true()
   gr(s(x),s(y)) -> gr(x,y)
   p(0()) -> 0()
   p(s(x)) -> x
  TDG Processor:
   DPs:
    cond1#(true(),x,y) -> gr#(y,0())
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
    cond2#(true(),x,y) -> p#(y)
    cond2#(true(),x,y) -> gr#(y,0())
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
    cond2#(false(),x,y) -> p#(x)
    cond2#(false(),x,y) -> gr#(x,0())
    cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
    gr#(s(x),s(y)) -> gr#(x,y)
   TRS:
    cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
    cond2(true(),x,y) -> cond2(gr(y,0()),x,p(y))
    cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
    gr(0(),x) -> false()
    gr(s(x),0()) -> true()
    gr(s(x),s(y)) -> gr(x,y)
    p(0()) -> 0()
    p(s(x)) -> x
   graph:
    cond2#(false(),x,y) -> gr#(x,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y) ->
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
    cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y) ->
    cond1#(true(),x,y) -> gr#(y,0())
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y)) ->
    cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y)) ->
    cond2#(false(),x,y) -> gr#(x,0())
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y)) ->
    cond2#(false(),x,y) -> p#(x)
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y)) ->
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y)) ->
    cond2#(true(),x,y) -> gr#(y,0())
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y)) ->
    cond2#(true(),x,y) -> p#(y)
    cond2#(true(),x,y) -> gr#(y,0()) -> gr#(s(x),s(y)) -> gr#(x,y)
    gr#(s(x),s(y)) -> gr#(x,y) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(false(),x,y) -> gr#(x,0())
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(false(),x,y) -> p#(x)
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(true(),x,y) -> gr#(y,0())
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(true(),x,y) -> p#(y)
    cond1#(true(),x,y) -> gr#(y,0()) -> gr#(s(x),s(y)) -> gr#(x,y)
   SCC Processor:
    #sccs: 2
    #rules: 4
    #arcs: 18/81
    DPs:
     cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
     cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
    TRS:
     cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
     cond2(true(),x,y) -> cond2(gr(y,0()),x,p(y))
     cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
     p(0()) -> 0()
     p(s(x)) -> x
    Usable Rule Processor:
     DPs:
      cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
      cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
      cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
     TRS:
      p(0()) -> 0()
      p(s(x)) -> x
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
     Matrix Interpretation Processor: dim=1
      
      usable rules:
       p(0()) -> 0()
       p(s(x)) -> x
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
      interpretation:
       [cond2#](x0, x1, x2) = x1 + 1,
       
       [cond1#](x0, x1, x2) = x0 + x1,
       
       [s](x0) = 2x0 + 1,
       
       [false] = 0,
       
       [p](x0) = 1/2x0,
       
       [gr](x0, x1) = 1/2x0 + 1,
       
       [0] = 0,
       
       [true] = 3/2
      orientation:
       cond2#(false(),x,y) = x + 1 >= x + 1 = cond1#(gr(x,0()),p(x),y)
       
       cond1#(true(),x,y) = x + 3/2 >= x + 1 = cond2#(gr(y,0()),x,y)
       
       cond2#(true(),x,y) = x + 1 >= x + 1 = cond2#(gr(y,0()),x,p(y))
       
       p(0()) = 0 >= 0 = 0()
       
       p(s(x)) = x + 1/2 >= x = x
       
       gr(0(),x) = 1 >= 0 = false()
       
       gr(s(x),0()) = x + 3/2 >= 3/2 = true()
      problem:
       DPs:
        cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
        cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
       TRS:
        p(0()) -> 0()
        p(s(x)) -> x
        gr(0(),x) -> false()
        gr(s(x),0()) -> true()
      Restore Modifier:
       DPs:
        cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
        cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
       TRS:
        cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
        cond2(true(),x,y) -> cond2(gr(y,0()),x,p(y))
        cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
        gr(0(),x) -> false()
        gr(s(x),0()) -> true()
        gr(s(x),s(y)) -> gr(x,y)
        p(0()) -> 0()
        p(s(x)) -> x
       SCC Processor:
        #sccs: 1
        #rules: 1
        #arcs: 5/4
        DPs:
         cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
        TRS:
         cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
         cond2(true(),x,y) -> cond2(gr(y,0()),x,p(y))
         cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
         gr(0(),x) -> false()
         gr(s(x),0()) -> true()
         gr(s(x),s(y)) -> gr(x,y)
         p(0()) -> 0()
         p(s(x)) -> x
        Usable Rule Processor:
         DPs:
          cond2#(true(),x,y) -> cond2#(gr(y,0()),x,p(y))
         TRS:
          p(0()) -> 0()
          p(s(x)) -> x
          gr(0(),x) -> false()
          gr(s(x),0()) -> true()
         Matrix Interpretation Processor: dim=1
          
          usable rules:
           p(0()) -> 0()
           p(s(x)) -> x
           gr(0(),x) -> false()
           gr(s(x),0()) -> true()
          interpretation:
           [cond2#](x0, x1, x2) = x0 + 2x2,
           
           [s](x0) = 2x0 + 1,
           
           [false] = 0,
           
           [p](x0) = 1/2x0,
           
           [gr](x0, x1) = 1/2x0 + 1/2x1,
           
           [0] = 0,
           
           [true] = 1/2
          orientation:
           cond2#(true(),x,y) = 2y + 1/2 >= 3/2y = cond2#(gr(y,0()),x,p(y))
           
           p(0()) = 0 >= 0 = 0()
           
           p(s(x)) = x + 1/2 >= x = x
           
           gr(0(),x) = 1/2x >= 0 = false()
           
           gr(s(x),0()) = x + 1/2 >= 1/2 = true()
          problem:
           DPs:
            
           TRS:
            p(0()) -> 0()
            p(s(x)) -> x
            gr(0(),x) -> false()
            gr(s(x),0()) -> true()
          Qed
    
    DPs:
     gr#(s(x),s(y)) -> gr#(x,y)
    TRS:
     cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
     cond2(true(),x,y) -> cond2(gr(y,0()),x,p(y))
     cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
     p(0()) -> 0()
     p(s(x)) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(gr#) = 0
     problem:
      DPs:
       
      TRS:
       cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
       cond2(true(),x,y) -> cond2(gr(y,0()),x,p(y))
       cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       p(0()) -> 0()
       p(s(x)) -> x
     Qed