/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:
 cond1(true(),x) -> cond2(even(x),x)
 cond2(true(),x) -> cond1(neq(x,0()),div2(x))
 cond2(false(),x) -> cond1(neq(x,0()),p(x))
 neq(0(),0()) -> false()
 neq(0(),s(x)) -> true()
 neq(s(x),0()) -> true()
 neq(s(x),s(y())) -> neq(x,y())
 even(0()) -> true()
 even(s(0())) -> false()
 even(s(s(x))) -> even(x)
 div2(0()) -> 0()
 div2(s(0())) -> 0()
 div2(s(s(x))) -> s(div2(x))
 p(0()) -> 0()
 p(s(x)) -> x

Proof:
 DP Processor:
  DPs:
   cond1#(true(),x) -> even#(x)
   cond1#(true(),x) -> cond2#(even(x),x)
   cond2#(true(),x) -> div2#(x)
   cond2#(true(),x) -> neq#(x,0())
   cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
   cond2#(false(),x) -> p#(x)
   cond2#(false(),x) -> neq#(x,0())
   cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
   neq#(s(x),s(y())) -> neq#(x,y())
   even#(s(s(x))) -> even#(x)
   div2#(s(s(x))) -> div2#(x)
  TRS:
   cond1(true(),x) -> cond2(even(x),x)
   cond2(true(),x) -> cond1(neq(x,0()),div2(x))
   cond2(false(),x) -> cond1(neq(x,0()),p(x))
   neq(0(),0()) -> false()
   neq(0(),s(x)) -> true()
   neq(s(x),0()) -> true()
   neq(s(x),s(y())) -> neq(x,y())
   even(0()) -> true()
   even(s(0())) -> false()
   even(s(s(x))) -> even(x)
   div2(0()) -> 0()
   div2(s(0())) -> 0()
   div2(s(s(x))) -> s(div2(x))
   p(0()) -> 0()
   p(s(x)) -> x
  TDG Processor:
   DPs:
    cond1#(true(),x) -> even#(x)
    cond1#(true(),x) -> cond2#(even(x),x)
    cond2#(true(),x) -> div2#(x)
    cond2#(true(),x) -> neq#(x,0())
    cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
    cond2#(false(),x) -> p#(x)
    cond2#(false(),x) -> neq#(x,0())
    cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
    neq#(s(x),s(y())) -> neq#(x,y())
    even#(s(s(x))) -> even#(x)
    div2#(s(s(x))) -> div2#(x)
   TRS:
    cond1(true(),x) -> cond2(even(x),x)
    cond2(true(),x) -> cond1(neq(x,0()),div2(x))
    cond2(false(),x) -> cond1(neq(x,0()),p(x))
    neq(0(),0()) -> false()
    neq(0(),s(x)) -> true()
    neq(s(x),0()) -> true()
    neq(s(x),s(y())) -> neq(x,y())
    even(0()) -> true()
    even(s(0())) -> false()
    even(s(s(x))) -> even(x)
    div2(0()) -> 0()
    div2(s(0())) -> 0()
    div2(s(s(x))) -> s(div2(x))
    p(0()) -> 0()
    p(s(x)) -> x
   graph:
    neq#(s(x),s(y())) -> neq#(x,y()) -> neq#(s(x),s(y())) -> neq#(x,y())
    div2#(s(s(x))) -> div2#(x) -> div2#(s(s(x))) -> div2#(x)
    cond2#(false(),x) -> neq#(x,0()) ->
    neq#(s(x),s(y())) -> neq#(x,y())
    cond2#(false(),x) -> cond1#(neq(x,0()),p(x)) ->
    cond1#(true(),x) -> cond2#(even(x),x)
    cond2#(false(),x) -> cond1#(neq(x,0()),p(x)) ->
    cond1#(true(),x) -> even#(x)
    cond2#(true(),x) -> neq#(x,0()) -> neq#(s(x),s(y())) -> neq#(x,y())
    cond2#(true(),x) -> div2#(x) ->
    div2#(s(s(x))) -> div2#(x)
    cond2#(true(),x) -> cond1#(neq(x,0()),div2(x)) ->
    cond1#(true(),x) -> cond2#(even(x),x)
    cond2#(true(),x) -> cond1#(neq(x,0()),div2(x)) ->
    cond1#(true(),x) -> even#(x)
    even#(s(s(x))) -> even#(x) -> even#(s(s(x))) -> even#(x)
    cond1#(true(),x) -> cond2#(even(x),x) ->
    cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
    cond1#(true(),x) -> cond2#(even(x),x) ->
    cond2#(false(),x) -> neq#(x,0())
    cond1#(true(),x) -> cond2#(even(x),x) ->
    cond2#(false(),x) -> p#(x)
    cond1#(true(),x) -> cond2#(even(x),x) ->
    cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
    cond1#(true(),x) -> cond2#(even(x),x) ->
    cond2#(true(),x) -> neq#(x,0())
    cond1#(true(),x) -> cond2#(even(x),x) ->
    cond2#(true(),x) -> div2#(x)
    cond1#(true(),x) -> even#(x) -> even#(s(s(x))) -> even#(x)
   SCC Processor:
    #sccs: 4
    #rules: 6
    #arcs: 17/121
    DPs:
     cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
     cond1#(true(),x) -> cond2#(even(x),x)
     cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
    TRS:
     cond1(true(),x) -> cond2(even(x),x)
     cond2(true(),x) -> cond1(neq(x,0()),div2(x))
     cond2(false(),x) -> cond1(neq(x,0()),p(x))
     neq(0(),0()) -> false()
     neq(0(),s(x)) -> true()
     neq(s(x),0()) -> true()
     neq(s(x),s(y())) -> neq(x,y())
     even(0()) -> true()
     even(s(0())) -> false()
     even(s(s(x))) -> even(x)
     div2(0()) -> 0()
     div2(s(0())) -> 0()
     div2(s(s(x))) -> s(div2(x))
     p(0()) -> 0()
     p(s(x)) -> x
    Usable Rule Processor:
     DPs:
      cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
      cond1#(true(),x) -> cond2#(even(x),x)
      cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
     TRS:
      p(0()) -> 0()
      p(s(x)) -> x
      neq(0(),0()) -> false()
      neq(s(x),0()) -> true()
      even(0()) -> true()
      even(s(0())) -> false()
      even(s(s(x))) -> even(x)
      div2(0()) -> 0()
      div2(s(0())) -> 0()
      div2(s(s(x))) -> s(div2(x))
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       p(0()) -> 0()
       p(s(x)) -> x
       even(0()) -> true()
       even(s(0())) -> false()
       even(s(s(x))) -> even(x)
       div2(0()) -> 0()
       div2(s(0())) -> 0()
       div2(s(s(x))) -> s(div2(x))
      interpretation:
       [cond2#](x0, x1) = x0 + x1 + 0,
       
       [cond1#](x0, x1) = x1 + 0,
       
       [s](x0) = 3x0 + 0,
       
       [p](x0) = -3x0 + 0,
       
       [false] = 2,
       
       [div2](x0) = x0 + 0,
       
       [neq](x0, x1) = x0 + 4x1 + 0,
       
       [0] = 0,
       
       [even](x0) = x0 + 0,
       
       [true] = 0
      orientation:
       cond2#(false(),x) = x + 2 >= -3x + 0 = cond1#(neq(x,0()),p(x))
       
       cond1#(true(),x) = x + 0 >= x + 0 = cond2#(even(x),x)
       
       cond2#(true(),x) = x + 0 >= x + 0 = cond1#(neq(x,0()),div2(x))
       
       p(0()) = 0 >= 0 = 0()
       
       p(s(x)) = x + 0 >= x = x
       
       neq(0(),0()) = 4 >= 2 = false()
       
       neq(s(x),0()) = 3x + 4 >= 0 = true()
       
       even(0()) = 0 >= 0 = true()
       
       even(s(0())) = 3 >= 2 = false()
       
       even(s(s(x))) = 6x + 3 >= x + 0 = even(x)
       
       div2(0()) = 0 >= 0 = 0()
       
       div2(s(0())) = 3 >= 0 = 0()
       
       div2(s(s(x))) = 6x + 3 >= 3x + 3 = s(div2(x))
      problem:
       DPs:
        cond1#(true(),x) -> cond2#(even(x),x)
        cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
       TRS:
        p(0()) -> 0()
        p(s(x)) -> x
        neq(0(),0()) -> false()
        neq(s(x),0()) -> true()
        even(0()) -> true()
        even(s(0())) -> false()
        even(s(s(x))) -> even(x)
        div2(0()) -> 0()
        div2(s(0())) -> 0()
        div2(s(s(x))) -> s(div2(x))
      Restore Modifier:
       DPs:
        cond1#(true(),x) -> cond2#(even(x),x)
        cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
       TRS:
        cond1(true(),x) -> cond2(even(x),x)
        cond2(true(),x) -> cond1(neq(x,0()),div2(x))
        cond2(false(),x) -> cond1(neq(x,0()),p(x))
        neq(0(),0()) -> false()
        neq(0(),s(x)) -> true()
        neq(s(x),0()) -> true()
        neq(s(x),s(y())) -> neq(x,y())
        even(0()) -> true()
        even(s(0())) -> false()
        even(s(s(x))) -> even(x)
        div2(0()) -> 0()
        div2(s(0())) -> 0()
        div2(s(s(x))) -> s(div2(x))
        p(0()) -> 0()
        p(s(x)) -> x
       Usable Rule Processor:
        DPs:
         cond1#(true(),x) -> cond2#(even(x),x)
         cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
        TRS:
         even(0()) -> true()
         even(s(0())) -> false()
         even(s(s(x))) -> even(x)
         div2(0()) -> 0()
         div2(s(0())) -> 0()
         div2(s(s(x))) -> s(div2(x))
         neq(0(),0()) -> false()
         neq(s(x),0()) -> true()
        Matrix Interpretation Processor: dim=1
         
         usable rules:
          div2(0()) -> 0()
          div2(s(0())) -> 0()
          div2(s(s(x))) -> s(div2(x))
          neq(0(),0()) -> false()
          neq(s(x),0()) -> true()
         interpretation:
          [cond2#](x0, x1) = 2x1 + 7/2,
          
          [cond1#](x0, x1) = x0 + 2x1 + 2,
          
          [s](x0) = 3x0 + 2,
          
          [false] = 0,
          
          [div2](x0) = 1/2x0,
          
          [neq](x0, x1) = x0 + 1,
          
          [0] = 0,
          
          [even](x0) = 1/2x0 + 1,
          
          [true] = 3/2
         orientation:
          cond1#(true(),x) = 2x + 7/2 >= 2x + 7/2 = cond2#(even(x),x)
          
          cond2#(true(),x) = 2x + 7/2 >= 2x + 3 = cond1#(neq(x,0()),div2(x))
          
          even(0()) = 1 >= 3/2 = true()
          
          even(s(0())) = 2 >= 0 = false()
          
          even(s(s(x))) = 9/2x + 5 >= 1/2x + 1 = even(x)
          
          div2(0()) = 0 >= 0 = 0()
          
          div2(s(0())) = 1 >= 0 = 0()
          
          div2(s(s(x))) = 9/2x + 4 >= 3/2x + 2 = s(div2(x))
          
          neq(0(),0()) = 1 >= 0 = false()
          
          neq(s(x),0()) = 3x + 3 >= 3/2 = true()
         problem:
          DPs:
           cond1#(true(),x) -> cond2#(even(x),x)
          TRS:
           even(0()) -> true()
           even(s(0())) -> false()
           even(s(s(x))) -> even(x)
           div2(0()) -> 0()
           div2(s(0())) -> 0()
           div2(s(s(x))) -> s(div2(x))
           neq(0(),0()) -> false()
           neq(s(x),0()) -> true()
         Restore Modifier:
          DPs:
           cond1#(true(),x) -> cond2#(even(x),x)
          TRS:
           cond1(true(),x) -> cond2(even(x),x)
           cond2(true(),x) -> cond1(neq(x,0()),div2(x))
           cond2(false(),x) -> cond1(neq(x,0()),p(x))
           neq(0(),0()) -> false()
           neq(0(),s(x)) -> true()
           neq(s(x),0()) -> true()
           neq(s(x),s(y())) -> neq(x,y())
           even(0()) -> true()
           even(s(0())) -> false()
           even(s(s(x))) -> even(x)
           div2(0()) -> 0()
           div2(s(0())) -> 0()
           div2(s(s(x))) -> s(div2(x))
           p(0()) -> 0()
           p(s(x)) -> x
          SCC Processor:
           #sccs: 0
           #rules: 0
           #arcs: 4/1
           
    
    DPs:
     even#(s(s(x))) -> even#(x)
    TRS:
     cond1(true(),x) -> cond2(even(x),x)
     cond2(true(),x) -> cond1(neq(x,0()),div2(x))
     cond2(false(),x) -> cond1(neq(x,0()),p(x))
     neq(0(),0()) -> false()
     neq(0(),s(x)) -> true()
     neq(s(x),0()) -> true()
     neq(s(x),s(y())) -> neq(x,y())
     even(0()) -> true()
     even(s(0())) -> false()
     even(s(s(x))) -> even(x)
     div2(0()) -> 0()
     div2(s(0())) -> 0()
     div2(s(s(x))) -> s(div2(x))
     p(0()) -> 0()
     p(s(x)) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(even#) = 0
     problem:
      DPs:
       
      TRS:
       cond1(true(),x) -> cond2(even(x),x)
       cond2(true(),x) -> cond1(neq(x,0()),div2(x))
       cond2(false(),x) -> cond1(neq(x,0()),p(x))
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y())) -> neq(x,y())
       even(0()) -> true()
       even(s(0())) -> false()
       even(s(s(x))) -> even(x)
       div2(0()) -> 0()
       div2(s(0())) -> 0()
       div2(s(s(x))) -> s(div2(x))
       p(0()) -> 0()
       p(s(x)) -> x
     Qed
    
    DPs:
     div2#(s(s(x))) -> div2#(x)
    TRS:
     cond1(true(),x) -> cond2(even(x),x)
     cond2(true(),x) -> cond1(neq(x,0()),div2(x))
     cond2(false(),x) -> cond1(neq(x,0()),p(x))
     neq(0(),0()) -> false()
     neq(0(),s(x)) -> true()
     neq(s(x),0()) -> true()
     neq(s(x),s(y())) -> neq(x,y())
     even(0()) -> true()
     even(s(0())) -> false()
     even(s(s(x))) -> even(x)
     div2(0()) -> 0()
     div2(s(0())) -> 0()
     div2(s(s(x))) -> s(div2(x))
     p(0()) -> 0()
     p(s(x)) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(div2#) = 0
     problem:
      DPs:
       
      TRS:
       cond1(true(),x) -> cond2(even(x),x)
       cond2(true(),x) -> cond1(neq(x,0()),div2(x))
       cond2(false(),x) -> cond1(neq(x,0()),p(x))
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y())) -> neq(x,y())
       even(0()) -> true()
       even(s(0())) -> false()
       even(s(s(x))) -> even(x)
       div2(0()) -> 0()
       div2(s(0())) -> 0()
       div2(s(s(x))) -> s(div2(x))
       p(0()) -> 0()
       p(s(x)) -> x
     Qed
    
    DPs:
     neq#(s(x),s(y())) -> neq#(x,y())
    TRS:
     cond1(true(),x) -> cond2(even(x),x)
     cond2(true(),x) -> cond1(neq(x,0()),div2(x))
     cond2(false(),x) -> cond1(neq(x,0()),p(x))
     neq(0(),0()) -> false()
     neq(0(),s(x)) -> true()
     neq(s(x),0()) -> true()
     neq(s(x),s(y())) -> neq(x,y())
     even(0()) -> true()
     even(s(0())) -> false()
     even(s(s(x))) -> even(x)
     div2(0()) -> 0()
     div2(s(0())) -> 0()
     div2(s(s(x))) -> s(div2(x))
     p(0()) -> 0()
     p(s(x)) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(neq#) = 0
     problem:
      DPs:
       
      TRS:
       cond1(true(),x) -> cond2(even(x),x)
       cond2(true(),x) -> cond1(neq(x,0()),div2(x))
       cond2(false(),x) -> cond1(neq(x,0()),p(x))
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y())) -> neq(x,y())
       even(0()) -> true()
       even(s(0())) -> false()
       even(s(s(x))) -> even(x)
       div2(0()) -> 0()
       div2(s(0())) -> 0()
       div2(s(s(x))) -> s(div2(x))
       p(0()) -> 0()
       p(s(x)) -> x
     Qed