/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:
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 inc(0()) -> 0()
 inc(s(x)) -> s(inc(x))
 zero(0()) -> true()
 zero(s(x)) -> false()
 p(0()) -> 0()
 p(s(x)) -> x
 bits(x) -> bitIter(x,0())
 bitIter(x,y) -> if(zero(x),x,inc(y))
 if(true(),x,y) -> p(y)
 if(false(),x,y) -> bitIter(half(x),y)

Proof:
 DP Processor:
  DPs:
   half#(s(s(x))) -> half#(x)
   inc#(s(x)) -> inc#(x)
   bits#(x) -> bitIter#(x,0())
   bitIter#(x,y) -> inc#(y)
   bitIter#(x,y) -> zero#(x)
   bitIter#(x,y) -> if#(zero(x),x,inc(y))
   if#(true(),x,y) -> p#(y)
   if#(false(),x,y) -> half#(x)
   if#(false(),x,y) -> bitIter#(half(x),y)
  TRS:
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   inc(0()) -> 0()
   inc(s(x)) -> s(inc(x))
   zero(0()) -> true()
   zero(s(x)) -> false()
   p(0()) -> 0()
   p(s(x)) -> x
   bits(x) -> bitIter(x,0())
   bitIter(x,y) -> if(zero(x),x,inc(y))
   if(true(),x,y) -> p(y)
   if(false(),x,y) -> bitIter(half(x),y)
  TDG Processor:
   DPs:
    half#(s(s(x))) -> half#(x)
    inc#(s(x)) -> inc#(x)
    bits#(x) -> bitIter#(x,0())
    bitIter#(x,y) -> inc#(y)
    bitIter#(x,y) -> zero#(x)
    bitIter#(x,y) -> if#(zero(x),x,inc(y))
    if#(true(),x,y) -> p#(y)
    if#(false(),x,y) -> half#(x)
    if#(false(),x,y) -> bitIter#(half(x),y)
   TRS:
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(x))) -> s(half(x))
    inc(0()) -> 0()
    inc(s(x)) -> s(inc(x))
    zero(0()) -> true()
    zero(s(x)) -> false()
    p(0()) -> 0()
    p(s(x)) -> x
    bits(x) -> bitIter(x,0())
    bitIter(x,y) -> if(zero(x),x,inc(y))
    if(true(),x,y) -> p(y)
    if(false(),x,y) -> bitIter(half(x),y)
   graph:
    if#(false(),x,y) -> bitIter#(half(x),y) ->
    bitIter#(x,y) -> if#(zero(x),x,inc(y))
    if#(false(),x,y) -> bitIter#(half(x),y) ->
    bitIter#(x,y) -> zero#(x)
    if#(false(),x,y) -> bitIter#(half(x),y) -> bitIter#(x,y) -> inc#(y)
    if#(false(),x,y) -> half#(x) ->
    half#(s(s(x))) -> half#(x)
    bitIter#(x,y) -> if#(zero(x),x,inc(y)) ->
    if#(false(),x,y) -> bitIter#(half(x),y)
    bitIter#(x,y) -> if#(zero(x),x,inc(y)) ->
    if#(false(),x,y) -> half#(x)
    bitIter#(x,y) -> if#(zero(x),x,inc(y)) -> if#(true(),x,y) -> p#(y)
    bitIter#(x,y) -> inc#(y) -> inc#(s(x)) -> inc#(x)
    bits#(x) -> bitIter#(x,0()) ->
    bitIter#(x,y) -> if#(zero(x),x,inc(y))
    bits#(x) -> bitIter#(x,0()) -> bitIter#(x,y) -> zero#(x)
    bits#(x) -> bitIter#(x,0()) -> bitIter#(x,y) -> inc#(y)
    inc#(s(x)) -> inc#(x) -> inc#(s(x)) -> inc#(x)
    half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
   SCC Processor:
    #sccs: 3
    #rules: 4
    #arcs: 13/81
    DPs:
     if#(false(),x,y) -> bitIter#(half(x),y)
     bitIter#(x,y) -> if#(zero(x),x,inc(y))
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     zero(0()) -> true()
     zero(s(x)) -> false()
     p(0()) -> 0()
     p(s(x)) -> x
     bits(x) -> bitIter(x,0())
     bitIter(x,y) -> if(zero(x),x,inc(y))
     if(true(),x,y) -> p(y)
     if(false(),x,y) -> bitIter(half(x),y)
    Usable Rule Processor:
     DPs:
      if#(false(),x,y) -> bitIter#(half(x),y)
      bitIter#(x,y) -> if#(zero(x),x,inc(y))
     TRS:
      half(0()) -> 0()
      half(s(0())) -> 0()
      half(s(s(x))) -> s(half(x))
      inc(0()) -> 0()
      inc(s(x)) -> s(inc(x))
      zero(0()) -> true()
      zero(s(x)) -> false()
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       zero(0()) -> true()
       zero(s(x)) -> false()
      interpretation:
       [if#](x0, x1, x2) = x0 + 1x1,
       
       [bitIter#](x0, x1) = 2x0 + 0,
       
       [false] = 4,
       
       [true] = 0,
       
       [zero](x0) = x0,
       
       [inc](x0) = x0 + 1,
       
       [s](x0) = 1x0 + 4,
       
       [half](x0) = -1x0 + 2,
       
       [0] = 0
      orientation:
       if#(false(),x,y) = 1x + 4 >= 1x + 4 = bitIter#(half(x),y)
       
       bitIter#(x,y) = 2x + 0 >= 1x = if#(zero(x),x,inc(y))
       
       half(0()) = 2 >= 0 = 0()
       
       half(s(0())) = 3 >= 0 = 0()
       
       half(s(s(x))) = 1x + 4 >= x + 4 = s(half(x))
       
       inc(0()) = 1 >= 0 = 0()
       
       inc(s(x)) = 1x + 4 >= 1x + 4 = s(inc(x))
       
       zero(0()) = 0 >= 0 = true()
       
       zero(s(x)) = 1x + 4 >= 4 = false()
      problem:
       DPs:
        if#(false(),x,y) -> bitIter#(half(x),y)
       TRS:
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
        inc(0()) -> 0()
        inc(s(x)) -> s(inc(x))
        zero(0()) -> true()
        zero(s(x)) -> false()
      Restore Modifier:
       DPs:
        if#(false(),x,y) -> bitIter#(half(x),y)
       TRS:
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
        inc(0()) -> 0()
        inc(s(x)) -> s(inc(x))
        zero(0()) -> true()
        zero(s(x)) -> false()
        p(0()) -> 0()
        p(s(x)) -> x
        bits(x) -> bitIter(x,0())
        bitIter(x,y) -> if(zero(x),x,inc(y))
        if(true(),x,y) -> p(y)
        if(false(),x,y) -> bitIter(half(x),y)
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 2/1
        
    
    DPs:
     half#(s(s(x))) -> half#(x)
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     zero(0()) -> true()
     zero(s(x)) -> false()
     p(0()) -> 0()
     p(s(x)) -> x
     bits(x) -> bitIter(x,0())
     bitIter(x,y) -> if(zero(x),x,inc(y))
     if(true(),x,y) -> p(y)
     if(false(),x,y) -> bitIter(half(x),y)
    Subterm Criterion Processor:
     simple projection:
      pi(half#) = 0
     problem:
      DPs:
       
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       zero(0()) -> true()
       zero(s(x)) -> false()
       p(0()) -> 0()
       p(s(x)) -> x
       bits(x) -> bitIter(x,0())
       bitIter(x,y) -> if(zero(x),x,inc(y))
       if(true(),x,y) -> p(y)
       if(false(),x,y) -> bitIter(half(x),y)
     Qed
    
    DPs:
     inc#(s(x)) -> inc#(x)
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     zero(0()) -> true()
     zero(s(x)) -> false()
     p(0()) -> 0()
     p(s(x)) -> x
     bits(x) -> bitIter(x,0())
     bitIter(x,y) -> if(zero(x),x,inc(y))
     if(true(),x,y) -> p(y)
     if(false(),x,y) -> bitIter(half(x),y)
    Subterm Criterion Processor:
     simple projection:
      pi(inc#) = 0
     problem:
      DPs:
       
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       zero(0()) -> true()
       zero(s(x)) -> false()
       p(0()) -> 0()
       p(s(x)) -> x
       bits(x) -> bitIter(x,0())
       bitIter(x,y) -> if(zero(x),x,inc(y))
       if(true(),x,y) -> p(y)
       if(false(),x,y) -> bitIter(half(x),y)
     Qed