/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:
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 inc(0()) -> 0()
 inc(s(x)) -> s(inc(x))
 log(x) -> log2(x,0())
 log2(x,y) -> if(le(x,s(0())),x,inc(y))
 if(true(),x,s(y)) -> y
 if(false(),x,y) -> log2(half(x),y)

Proof:
 DP Processor:
  DPs:
   half#(s(s(x))) -> half#(x)
   le#(s(x),s(y)) -> le#(x,y)
   inc#(s(x)) -> inc#(x)
   log#(x) -> log2#(x,0())
   log2#(x,y) -> inc#(y)
   log2#(x,y) -> le#(x,s(0()))
   log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
   if#(false(),x,y) -> half#(x)
   if#(false(),x,y) -> log2#(half(x),y)
  TRS:
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   inc(0()) -> 0()
   inc(s(x)) -> s(inc(x))
   log(x) -> log2(x,0())
   log2(x,y) -> if(le(x,s(0())),x,inc(y))
   if(true(),x,s(y)) -> y
   if(false(),x,y) -> log2(half(x),y)
  TDG Processor:
   DPs:
    half#(s(s(x))) -> half#(x)
    le#(s(x),s(y)) -> le#(x,y)
    inc#(s(x)) -> inc#(x)
    log#(x) -> log2#(x,0())
    log2#(x,y) -> inc#(y)
    log2#(x,y) -> le#(x,s(0()))
    log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
    if#(false(),x,y) -> half#(x)
    if#(false(),x,y) -> log2#(half(x),y)
   TRS:
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(x))) -> s(half(x))
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    inc(0()) -> 0()
    inc(s(x)) -> s(inc(x))
    log(x) -> log2(x,0())
    log2(x,y) -> if(le(x,s(0())),x,inc(y))
    if(true(),x,s(y)) -> y
    if(false(),x,y) -> log2(half(x),y)
   graph:
    if#(false(),x,y) -> log2#(half(x),y) ->
    log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
    if#(false(),x,y) -> log2#(half(x),y) ->
    log2#(x,y) -> le#(x,s(0()))
    if#(false(),x,y) -> log2#(half(x),y) -> log2#(x,y) -> inc#(y)
    if#(false(),x,y) -> half#(x) ->
    half#(s(s(x))) -> half#(x)
    log2#(x,y) -> if#(le(x,s(0())),x,inc(y)) ->
    if#(false(),x,y) -> log2#(half(x),y)
    log2#(x,y) -> if#(le(x,s(0())),x,inc(y)) -> if#(false(),x,y) -> half#(x)
    log2#(x,y) -> inc#(y) -> inc#(s(x)) -> inc#(x)
    log2#(x,y) -> le#(x,s(0())) -> le#(s(x),s(y)) -> le#(x,y)
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> le#(x,s(0()))
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> inc#(y)
    inc#(s(x)) -> inc#(x) -> inc#(s(x)) -> inc#(x)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
    half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
   SCC Processor:
    #sccs: 4
    #rules: 5
    #arcs: 14/81
    DPs:
     if#(false(),x,y) -> log2#(half(x),y)
     log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,s(0())),x,inc(y))
     if(true(),x,s(y)) -> y
     if(false(),x,y) -> log2(half(x),y)
    Usable Rule Processor:
     DPs:
      if#(false(),x,y) -> log2#(half(x),y)
      log2#(x,y) -> if#(le(x,s(0())),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))
      le(0(),y) -> true()
      le(s(x),s(y)) -> le(x,y)
      le(s(x),0()) -> false()
     Matrix Interpretation Processor: dim=1
      
      usable rules:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       le(0(),y) -> true()
       le(s(x),s(y)) -> le(x,y)
       le(s(x),0()) -> false()
      interpretation:
       [if#](x0, x1, x2) = 1/2x0 + x1,
       
       [log2#](x0, x1) = 2x0 + 3/2,
       
       [inc](x0) = 0,
       
       [false] = 3,
       
       [true] = 0,
       
       [le](x0, x1) = 2x0 + 1,
       
       [s](x0) = x0 + 2,
       
       [half](x0) = 1/2x0,
       
       [0] = 0
      orientation:
       if#(false(),x,y) = x + 3/2 >= x + 3/2 = log2#(half(x),y)
       
       log2#(x,y) = 2x + 3/2 >= 2x + 1/2 = if#(le(x,s(0())),x,inc(y))
       
       half(0()) = 0 >= 0 = 0()
       
       half(s(0())) = 1 >= 0 = 0()
       
       half(s(s(x))) = 1/2x + 2 >= 1/2x + 2 = s(half(x))
       
       inc(0()) = 0 >= 0 = 0()
       
       inc(s(x)) = 0 >= 2 = s(inc(x))
       
       le(0(),y) = 1 >= 0 = true()
       
       le(s(x),s(y)) = 2x + 5 >= 2x + 1 = le(x,y)
       
       le(s(x),0()) = 2x + 5 >= 3 = false()
      problem:
       DPs:
        if#(false(),x,y) -> log2#(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))
        le(0(),y) -> true()
        le(s(x),s(y)) -> le(x,y)
        le(s(x),0()) -> false()
      Restore Modifier:
       DPs:
        if#(false(),x,y) -> log2#(half(x),y)
       TRS:
        half(0()) -> 0()
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
        le(0(),y) -> true()
        le(s(x),0()) -> false()
        le(s(x),s(y)) -> le(x,y)
        inc(0()) -> 0()
        inc(s(x)) -> s(inc(x))
        log(x) -> log2(x,0())
        log2(x,y) -> if(le(x,s(0())),x,inc(y))
        if(true(),x,s(y)) -> y
        if(false(),x,y) -> log2(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))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,s(0())),x,inc(y))
     if(true(),x,s(y)) -> y
     if(false(),x,y) -> log2(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))
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,s(0())),x,inc(y))
       if(true(),x,s(y)) -> y
       if(false(),x,y) -> log2(half(x),y)
     Qed
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,s(0())),x,inc(y))
     if(true(),x,s(y)) -> y
     if(false(),x,y) -> log2(half(x),y)
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 0
     problem:
      DPs:
       
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,s(0())),x,inc(y))
       if(true(),x,s(y)) -> y
       if(false(),x,y) -> log2(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))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,s(0())),x,inc(y))
     if(true(),x,s(y)) -> y
     if(false(),x,y) -> log2(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))
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,s(0())),x,inc(y))
       if(true(),x,s(y)) -> y
       if(false(),x,y) -> log2(half(x),y)
     Qed