MAYBE

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)
    Open
    
    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)
    Open
    
    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)
    Open
    
    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)
    Open