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(s(x)) -> s(inc(x))
 inc(0()) -> s(0())
 logarithm(x) -> logIter(x,0())
 logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
 if(false(),b,x,y) -> logZeroError()
 if(true(),false(),x,s(y)) -> y
 if(true(),true(),x,y) -> logIter(x,y)
 f() -> g()
 f() -> h()

Proof:
 DP Processor:
  DPs:
   half#(s(s(x))) -> half#(x)
   le#(s(x),s(y)) -> le#(x,y)
   inc#(s(x)) -> inc#(x)
   logarithm#(x) -> logIter#(x,0())
   logIter#(x,y) -> inc#(y)
   logIter#(x,y) -> half#(x)
   logIter#(x,y) -> le#(s(s(0())),x)
   logIter#(x,y) -> le#(s(0()),x)
   logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
   if#(true(),true(),x,y) -> logIter#(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(s(x)) -> s(inc(x))
   inc(0()) -> s(0())
   logarithm(x) -> logIter(x,0())
   logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
   if(false(),b,x,y) -> logZeroError()
   if(true(),false(),x,s(y)) -> y
   if(true(),true(),x,y) -> logIter(x,y)
   f() -> g()
   f() -> h()
  TDG Processor:
   DPs:
    half#(s(s(x))) -> half#(x)
    le#(s(x),s(y)) -> le#(x,y)
    inc#(s(x)) -> inc#(x)
    logarithm#(x) -> logIter#(x,0())
    logIter#(x,y) -> inc#(y)
    logIter#(x,y) -> half#(x)
    logIter#(x,y) -> le#(s(s(0())),x)
    logIter#(x,y) -> le#(s(0()),x)
    logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
    if#(true(),true(),x,y) -> logIter#(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(s(x)) -> s(inc(x))
    inc(0()) -> s(0())
    logarithm(x) -> logIter(x,0())
    logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
    if(false(),b,x,y) -> logZeroError()
    if(true(),false(),x,s(y)) -> y
    if(true(),true(),x,y) -> logIter(x,y)
    f() -> g()
    f() -> h()
   graph:
    if#(true(),true(),x,y) -> logIter#(x,y) ->
    logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
    if#(true(),true(),x,y) -> logIter#(x,y) ->
    logIter#(x,y) -> le#(s(0()),x)
    if#(true(),true(),x,y) -> logIter#(x,y) ->
    logIter#(x,y) -> le#(s(s(0())),x)
    if#(true(),true(),x,y) -> logIter#(x,y) ->
    logIter#(x,y) -> half#(x)
    if#(true(),true(),x,y) -> logIter#(x,y) ->
    logIter#(x,y) -> inc#(y)
    logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ->
    if#(true(),true(),x,y) -> logIter#(x,y)
    logIter#(x,y) -> inc#(y) -> inc#(s(x)) -> inc#(x)
    logIter#(x,y) -> le#(s(s(0())),x) -> le#(s(x),s(y)) -> le#(x,y)
    logIter#(x,y) -> le#(s(0()),x) -> le#(s(x),s(y)) -> le#(x,y)
    logIter#(x,y) -> half#(x) -> half#(s(s(x))) -> half#(x)
    logarithm#(x) -> logIter#(x,0()) ->
    logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
    logarithm#(x) -> logIter#(x,0()) ->
    logIter#(x,y) -> le#(s(0()),x)
    logarithm#(x) -> logIter#(x,0()) ->
    logIter#(x,y) -> le#(s(s(0())),x)
    logarithm#(x) -> logIter#(x,0()) -> logIter#(x,y) -> half#(x)
    logarithm#(x) -> logIter#(x,0()) -> logIter#(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: 18/100
    DPs:
     if#(true(),true(),x,y) -> logIter#(x,y)
     logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),half(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(s(x)) -> s(inc(x))
     inc(0()) -> s(0())
     logarithm(x) -> logIter(x,0())
     logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
     if(false(),b,x,y) -> logZeroError()
     if(true(),false(),x,s(y)) -> y
     if(true(),true(),x,y) -> logIter(x,y)
     f() -> g()
     f() -> h()
    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(s(x)) -> s(inc(x))
     inc(0()) -> s(0())
     logarithm(x) -> logIter(x,0())
     logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
     if(false(),b,x,y) -> logZeroError()
     if(true(),false(),x,s(y)) -> y
     if(true(),true(),x,y) -> logIter(x,y)
     f() -> g()
     f() -> h()
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 1
     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(s(x)) -> s(inc(x))
       inc(0()) -> s(0())
       logarithm(x) -> logIter(x,0())
       logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
       if(false(),b,x,y) -> logZeroError()
       if(true(),false(),x,s(y)) -> y
       if(true(),true(),x,y) -> logIter(x,y)
       f() -> g()
       f() -> h()
     Qed
    
    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(s(x)) -> s(inc(x))
     inc(0()) -> s(0())
     logarithm(x) -> logIter(x,0())
     logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
     if(false(),b,x,y) -> logZeroError()
     if(true(),false(),x,s(y)) -> y
     if(true(),true(),x,y) -> logIter(x,y)
     f() -> g()
     f() -> h()
    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(s(x)) -> s(inc(x))
       inc(0()) -> s(0())
       logarithm(x) -> logIter(x,0())
       logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
       if(false(),b,x,y) -> logZeroError()
       if(true(),false(),x,s(y)) -> y
       if(true(),true(),x,y) -> logIter(x,y)
       f() -> g()
       f() -> h()
     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(s(x)) -> s(inc(x))
     inc(0()) -> s(0())
     logarithm(x) -> logIter(x,0())
     logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
     if(false(),b,x,y) -> logZeroError()
     if(true(),false(),x,s(y)) -> y
     if(true(),true(),x,y) -> logIter(x,y)
     f() -> g()
     f() -> h()
    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(s(x)) -> s(inc(x))
       inc(0()) -> s(0())
       logarithm(x) -> logIter(x,0())
       logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
       if(false(),b,x,y) -> logZeroError()
       if(true(),false(),x,s(y)) -> y
       if(true(),true(),x,y) -> logIter(x,y)
       f() -> g()
       f() -> h()
     Qed