MAYBE

Problem:
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 quot(0(),s(y)) -> 0()
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
 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())
 log(x) -> logIter(x,0())
 logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
 if(false(),b,x,y) -> logZeroError()
 if(true(),false(),x,s(y)) -> y
 if(true(),true(),x,y) -> logIter(x,y)

Proof:
 DP Processor:
  DPs:
   minus#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
   le#(s(x),s(y)) -> le#(x,y)
   inc#(s(x)) -> inc#(x)
   log#(x) -> logIter#(x,0())
   logIter#(x,y) -> inc#(y)
   logIter#(x,y) -> quot#(x,s(s(0())))
   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),quot(x,s(s(0()))),inc(y))
   if#(true(),true(),x,y) -> logIter#(x,y)
  TRS:
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   quot(0(),s(y)) -> 0()
   quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
   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())
   log(x) -> logIter(x,0())
   logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
   if(false(),b,x,y) -> logZeroError()
   if(true(),false(),x,s(y)) -> y
   if(true(),true(),x,y) -> logIter(x,y)
  Usable Rule Processor:
   DPs:
    minus#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    le#(s(x),s(y)) -> le#(x,y)
    inc#(s(x)) -> inc#(x)
    log#(x) -> logIter#(x,0())
    logIter#(x,y) -> inc#(y)
    logIter#(x,y) -> quot#(x,s(s(0())))
    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),quot(x,s(s(0()))),inc(y))
    if#(true(),true(),x,y) -> logIter#(x,y)
   TRS:
    f19(x,y) -> x
    f19(x,y) -> y
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    inc(s(x)) -> s(inc(x))
    inc(0()) -> s(0())
    quot(0(),s(y)) -> 0()
    quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    le(0(),y) -> true()
   EDG Processor:
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
     quot#(s(x),s(y)) -> minus#(x,y)
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
     le#(s(x),s(y)) -> le#(x,y)
     inc#(s(x)) -> inc#(x)
     log#(x) -> logIter#(x,0())
     logIter#(x,y) -> inc#(y)
     logIter#(x,y) -> quot#(x,s(s(0())))
     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),quot(x,s(s(0()))),inc(y))
     if#(true(),true(),x,y) -> logIter#(x,y)
    TRS:
     f19(x,y) -> x
     f19(x,y) -> y
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     inc(s(x)) -> s(inc(x))
     inc(0()) -> s(0())
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     le(0(),y) -> true()
    graph:
     if#(true(),true(),x,y) -> logIter#(x,y) ->
     logIter#(x,y) -> inc#(y)
     if#(true(),true(),x,y) -> logIter#(x,y) ->
     logIter#(x,y) -> quot#(x,s(s(0())))
     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) -> le#(s(0()),x)
     if#(true(),true(),x,y) -> logIter#(x,y) ->
     logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
     logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),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) -> quot#(x,s(s(0()))) ->
     quot#(s(x),s(y)) -> minus#(x,y)
     logIter#(x,y) -> quot#(x,s(s(0()))) ->
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
     log#(x) -> logIter#(x,0()) -> logIter#(x,y) -> inc#(y)
     log#(x) -> logIter#(x,0()) -> logIter#(x,y) -> quot#(x,s(s(0())))
     log#(x) -> logIter#(x,0()) -> logIter#(x,y) -> le#(s(s(0())),x)
     log#(x) -> logIter#(x,0()) -> logIter#(x,y) -> le#(s(0()),x)
     log#(x) -> logIter#(x,0()) ->
     logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),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)
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
     quot#(s(x),s(y)) -> minus#(x,y)
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
     quot#(s(x),s(y)) -> minus#(x,y) ->
     minus#(s(x),s(y)) -> minus#(x,y)
     minus#(s(x),s(y)) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(x,y)
    Restore Modifier:
     DPs:
      minus#(s(x),s(y)) -> minus#(x,y)
      quot#(s(x),s(y)) -> minus#(x,y)
      quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
      le#(s(x),s(y)) -> le#(x,y)
      inc#(s(x)) -> inc#(x)
      log#(x) -> logIter#(x,0())
      logIter#(x,y) -> inc#(y)
      logIter#(x,y) -> quot#(x,s(s(0())))
      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),quot(x,s(s(0()))),inc(y))
      if#(true(),true(),x,y) -> logIter#(x,y)
     TRS:
      minus(x,0()) -> x
      minus(s(x),s(y)) -> minus(x,y)
      quot(0(),s(y)) -> 0()
      quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      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())
      log(x) -> logIter(x,0())
      logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
      if(false(),b,x,y) -> logZeroError()
      if(true(),false(),x,s(y)) -> y
      if(true(),true(),x,y) -> logIter(x,y)
     SCC Processor:
      #sccs: 5
      #rules: 6
      #arcs: 22/144
      DPs:
       if#(true(),true(),x,y) -> logIter#(x,y)
       logIter#(x,y) -> if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       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())
       log(x) -> logIter(x,0())
       logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
       if(false(),b,x,y) -> logZeroError()
       if(true(),false(),x,s(y)) -> y
       if(true(),true(),x,y) -> logIter(x,y)
      Open
      
      DPs:
       inc#(s(x)) -> inc#(x)
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       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())
       log(x) -> logIter(x,0())
       logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
       if(false(),b,x,y) -> logZeroError()
       if(true(),false(),x,s(y)) -> y
       if(true(),true(),x,y) -> logIter(x,y)
      Open
      
      DPs:
       quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       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())
       log(x) -> logIter(x,0())
       logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
       if(false(),b,x,y) -> logZeroError()
       if(true(),false(),x,s(y)) -> y
       if(true(),true(),x,y) -> logIter(x,y)
      Open
      
      DPs:
       minus#(s(x),s(y)) -> minus#(x,y)
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       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())
       log(x) -> logIter(x,0())
       logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
       if(false(),b,x,y) -> logZeroError()
       if(true(),false(),x,s(y)) -> y
       if(true(),true(),x,y) -> logIter(x,y)
      Open
      
      DPs:
       le#(s(x),s(y)) -> le#(x,y)
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       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())
       log(x) -> logIter(x,0())
       logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y))
       if(false(),b,x,y) -> logZeroError()
       if(true(),false(),x,s(y)) -> y
       if(true(),true(),x,y) -> logIter(x,y)
      Open