MAYBE

Problem:
 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))
 minus(0(),y) -> 0()
 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)))
 log(x) -> log2(x,0())
 log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
 if(true(),b,x,y) -> log_undefined()
 if(false(),b,x,y) -> if2(b,x,y)
 if2(true(),x,s(y)) -> y
 if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   inc#(s(x)) -> inc#(x)
   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))
   log#(x) -> log2#(x,0())
   log2#(x,y) -> inc#(y)
   log2#(x,y) -> le#(x,s(0()))
   log2#(x,y) -> le#(x,0())
   log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
   if#(false(),b,x,y) -> if2#(b,x,y)
   if2#(false(),x,y) -> quot#(x,s(s(0())))
   if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y)
  TRS:
   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))
   minus(0(),y) -> 0()
   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)))
   log(x) -> log2(x,0())
   log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
   if(true(),b,x,y) -> log_undefined()
   if(false(),b,x,y) -> if2(b,x,y)
   if2(true(),x,s(y)) -> y
   if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    inc#(s(x)) -> inc#(x)
    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))
    log#(x) -> log2#(x,0())
    log2#(x,y) -> inc#(y)
    log2#(x,y) -> le#(x,s(0()))
    log2#(x,y) -> le#(x,0())
    log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
    if#(false(),b,x,y) -> if2#(b,x,y)
    if2#(false(),x,y) -> quot#(x,s(s(0())))
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y)
   TRS:
    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))
    minus(0(),y) -> 0()
    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)))
    log(x) -> log2(x,0())
    log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
    if(true(),b,x,y) -> log_undefined()
    if(false(),b,x,y) -> if2(b,x,y)
    if2(true(),x,s(y)) -> y
    if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
   graph:
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
    log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
    log2#(x,y) -> le#(x,0())
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
    log2#(x,y) -> le#(x,s(0()))
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
    log2#(x,y) -> inc#(y)
    if2#(false(),x,y) -> quot#(x,s(s(0()))) ->
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    if2#(false(),x,y) -> quot#(x,s(s(0()))) ->
    quot#(s(x),s(y)) -> minus#(x,y)
    if#(false(),b,x,y) -> if2#(b,x,y) ->
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y)
    if#(false(),b,x,y) -> if2#(b,x,y) ->
    if2#(false(),x,y) -> quot#(x,s(s(0())))
    log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y)) ->
    if#(false(),b,x,y) -> if2#(b,x,y)
    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)
    log2#(x,y) -> le#(x,0()) -> le#(s(x),s(y)) -> le#(x,y)
    log#(x) -> log2#(x,0()) ->
    log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> le#(x,0())
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> le#(x,s(0()))
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> inc#(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)) -> quot#(minus(x,y),s(y)) ->
    quot#(s(x),s(y)) -> minus#(x,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)
    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)
   SCC Processor:
    #sccs: 5
    #rules: 7
    #arcs: 22/169
    DPs:
     if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y)
     log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
     if#(false(),b,x,y) -> if2#(b,x,y)
    TRS:
     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))
     minus(0(),y) -> 0()
     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)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Open
    
    DPs:
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    TRS:
     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))
     minus(0(),y) -> 0()
     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)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Open
    
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     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))
     minus(0(),y) -> 0()
     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)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Open
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     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))
     minus(0(),y) -> 0()
     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)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Open
    
    DPs:
     inc#(s(x)) -> inc#(x)
    TRS:
     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))
     minus(0(),y) -> 0()
     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)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Open