MAYBE

Problem:
 lt(x,0()) -> false()
 lt(0(),s(y)) -> true()
 lt(s(x),s(y)) -> lt(x,y)
 plus(x,0()) -> x
 plus(x,s(y)) -> s(plus(x,y))
 quot(x,s(y)) -> help(x,s(y),0())
 help(x,s(y),c) -> if(lt(c,x),x,s(y),c)
 if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y))))
 if(false(),x,s(y),c) -> 0()

Proof:
 DP Processor:
  DPs:
   lt#(s(x),s(y)) -> lt#(x,y)
   plus#(x,s(y)) -> plus#(x,y)
   quot#(x,s(y)) -> help#(x,s(y),0())
   help#(x,s(y),c) -> lt#(c,x)
   help#(x,s(y),c) -> if#(lt(c,x),x,s(y),c)
   if#(true(),x,s(y),c) -> plus#(c,s(y))
   if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y)))
  TRS:
   lt(x,0()) -> false()
   lt(0(),s(y)) -> true()
   lt(s(x),s(y)) -> lt(x,y)
   plus(x,0()) -> x
   plus(x,s(y)) -> s(plus(x,y))
   quot(x,s(y)) -> help(x,s(y),0())
   help(x,s(y),c) -> if(lt(c,x),x,s(y),c)
   if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y))))
   if(false(),x,s(y),c) -> 0()
  Usable Rule Processor:
   DPs:
    lt#(s(x),s(y)) -> lt#(x,y)
    plus#(x,s(y)) -> plus#(x,y)
    quot#(x,s(y)) -> help#(x,s(y),0())
    help#(x,s(y),c) -> lt#(c,x)
    help#(x,s(y),c) -> if#(lt(c,x),x,s(y),c)
    if#(true(),x,s(y),c) -> plus#(c,s(y))
    if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y)))
   TRS:
    lt(x,0()) -> false()
    lt(0(),s(y)) -> true()
    lt(s(x),s(y)) -> lt(x,y)
    plus(x,s(y)) -> s(plus(x,y))
    plus(x,0()) -> x
   Open