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)))
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 minus(minus(x,y),z) -> minus(x,plus(y,z))
 app(nil(),k) -> k
 app(l,nil()) -> l
 app(cons(x,l),k) -> cons(x,app(l,k))
 sum(cons(x,nil())) -> cons(x,nil())
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
 plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
 plus(zero(),y) -> y
 plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
 id(x) -> x
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 not(x) -> if(x,false(),true())
 gt(s(x),zero()) -> true()
 gt(zero(),y) -> false()
 gt(s(x),s(y)) -> gt(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))
   plus#(s(x),y) -> plus#(x,y)
   minus#(minus(x,y),z) -> plus#(y,z)
   minus#(minus(x,y),z) -> minus#(x,plus(y,z))
   app#(cons(x,l),k) -> app#(l,k)
   sum#(cons(x,cons(y,l))) -> plus#(x,y)
   sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
   sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
   sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
   sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
   plus#(s(x),s(y)) -> id#(y)
   plus#(s(x),s(y)) -> id#(x)
   plus#(s(x),s(y)) -> not#(gt(x,y))
   plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
   plus#(s(x),s(y)) -> gt#(x,y)
   plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
   plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
   plus#(s(x),x) -> id#(x)
   plus#(s(x),x) -> gt#(x,x)
   plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
   plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
   plus#(id(x),s(y)) -> gt#(s(y),y)
   plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
   plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
   not#(x) -> if#(x,false(),true())
   gt#(s(x),s(y)) -> gt#(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)))
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   minus(minus(x,y),z) -> minus(x,plus(y,z))
   app(nil(),k) -> k
   app(l,nil()) -> l
   app(cons(x,l),k) -> cons(x,app(l,k))
   sum(cons(x,nil())) -> cons(x,nil())
   sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
   sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
   plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
   plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
   plus(zero(),y) -> y
   plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
   id(x) -> x
   if(true(),x,y) -> x
   if(false(),x,y) -> y
   not(x) -> if(x,false(),true())
   gt(s(x),zero()) -> true()
   gt(zero(),y) -> false()
   gt(s(x),s(y)) -> gt(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))
    plus#(s(x),y) -> plus#(x,y)
    minus#(minus(x,y),z) -> plus#(y,z)
    minus#(minus(x,y),z) -> minus#(x,plus(y,z))
    app#(cons(x,l),k) -> app#(l,k)
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    plus#(s(x),s(y)) -> id#(y)
    plus#(s(x),s(y)) -> id#(x)
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(s(x),x) -> id#(x)
    plus#(s(x),x) -> gt#(x,x)
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    not#(x) -> if#(x,false(),true())
    gt#(s(x),s(y)) -> gt#(x,y)
   TRS:
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    minus(minus(x,y),z) -> minus(x,plus(y,z))
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
    plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
    plus(zero(),y) -> y
    plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
    id(x) -> x
    not(x) -> if(x,false(),true())
    gt(s(x),zero()) -> true()
    gt(zero(),y) -> false()
    gt(s(x),s(y)) -> gt(x,y)
    if(true(),x,y) -> x
    if(false(),x,y) -> y
    sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
    sum(cons(x,nil())) -> cons(x,nil())
    app(nil(),k) -> k
    app(l,nil()) -> l
    app(cons(x,l),k) -> cons(x,app(l,k))
   Open