MAYBE

Problem:
 p(s(x)) -> x
 plus(x,0()) -> x
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(s(x),y) -> s(plus(p(s(x)),y))
 plus(x,s(y)) -> s(plus(x,p(s(y))))
 times(0(),y) -> 0()
 times(s(0()),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
 div(0(),y) -> 0()
 div(x,y) -> quot(x,y,y)
 quot(0(),s(y),z) -> 0()
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0(),s(z)) -> s(div(x,s(z)))
 div(div(x,y),z) -> div(x,times(y,z))
 eq(0(),0()) -> true()
 eq(s(x),0()) -> false()
 eq(0(),s(y)) -> false()
 eq(s(x),s(y)) -> eq(x,y)
 divides(y,x) -> eq(x,times(div(x,y),y))
 prime(s(s(x))) -> pr(s(s(x)),s(x))
 pr(x,s(0())) -> true()
 pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
 if(true(),x,y) -> false()
 if(false(),x,y) -> pr(x,y)

Proof:
 DP Processor:
  DPs:
   plus#(s(x),y) -> plus#(x,y)
   plus#(s(x),y) -> p#(s(x))
   plus#(s(x),y) -> plus#(p(s(x)),y)
   plus#(x,s(y)) -> p#(s(y))
   plus#(x,s(y)) -> plus#(x,p(s(y)))
   times#(s(x),y) -> times#(x,y)
   times#(s(x),y) -> plus#(y,times(x,y))
   div#(x,y) -> quot#(x,y,y)
   quot#(s(x),s(y),z) -> quot#(x,y,z)
   quot#(x,0(),s(z)) -> div#(x,s(z))
   div#(div(x,y),z) -> times#(y,z)
   div#(div(x,y),z) -> div#(x,times(y,z))
   eq#(s(x),s(y)) -> eq#(x,y)
   divides#(y,x) -> div#(x,y)
   divides#(y,x) -> times#(div(x,y),y)
   divides#(y,x) -> eq#(x,times(div(x,y),y))
   prime#(s(s(x))) -> pr#(s(s(x)),s(x))
   pr#(x,s(s(y))) -> divides#(s(s(y)),x)
   pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
   if#(false(),x,y) -> pr#(x,y)
  TRS:
   p(s(x)) -> x
   plus(x,0()) -> x
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   plus(s(x),y) -> s(plus(p(s(x)),y))
   plus(x,s(y)) -> s(plus(x,p(s(y))))
   times(0(),y) -> 0()
   times(s(0()),y) -> y
   times(s(x),y) -> plus(y,times(x,y))
   div(0(),y) -> 0()
   div(x,y) -> quot(x,y,y)
   quot(0(),s(y),z) -> 0()
   quot(s(x),s(y),z) -> quot(x,y,z)
   quot(x,0(),s(z)) -> s(div(x,s(z)))
   div(div(x,y),z) -> div(x,times(y,z))
   eq(0(),0()) -> true()
   eq(s(x),0()) -> false()
   eq(0(),s(y)) -> false()
   eq(s(x),s(y)) -> eq(x,y)
   divides(y,x) -> eq(x,times(div(x,y),y))
   prime(s(s(x))) -> pr(s(s(x)),s(x))
   pr(x,s(0())) -> true()
   pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
   if(true(),x,y) -> false()
   if(false(),x,y) -> pr(x,y)
  Usable Rule Processor:
   DPs:
    plus#(s(x),y) -> plus#(x,y)
    plus#(s(x),y) -> p#(s(x))
    plus#(s(x),y) -> plus#(p(s(x)),y)
    plus#(x,s(y)) -> p#(s(y))
    plus#(x,s(y)) -> plus#(x,p(s(y)))
    times#(s(x),y) -> times#(x,y)
    times#(s(x),y) -> plus#(y,times(x,y))
    div#(x,y) -> quot#(x,y,y)
    quot#(s(x),s(y),z) -> quot#(x,y,z)
    quot#(x,0(),s(z)) -> div#(x,s(z))
    div#(div(x,y),z) -> times#(y,z)
    div#(div(x,y),z) -> div#(x,times(y,z))
    eq#(s(x),s(y)) -> eq#(x,y)
    divides#(y,x) -> div#(x,y)
    divides#(y,x) -> times#(div(x,y),y)
    divides#(y,x) -> eq#(x,times(div(x,y),y))
    prime#(s(s(x))) -> pr#(s(s(x)),s(x))
    pr#(x,s(s(y))) -> divides#(s(s(y)),x)
    pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
    if#(false(),x,y) -> pr#(x,y)
   TRS:
    p(s(x)) -> x
    times(0(),y) -> 0()
    times(s(0()),y) -> y
    times(s(x),y) -> plus(y,times(x,y))
    plus(x,0()) -> x
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    plus(s(x),y) -> s(plus(p(s(x)),y))
    plus(x,s(y)) -> s(plus(x,p(s(y))))
    div(0(),y) -> 0()
    div(x,y) -> quot(x,y,y)
    div(div(x,y),z) -> div(x,times(y,z))
    quot(0(),s(y),z) -> 0()
    quot(s(x),s(y),z) -> quot(x,y,z)
    quot(x,0(),s(z)) -> s(div(x,s(z)))
    divides(y,x) -> eq(x,times(div(x,y),y))
    eq(0(),0()) -> true()
    eq(s(x),0()) -> false()
    eq(0(),s(y)) -> false()
    eq(s(x),s(y)) -> eq(x,y)
   Open