YES

Problem:
 plus(x,0()) -> x
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,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)
   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:
   plus(x,0()) -> x
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,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)
  TDG Processor:
   DPs:
    plus#(s(x),y) -> plus#(x,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:
    plus(x,0()) -> x
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,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)
   graph:
    if#(false(),x,y) -> pr#(x,y) ->
    pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
    if#(false(),x,y) -> pr#(x,y) ->
    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)
    pr#(x,s(s(y))) -> divides#(s(s(y)),x) ->
    divides#(y,x) -> eq#(x,times(div(x,y),y))
    pr#(x,s(s(y))) -> divides#(s(s(y)),x) ->
    divides#(y,x) -> times#(div(x,y),y)
    pr#(x,s(s(y))) -> divides#(s(s(y)),x) ->
    divides#(y,x) -> div#(x,y)
    prime#(s(s(x))) -> pr#(s(s(x)),s(x)) ->
    pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
    prime#(s(s(x))) -> pr#(s(s(x)),s(x)) ->
    pr#(x,s(s(y))) -> divides#(s(s(y)),x)
    divides#(y,x) -> eq#(x,times(div(x,y),y)) ->
    eq#(s(x),s(y)) -> eq#(x,y)
    divides#(y,x) -> div#(x,y) -> div#(div(x,y),z) -> div#(x,times(y,z))
    divides#(y,x) -> div#(x,y) -> div#(div(x,y),z) -> times#(y,z)
    divides#(y,x) -> div#(x,y) -> div#(x,y) -> quot#(x,y,y)
    divides#(y,x) -> times#(div(x,y),y) ->
    times#(s(x),y) -> plus#(y,times(x,y))
    divides#(y,x) -> times#(div(x,y),y) -> times#(s(x),y) -> times#(x,y)
    eq#(s(x),s(y)) -> eq#(x,y) -> eq#(s(x),s(y)) -> eq#(x,y)
    quot#(s(x),s(y),z) -> quot#(x,y,z) ->
    quot#(x,0(),s(z)) -> div#(x,s(z))
    quot#(s(x),s(y),z) -> quot#(x,y,z) ->
    quot#(s(x),s(y),z) -> quot#(x,y,z)
    quot#(x,0(),s(z)) -> div#(x,s(z)) ->
    div#(div(x,y),z) -> div#(x,times(y,z))
    quot#(x,0(),s(z)) -> div#(x,s(z)) ->
    div#(div(x,y),z) -> times#(y,z)
    quot#(x,0(),s(z)) -> div#(x,s(z)) ->
    div#(x,y) -> quot#(x,y,y)
    div#(div(x,y),z) -> div#(x,times(y,z)) ->
    div#(div(x,y),z) -> div#(x,times(y,z))
    div#(div(x,y),z) -> div#(x,times(y,z)) ->
    div#(div(x,y),z) -> times#(y,z)
    div#(div(x,y),z) -> div#(x,times(y,z)) ->
    div#(x,y) -> quot#(x,y,y)
    div#(div(x,y),z) -> times#(y,z) ->
    times#(s(x),y) -> plus#(y,times(x,y))
    div#(div(x,y),z) -> times#(y,z) -> times#(s(x),y) -> times#(x,y)
    div#(x,y) -> quot#(x,y,y) -> quot#(x,0(),s(z)) -> div#(x,s(z))
    div#(x,y) -> quot#(x,y,y) -> quot#(s(x),s(y),z) -> quot#(x,y,z)
    times#(s(x),y) -> times#(x,y) ->
    times#(s(x),y) -> plus#(y,times(x,y))
    times#(s(x),y) -> times#(x,y) ->
    times#(s(x),y) -> times#(x,y)
    times#(s(x),y) -> plus#(y,times(x,y)) -> plus#(s(x),y) -> plus#(x,y)
    plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y)
   SCC Processor:
    #sccs: 5
    #rules: 9
    #arcs: 31/256
    DPs:
     if#(false(),x,y) -> pr#(x,y)
     pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
    TRS:
     plus(x,0()) -> x
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,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)
    Subterm Criterion Processor:
     simple projection:
      pi(pr#) = 1
      pi(if#) = 2
     problem:
      DPs:
       if#(false(),x,y) -> pr#(x,y)
      TRS:
       plus(x,0()) -> x
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,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)
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    DPs:
     eq#(s(x),s(y)) -> eq#(x,y)
    TRS:
     plus(x,0()) -> x
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,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)
    Subterm Criterion Processor:
     simple projection:
      pi(eq#) = 1
     problem:
      DPs:
       
      TRS:
       plus(x,0()) -> x
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,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)
     Qed
    
    DPs:
     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) -> div#(x,times(y,z))
    TRS:
     plus(x,0()) -> x
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,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)
    Subterm Criterion Processor:
     simple projection:
      pi(div#) = 0
      pi(quot#) = 0
     problem:
      DPs:
       div#(x,y) -> quot#(x,y,y)
       quot#(x,0(),s(z)) -> div#(x,s(z))
      TRS:
       plus(x,0()) -> x
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,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)
     EDG Processor:
      DPs:
       div#(x,y) -> quot#(x,y,y)
       quot#(x,0(),s(z)) -> div#(x,s(z))
      TRS:
       plus(x,0()) -> x
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,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)
      graph:
       quot#(x,0(),s(z)) -> div#(x,s(z)) -> div#(x,y) -> quot#(x,y,y)
       div#(x,y) -> quot#(x,y,y) -> quot#(x,0(),s(z)) -> div#(x,s(z))
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [quot#](x0, x1, x2) = x1 + x2 + 0,
        
        [div#](x0, x1) = 1x1 + 1,
        
        [if](x0, x1, x2) = x1 + x2 + 0,
        
        [pr](x0, x1) = x0 + x1 + 0,
        
        [prime](x0) = x0 + 0,
        
        [divides](x0, x1) = x0 + 6x1 + 0,
        
        [false] = 0,
        
        [true] = 0,
        
        [eq](x0, x1) = 0,
        
        [quot](x0, x1, x2) = 1,
        
        [div](x0, x1) = x0 + 2,
        
        [times](x0, x1) = x1 + 1,
        
        [s](x0) = 0,
        
        [plus](x0, x1) = x0 + x1 + 0,
        
        [0] = 1
       orientation:
        div#(x,y) = 1y + 1 >= y + 0 = quot#(x,y,y)
        
        quot#(x,0(),s(z)) = 1 >= 1 = div#(x,s(z))
        
        plus(x,0()) = x + 1 >= x = x
        
        plus(0(),y) = y + 1 >= y = y
        
        plus(s(x),y) = y + 0 >= 0 = s(plus(x,y))
        
        times(0(),y) = y + 1 >= 1 = 0()
        
        times(s(0()),y) = y + 1 >= y = y
        
        times(s(x),y) = y + 1 >= y + 1 = plus(y,times(x,y))
        
        div(0(),y) = 2 >= 1 = 0()
        
        div(x,y) = x + 2 >= 1 = quot(x,y,y)
        
        quot(0(),s(y),z) = 1 >= 1 = 0()
        
        quot(s(x),s(y),z) = 1 >= 1 = quot(x,y,z)
        
        quot(x,0(),s(z)) = 1 >= 0 = s(div(x,s(z)))
        
        div(div(x,y),z) = x + 2 >= x + 2 = div(x,times(y,z))
        
        eq(0(),0()) = 0 >= 0 = true()
        
        eq(s(x),0()) = 0 >= 0 = false()
        
        eq(0(),s(y)) = 0 >= 0 = false()
        
        eq(s(x),s(y)) = 0 >= 0 = eq(x,y)
        
        divides(y,x) = 6x + y + 0 >= 0 = eq(x,times(div(x,y),y))
        
        prime(s(s(x))) = 0 >= 0 = pr(s(s(x)),s(x))
        
        pr(x,s(0())) = x + 0 >= 0 = true()
        
        pr(x,s(s(y))) = x + 0 >= x + 0 = if(divides(s(s(y)),x),x,s(y))
        
        if(true(),x,y) = x + y + 0 >= 0 = false()
        
        if(false(),x,y) = x + y + 0 >= x + y + 0 = pr(x,y)
       problem:
        DPs:
         quot#(x,0(),s(z)) -> div#(x,s(z))
        TRS:
         plus(x,0()) -> x
         plus(0(),y) -> y
         plus(s(x),y) -> s(plus(x,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)
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 2/1
        
    
    DPs:
     times#(s(x),y) -> times#(x,y)
    TRS:
     plus(x,0()) -> x
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,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)
    Subterm Criterion Processor:
     simple projection:
      pi(times#) = 0
     problem:
      DPs:
       
      TRS:
       plus(x,0()) -> x
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,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)
     Qed
    
    DPs:
     plus#(s(x),y) -> plus#(x,y)
    TRS:
     plus(x,0()) -> x
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,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)
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 0
     problem:
      DPs:
       
      TRS:
       plus(x,0()) -> x
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,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)
     Qed