/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

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)
  TDG 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)
   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#(x,s(y)) -> plus#(x,p(s(y)))
    times#(s(x),y) -> plus#(y,times(x,y)) ->
    plus#(x,s(y)) -> p#(s(y))
    times#(s(x),y) -> plus#(y,times(x,y)) ->
    plus#(s(x),y) -> plus#(p(s(x)),y)
    times#(s(x),y) -> plus#(y,times(x,y)) ->
    plus#(s(x),y) -> p#(s(x))
    times#(s(x),y) -> plus#(y,times(x,y)) ->
    plus#(s(x),y) -> plus#(x,y)
    plus#(s(x),y) -> plus#(p(s(x)),y) ->
    plus#(x,s(y)) -> plus#(x,p(s(y)))
    plus#(s(x),y) -> plus#(p(s(x)),y) -> plus#(x,s(y)) -> p#(s(y))
    plus#(s(x),y) -> plus#(p(s(x)),y) ->
    plus#(s(x),y) -> plus#(p(s(x)),y)
    plus#(s(x),y) -> plus#(p(s(x)),y) -> plus#(s(x),y) -> p#(s(x))
    plus#(s(x),y) -> plus#(p(s(x)),y) -> plus#(s(x),y) -> plus#(x,y)
    plus#(s(x),y) -> plus#(x,y) -> plus#(x,s(y)) -> plus#(x,p(s(y)))
    plus#(s(x),y) -> plus#(x,y) -> plus#(x,s(y)) -> p#(s(y))
    plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(p(s(x)),y)
    plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> p#(s(x))
    plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y)
    plus#(x,s(y)) -> plus#(x,p(s(y))) ->
    plus#(x,s(y)) -> plus#(x,p(s(y)))
    plus#(x,s(y)) -> plus#(x,p(s(y))) -> plus#(x,s(y)) -> p#(s(y))
    plus#(x,s(y)) -> plus#(x,p(s(y))) ->
    plus#(s(x),y) -> plus#(p(s(x)),y)
    plus#(x,s(y)) -> plus#(x,p(s(y))) -> plus#(s(x),y) -> p#(s(x))
    plus#(x,s(y)) -> plus#(x,p(s(y))) -> plus#(s(x),y) -> plus#(x,y)
   SCC Processor:
    #sccs: 5
    #rules: 11
    #arcs: 49/400
    DPs:
     if#(false(),x,y) -> pr#(x,y)
     pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(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)
    Subterm Criterion Processor:
     simple projection:
      pi(pr#) = 1
      pi(if#) = 2
     problem:
      DPs:
       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)
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    DPs:
     eq#(s(x),s(y)) -> eq#(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)
    Subterm Criterion Processor:
     simple projection:
      pi(eq#) = 0
     problem:
      DPs:
       
      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)
     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:
     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)
    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:
       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)
     EDG Processor:
      DPs:
       div#(x,y) -> quot#(x,y,y)
       quot#(x,0(),s(z)) -> div#(x,s(z))
      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)
      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))
      Bounds Processor:
       bound: 3
       enrichment: top-dp
       automaton:
        final states: {3,2}
        transitions:
         quot0(1,1,1) -> 1*
         eq0(1,1) -> 1*
         true0() -> 1*
         false0() -> 1*
         divides0(1,1) -> 1*
         prime0(1) -> 1*
         pr0(1,1) -> 1*
         if0(1,1,1) -> 1*
         div{#,1}(1,8) -> 3*
         s1(1) -> 8*
         quot{#,1}(1,1,1) -> 2*
         div{#,2}(1,10) -> 2*
         s2(1) -> 10*
         quot{#,2}(1,8,8) -> 3*
         quot{#,3}(1,10,10) -> 2*
         div{#,0}(1,1) -> 2*
         quot{#,0}(1,1,1) -> 3*
         00() -> 1*
         s0(1) -> 1*
         p0(1) -> 1*
         plus0(1,1) -> 1*
         times0(1,1) -> 1*
         div0(1,1) -> 1*
       problem:
        DPs:
         
        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)
       Qed
    
    DPs:
     times#(s(x),y) -> times#(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)
    Subterm Criterion Processor:
     simple projection:
      pi(times#) = 0
     problem:
      DPs:
       
      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)
     Qed
    
    DPs:
     plus#(s(x),y) -> plus#(x,y)
     plus#(s(x),y) -> plus#(p(s(x)),y)
     plus#(x,s(y)) -> plus#(x,p(s(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)
    Subterm Criterion Processor:
     simple projection:
      pi(p) = 0
      pi(plus#) = 0
     problem:
      DPs:
       plus#(s(x),y) -> plus#(p(s(x)),y)
       plus#(x,s(y)) -> plus#(x,p(s(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#(p(s(x)),y)
       plus#(x,s(y)) -> plus#(x,p(s(y)))
      TRS:
       p(s(x)) -> x
      Semantic Labeling Processor:
       dimension: 2
       usable rules:
        
       interpretation:
                  [0 1]  
        [p](x0) = [1 0]x0,
        
                  [1 1]     [0]
        [s](x0) = [1 1]x0 + [1]
        labeled:
         plus#
        usable (for model):
         plus#
         s
         p
        argument filtering:
         pi(s) = 0
         pi(p) = []
         pi(plus#) = []
       precedence:
        plus# ~ p ~ s
       problem:
        DPs:
         plus#(s(x),y) -> plus#(p(s(x)),y)
        TRS:
         p(s(x)) -> x
       Restore Modifier:
        DPs:
         plus#(s(x),y) -> plus#(p(s(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#(p(s(x)),y)
         TRS:
          p(s(x)) -> x
         Semantic Labeling Processor:
          dimension: 2
          usable rules:
           
          interpretation:
                     [0 1]     [1]
           [p](x0) = [1 0]x0 + [0],
           
                     [0 1]     [0]
           [s](x0) = [1 1]x0 + [1]
           labeled:
            plus#
           usable (for model):
            plus#
            s
            p
           argument filtering:
            pi(s) = 0
            pi(p) = []
            pi(plus#) = []
          precedence:
           plus# ~ p ~ s
          problem:
           DPs:
            
           TRS:
            p(s(x)) -> x
          Qed