/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(0()) -> 0()
 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(zero(y),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(zero(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)
 zero(div(x,x)) -> x
 zero(divides(x,x)) -> x
 zero(times(x,x)) -> x
 zero(quot(x,x,x)) -> x
 zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))

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) -> zero#(y)
   div#(div(x,y),z) -> times#(zero(y),z)
   div#(div(x,y),z) -> div#(x,times(zero(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)
   zero#(s(x)) -> plus#(0(),zero(0()))
   zero#(s(x)) -> zero#(0())
   zero#(s(x)) -> plus#(zero(0()),0())
   zero#(s(x)) -> eq#(x,s(0()))
   zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
  TRS:
   p(0()) -> 0()
   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(zero(y),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(zero(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)
   zero(div(x,x)) -> x
   zero(divides(x,x)) -> x
   zero(times(x,x)) -> x
   zero(quot(x,x,x)) -> x
   zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
  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) -> zero#(y)
    div#(div(x,y),z) -> times#(zero(y),z)
    div#(div(x,y),z) -> div#(x,times(zero(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)
    zero#(s(x)) -> plus#(0(),zero(0()))
    zero#(s(x)) -> zero#(0())
    zero#(s(x)) -> plus#(zero(0()),0())
    zero#(s(x)) -> eq#(x,s(0()))
    zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
   TRS:
    p(0()) -> 0()
    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(zero(y),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(zero(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)
    zero(div(x,x)) -> x
    zero(divides(x,x)) -> x
    zero(times(x,x)) -> x
    zero(quot(x,x,x)) -> x
    zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
   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(zero(y),z))
    divides#(y,x) -> div#(x,y) -> div#(div(x,y),z) -> times#(zero(y),z)
    divides#(y,x) -> div#(x,y) -> div#(div(x,y),z) -> zero#(y)
    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)
    zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0())))) ->
    if#(false(),x,y) -> pr#(x,y)
    zero#(s(x)) -> eq#(x,s(0())) -> eq#(s(x),s(y)) -> eq#(x,y)
    zero#(s(x)) -> zero#(0()) ->
    zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
    zero#(s(x)) -> zero#(0()) -> zero#(s(x)) -> eq#(x,s(0()))
    zero#(s(x)) -> zero#(0()) -> zero#(s(x)) -> plus#(zero(0()),0())
    zero#(s(x)) -> zero#(0()) -> zero#(s(x)) -> zero#(0())
    zero#(s(x)) -> zero#(0()) ->
    zero#(s(x)) -> plus#(0(),zero(0()))
    zero#(s(x)) -> plus#(zero(0()),0()) ->
    plus#(x,s(y)) -> plus#(x,p(s(y)))
    zero#(s(x)) -> plus#(zero(0()),0()) ->
    plus#(x,s(y)) -> p#(s(y))
    zero#(s(x)) -> plus#(zero(0()),0()) ->
    plus#(s(x),y) -> plus#(p(s(x)),y)
    zero#(s(x)) -> plus#(zero(0()),0()) ->
    plus#(s(x),y) -> p#(s(x))
    zero#(s(x)) -> plus#(zero(0()),0()) ->
    plus#(s(x),y) -> plus#(x,y)
    zero#(s(x)) -> plus#(0(),zero(0())) ->
    plus#(x,s(y)) -> plus#(x,p(s(y)))
    zero#(s(x)) -> plus#(0(),zero(0())) ->
    plus#(x,s(y)) -> p#(s(y))
    zero#(s(x)) -> plus#(0(),zero(0())) ->
    plus#(s(x),y) -> plus#(p(s(x)),y)
    zero#(s(x)) -> plus#(0(),zero(0())) ->
    plus#(s(x),y) -> p#(s(x))
    zero#(s(x)) -> plus#(0(),zero(0())) ->
    plus#(s(x),y) -> plus#(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(zero(y),z))
    quot#(x,0(),s(z)) -> div#(x,s(z)) ->
    div#(div(x,y),z) -> times#(zero(y),z)
    quot#(x,0(),s(z)) -> div#(x,s(z)) ->
    div#(div(x,y),z) -> zero#(y)
    quot#(x,0(),s(z)) -> div#(x,s(z)) -> div#(x,y) -> quot#(x,y,y)
    div#(div(x,y),z) -> zero#(y) ->
    zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
    div#(div(x,y),z) -> zero#(y) -> zero#(s(x)) -> eq#(x,s(0()))
    div#(div(x,y),z) -> zero#(y) -> zero#(s(x)) -> plus#(zero(0()),0())
    div#(div(x,y),z) -> zero#(y) -> zero#(s(x)) -> zero#(0())
    div#(div(x,y),z) -> zero#(y) ->
    zero#(s(x)) -> plus#(0(),zero(0()))
    div#(div(x,y),z) -> div#(x,times(zero(y),z)) ->
    div#(div(x,y),z) -> div#(x,times(zero(y),z))
    div#(div(x,y),z) -> div#(x,times(zero(y),z)) ->
    div#(div(x,y),z) -> times#(zero(y),z)
    div#(div(x,y),z) -> div#(x,times(zero(y),z)) ->
    div#(div(x,y),z) -> zero#(y)
    div#(div(x,y),z) -> div#(x,times(zero(y),z)) ->
    div#(x,y) -> quot#(x,y,y)
    div#(div(x,y),z) -> times#(zero(y),z) ->
    times#(s(x),y) -> plus#(y,times(x,y))
    div#(div(x,y),z) -> times#(zero(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: 4
    #rules: 16
    #arcs: 74/676
    DPs:
     if#(false(),x,y) -> pr#(x,y)
     pr#(x,s(s(y))) -> divides#(s(s(y)),x)
     divides#(y,x) -> div#(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) -> zero#(y)
     zero#(s(x)) -> zero#(0())
     zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
     div#(div(x,y),z) -> div#(x,times(zero(y),z))
     pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
    TRS:
     p(0()) -> 0()
     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(zero(y),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(zero(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)
     zero(div(x,x)) -> x
     zero(divides(x,x)) -> x
     zero(times(x,x)) -> x
     zero(quot(x,x,x)) -> x
     zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
    EDG Processor:
     DPs:
      if#(false(),x,y) -> pr#(x,y)
      pr#(x,s(s(y))) -> divides#(s(s(y)),x)
      divides#(y,x) -> div#(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) -> zero#(y)
      zero#(s(x)) -> zero#(0())
      zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
      div#(div(x,y),z) -> div#(x,times(zero(y),z))
      pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
     TRS:
      p(0()) -> 0()
      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(zero(y),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(zero(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)
      zero(div(x,x)) -> x
      zero(divides(x,x)) -> x
      zero(times(x,x)) -> x
      zero(quot(x,x,x)) -> x
      zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
     graph:
      if#(false(),x,y) -> pr#(x,y) ->
      pr#(x,s(s(y))) -> divides#(s(s(y)),x)
      if#(false(),x,y) -> pr#(x,y) ->
      pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(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) -> divides#(y,x) -> div#(x,y)
      divides#(y,x) -> div#(x,y) -> div#(x,y) -> quot#(x,y,y)
      divides#(y,x) -> div#(x,y) -> div#(div(x,y),z) -> zero#(y)
      divides#(y,x) -> div#(x,y) ->
      div#(div(x,y),z) -> div#(x,times(zero(y),z))
      zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0())))) ->
      if#(false(),x,y) -> pr#(x,y)
      quot#(s(x),s(y),z) -> quot#(x,y,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))
      quot#(x,0(),s(z)) -> div#(x,s(z)) ->
      div#(x,y) -> quot#(x,y,y)
      quot#(x,0(),s(z)) -> div#(x,s(z)) ->
      div#(div(x,y),z) -> zero#(y)
      quot#(x,0(),s(z)) -> div#(x,s(z)) ->
      div#(div(x,y),z) -> div#(x,times(zero(y),z))
      div#(div(x,y),z) -> zero#(y) -> zero#(s(x)) -> zero#(0())
      div#(div(x,y),z) -> zero#(y) ->
      zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
      div#(div(x,y),z) -> div#(x,times(zero(y),z)) ->
      div#(x,y) -> quot#(x,y,y)
      div#(div(x,y),z) -> div#(x,times(zero(y),z)) ->
      div#(div(x,y),z) -> zero#(y)
      div#(div(x,y),z) -> div#(x,times(zero(y),z)) ->
      div#(div(x,y),z) -> div#(x,times(zero(y),z))
      div#(x,y) -> quot#(x,y,y) -> quot#(s(x),s(y),z) -> quot#(x,y,z)
      div#(x,y) -> quot#(x,y,y) -> quot#(x,0(),s(z)) -> div#(x,s(z))
     SCC Processor:
      #sccs: 1
      #rules: 10
      #arcs: 20/121
      DPs:
       if#(false(),x,y) -> pr#(x,y)
       pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
       pr#(x,s(s(y))) -> divides#(s(s(y)),x)
       divides#(y,x) -> div#(x,y)
       div#(div(x,y),z) -> div#(x,times(zero(y),z))
       div#(div(x,y),z) -> zero#(y)
       zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
       div#(x,y) -> quot#(x,y,y)
       quot#(x,0(),s(z)) -> div#(x,s(z))
       quot#(s(x),s(y),z) -> quot#(x,y,z)
      TRS:
       p(0()) -> 0()
       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(zero(y),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(zero(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)
       zero(div(x,x)) -> x
       zero(divides(x,x)) -> x
       zero(times(x,x)) -> x
       zero(quot(x,x,x)) -> x
       zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
      Usable Rule Processor:
       DPs:
        if#(false(),x,y) -> pr#(x,y)
        pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
        pr#(x,s(s(y))) -> divides#(s(s(y)),x)
        divides#(y,x) -> div#(x,y)
        div#(div(x,y),z) -> div#(x,times(zero(y),z))
        div#(div(x,y),z) -> zero#(y)
        zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
        div#(x,y) -> quot#(x,y,y)
        quot#(x,0(),s(z)) -> div#(x,s(z))
        quot#(s(x),s(y),z) -> quot#(x,y,z)
       TRS:
        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)
        div(0(),y) -> 0()
        div(x,y) -> quot(x,y,y)
        div(div(x,y),z) -> div(x,times(zero(y),z))
        quot(zero(y),s(y),z) -> 0()
        quot(s(x),s(y),z) -> quot(x,y,z)
        quot(x,0(),s(z)) -> s(div(x,s(z)))
        zero(div(x,x)) -> x
        zero(divides(x,x)) -> x
        zero(times(x,x)) -> x
        zero(quot(x,x,x)) -> x
        zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
        if(true(),x,y) -> false()
        if(false(),x,y) -> pr(x,y)
        plus(0(),y) -> y
        plus(x,0()) -> x
        pr(x,s(0())) -> true()
        pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
        times(0(),y) -> 0()
        times(s(0()),y) -> y
        times(s(x),y) -> plus(y,times(x,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))))
        p(s(x)) -> x
       Matrix Interpretation Processor: dim=1
        
        usable rules:
         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)
         zero(div(x,x)) -> x
         zero(divides(x,x)) -> x
         zero(times(x,x)) -> x
         zero(quot(x,x,x)) -> x
         zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
         if(true(),x,y) -> false()
         if(false(),x,y) -> pr(x,y)
         plus(0(),y) -> y
         plus(x,0()) -> x
         pr(x,s(0())) -> true()
         pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(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))))
         p(s(x)) -> x
        interpretation:
         [if#](x0, x1, x2) = 4x1,
         
         [pr#](x0, x1) = 4x0,
         
         [divides#](x0, x1) = 2x1,
         
         [zero#](x0) = 4,
         
         [quot#](x0, x1, x2) = 2x0,
         
         [div#](x0, x1) = 2x0,
         
         [if](x0, x1, x2) = 0,
         
         [pr](x0, x1) = 0,
         
         [divides](x0, x1) = x1 + 1,
         
         [false] = 0,
         
         [true] = 0,
         
         [eq](x0, x1) = 0,
         
         [zero](x0) = 4x0 + 1,
         
         [quot](x0, x1, x2) = 2x2,
         
         [div](x0, x1) = x0 + 2,
         
         [times](x0, x1) = 4x0,
         
         [plus](x0, x1) = x0 + x1,
         
         [s](x0) = x0,
         
         [p](x0) = x0,
         
         [0] = 0
        orientation:
         if#(false(),x,y) = 4x >= 4x = pr#(x,y)
         
         pr#(x,s(s(y))) = 4x >= 4x = if#(divides(s(s(y)),x),x,s(y))
         
         pr#(x,s(s(y))) = 4x >= 2x = divides#(s(s(y)),x)
         
         divides#(y,x) = 2x >= 2x = div#(x,y)
         
         div#(div(x,y),z) = 2x + 4 >= 2x = div#(x,times(zero(y),z))
         
         div#(div(x,y),z) = 2x + 4 >= 4 = zero#(y)
         
         zero#(s(x)) = 4 >= 4 = if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
         
         div#(x,y) = 2x >= 2x = quot#(x,y,y)
         
         quot#(x,0(),s(z)) = 2x >= 2x = div#(x,s(z))
         
         quot#(s(x),s(y),z) = 2x >= 2x = quot#(x,y,z)
         
         divides(y,x) = x + 1 >= 0 = eq(x,times(div(x,y),y))
         
         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)
         
         div(0(),y) = 2 >= 0 = 0()
         
         div(x,y) = x + 2 >= 2y = quot(x,y,y)
         
         div(div(x,y),z) = x + 4 >= x + 2 = div(x,times(zero(y),z))
         
         quot(zero(y),s(y),z) = 2z >= 0 = 0()
         
         quot(s(x),s(y),z) = 2z >= 2z = quot(x,y,z)
         
         quot(x,0(),s(z)) = 2z >= x + 2 = s(div(x,s(z)))
         
         zero(div(x,x)) = 4x + 9 >= x = x
         
         zero(divides(x,x)) = 4x + 5 >= x = x
         
         zero(times(x,x)) = 16x + 1 >= x = x
         
         zero(quot(x,x,x)) = 8x + 1 >= x = x
         
         zero(s(x)) = 4x + 1 >= 0 = if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
         
         if(true(),x,y) = 0 >= 0 = false()
         
         if(false(),x,y) = 0 >= 0 = pr(x,y)
         
         plus(0(),y) = y >= y = y
         
         plus(x,0()) = x >= x = x
         
         pr(x,s(0())) = 0 >= 0 = true()
         
         pr(x,s(s(y))) = 0 >= 0 = if(divides(s(s(y)),x),x,s(y))
         
         times(0(),y) = 0 >= 0 = 0()
         
         times(s(0()),y) = 0 >= y = y
         
         times(s(x),y) = 4x >= 4x + y = plus(y,times(x,y))
         
         plus(s(x),y) = x + y >= x + y = s(plus(x,y))
         
         plus(s(x),y) = x + y >= x + y = s(plus(p(s(x)),y))
         
         plus(x,s(y)) = x + y >= x + y = s(plus(x,p(s(y))))
         
         p(s(x)) = x >= x = x
        problem:
         DPs:
          if#(false(),x,y) -> pr#(x,y)
          pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
          pr#(x,s(s(y))) -> divides#(s(s(y)),x)
          divides#(y,x) -> div#(x,y)
          div#(div(x,y),z) -> zero#(y)
          zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
          div#(x,y) -> quot#(x,y,y)
          quot#(x,0(),s(z)) -> div#(x,s(z))
          quot#(s(x),s(y),z) -> quot#(x,y,z)
         TRS:
          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)
          div(0(),y) -> 0()
          div(x,y) -> quot(x,y,y)
          div(div(x,y),z) -> div(x,times(zero(y),z))
          quot(zero(y),s(y),z) -> 0()
          quot(s(x),s(y),z) -> quot(x,y,z)
          quot(x,0(),s(z)) -> s(div(x,s(z)))
          zero(div(x,x)) -> x
          zero(divides(x,x)) -> x
          zero(times(x,x)) -> x
          zero(quot(x,x,x)) -> x
          zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
          if(true(),x,y) -> false()
          if(false(),x,y) -> pr(x,y)
          plus(0(),y) -> y
          plus(x,0()) -> x
          pr(x,s(0())) -> true()
          pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
          times(0(),y) -> 0()
          times(s(0()),y) -> y
          times(s(x),y) -> plus(y,times(x,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))))
          p(s(x)) -> x
        Restore Modifier:
         DPs:
          if#(false(),x,y) -> pr#(x,y)
          pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
          pr#(x,s(s(y))) -> divides#(s(s(y)),x)
          divides#(y,x) -> div#(x,y)
          div#(div(x,y),z) -> zero#(y)
          zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
          div#(x,y) -> quot#(x,y,y)
          quot#(x,0(),s(z)) -> div#(x,s(z))
          quot#(s(x),s(y),z) -> quot#(x,y,z)
         TRS:
          p(0()) -> 0()
          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(zero(y),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(zero(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)
          zero(div(x,x)) -> x
          zero(divides(x,x)) -> x
          zero(times(x,x)) -> x
          zero(quot(x,x,x)) -> x
          zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
         Usable Rule Processor:
          DPs:
           if#(false(),x,y) -> pr#(x,y)
           pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
           pr#(x,s(s(y))) -> divides#(s(s(y)),x)
           divides#(y,x) -> div#(x,y)
           div#(div(x,y),z) -> zero#(y)
           zero#(s(x)) -> if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
           div#(x,y) -> quot#(x,y,y)
           quot#(x,0(),s(z)) -> div#(x,s(z))
           quot#(s(x),s(y),z) -> quot#(x,y,z)
          TRS:
           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)
           div(0(),y) -> 0()
           div(x,y) -> quot(x,y,y)
           div(div(x,y),z) -> div(x,times(zero(y),z))
           quot(zero(y),s(y),z) -> 0()
           quot(s(x),s(y),z) -> quot(x,y,z)
           quot(x,0(),s(z)) -> s(div(x,s(z)))
           zero(div(x,x)) -> x
           zero(divides(x,x)) -> x
           zero(times(x,x)) -> x
           zero(quot(x,x,x)) -> x
           zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
           if(true(),x,y) -> false()
           if(false(),x,y) -> pr(x,y)
           plus(0(),y) -> y
           plus(x,0()) -> x
           pr(x,s(0())) -> true()
           pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
           times(0(),y) -> 0()
           times(s(0()),y) -> y
           times(s(x),y) -> plus(y,times(x,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))))
           p(s(x)) -> x
          Matrix Interpretation Processor: dim=1
           
           usable rules:
            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)
            zero(div(x,x)) -> x
            zero(divides(x,x)) -> x
            zero(times(x,x)) -> x
            zero(quot(x,x,x)) -> x
            zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
            if(true(),x,y) -> false()
            if(false(),x,y) -> pr(x,y)
            plus(0(),y) -> y
            plus(x,0()) -> x
            pr(x,s(0())) -> true()
            pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(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))))
            p(s(x)) -> x
           interpretation:
            [if#](x0, x1, x2) = x1 + 4x2,
            
            [pr#](x0, x1) = x0 + 4x1,
            
            [divides#](x0, x1) = x1,
            
            [zero#](x0) = 6,
            
            [quot#](x0, x1, x2) = x0,
            
            [div#](x0, x1) = x0,
            
            [if](x0, x1, x2) = x1,
            
            [pr](x0, x1) = x0,
            
            [divides](x0, x1) = 2x0 + 2,
            
            [false] = 0,
            
            [true] = 0,
            
            [eq](x0, x1) = 0,
            
            [zero](x0) = 2x0 + 1,
            
            [quot](x0, x1, x2) = 2x0 + 4,
            
            [div](x0, x1) = x0 + 6,
            
            [times](x0, x1) = 2x1,
            
            [plus](x0, x1) = x0 + x1,
            
            [s](x0) = x0,
            
            [p](x0) = x0,
            
            [0] = 0
           orientation:
            if#(false(),x,y) = x + 4y >= x + 4y = pr#(x,y)
            
            pr#(x,s(s(y))) = x + 4y >= x + 4y = if#(divides(s(s(y)),x),x,s(y))
            
            pr#(x,s(s(y))) = x + 4y >= x = divides#(s(s(y)),x)
            
            divides#(y,x) = x >= x = div#(x,y)
            
            div#(div(x,y),z) = x + 6 >= 6 = zero#(y)
            
            zero#(s(x)) = 6 >= 5 = if#(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
            
            div#(x,y) = x >= x = quot#(x,y,y)
            
            quot#(x,0(),s(z)) = x >= x = div#(x,s(z))
            
            quot#(s(x),s(y),z) = x >= x = quot#(x,y,z)
            
            divides(y,x) = 2y + 2 >= 0 = eq(x,times(div(x,y),y))
            
            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)
            
            div(0(),y) = 6 >= 0 = 0()
            
            div(x,y) = x + 6 >= 2x + 4 = quot(x,y,y)
            
            div(div(x,y),z) = x + 12 >= x + 6 = div(x,times(zero(y),z))
            
            quot(zero(y),s(y),z) = 4y + 6 >= 0 = 0()
            
            quot(s(x),s(y),z) = 2x + 4 >= 2x + 4 = quot(x,y,z)
            
            quot(x,0(),s(z)) = 2x + 4 >= x + 6 = s(div(x,s(z)))
            
            zero(div(x,x)) = 2x + 13 >= x = x
            
            zero(divides(x,x)) = 4x + 5 >= x = x
            
            zero(times(x,x)) = 4x + 1 >= x = x
            
            zero(quot(x,x,x)) = 4x + 9 >= x = x
            
            zero(s(x)) = 2x + 1 >= 1 = if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
            
            if(true(),x,y) = x >= 0 = false()
            
            if(false(),x,y) = x >= x = pr(x,y)
            
            plus(0(),y) = y >= y = y
            
            plus(x,0()) = x >= x = x
            
            pr(x,s(0())) = x >= 0 = true()
            
            pr(x,s(s(y))) = x >= x = if(divides(s(s(y)),x),x,s(y))
            
            times(0(),y) = 2y >= 0 = 0()
            
            times(s(0()),y) = 2y >= y = y
            
            times(s(x),y) = 2y >= 3y = plus(y,times(x,y))
            
            plus(s(x),y) = x + y >= x + y = s(plus(x,y))
            
            plus(s(x),y) = x + y >= x + y = s(plus(p(s(x)),y))
            
            plus(x,s(y)) = x + y >= x + y = s(plus(x,p(s(y))))
            
            p(s(x)) = x >= x = x
           problem:
            DPs:
             if#(false(),x,y) -> pr#(x,y)
             pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
             pr#(x,s(s(y))) -> divides#(s(s(y)),x)
             divides#(y,x) -> div#(x,y)
             div#(div(x,y),z) -> zero#(y)
             div#(x,y) -> quot#(x,y,y)
             quot#(x,0(),s(z)) -> div#(x,s(z))
             quot#(s(x),s(y),z) -> quot#(x,y,z)
            TRS:
             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)
             div(0(),y) -> 0()
             div(x,y) -> quot(x,y,y)
             div(div(x,y),z) -> div(x,times(zero(y),z))
             quot(zero(y),s(y),z) -> 0()
             quot(s(x),s(y),z) -> quot(x,y,z)
             quot(x,0(),s(z)) -> s(div(x,s(z)))
             zero(div(x,x)) -> x
             zero(divides(x,x)) -> x
             zero(times(x,x)) -> x
             zero(quot(x,x,x)) -> x
             zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
             if(true(),x,y) -> false()
             if(false(),x,y) -> pr(x,y)
             plus(0(),y) -> y
             plus(x,0()) -> x
             pr(x,s(0())) -> true()
             pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
             times(0(),y) -> 0()
             times(s(0()),y) -> y
             times(s(x),y) -> plus(y,times(x,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))))
             p(s(x)) -> x
           Restore Modifier:
            DPs:
             if#(false(),x,y) -> pr#(x,y)
             pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
             pr#(x,s(s(y))) -> divides#(s(s(y)),x)
             divides#(y,x) -> div#(x,y)
             div#(div(x,y),z) -> zero#(y)
             div#(x,y) -> quot#(x,y,y)
             quot#(x,0(),s(z)) -> div#(x,s(z))
             quot#(s(x),s(y),z) -> quot#(x,y,z)
            TRS:
             p(0()) -> 0()
             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(zero(y),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(zero(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)
             zero(div(x,x)) -> x
             zero(divides(x,x)) -> x
             zero(times(x,x)) -> x
             zero(quot(x,x,x)) -> x
             zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
            SCC Processor:
             #sccs: 2
             #rules: 5
             #arcs: 19/64
             DPs:
              if#(false(),x,y) -> pr#(x,y)
              pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y))
             TRS:
              p(0()) -> 0()
              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(zero(y),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(zero(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)
              zero(div(x,x)) -> x
              zero(divides(x,x)) -> x
              zero(times(x,x)) -> x
              zero(quot(x,x,x)) -> x
              zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
             Size-Change Termination Processor:
              DPs:
               
              TRS:
               p(0()) -> 0()
               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(zero(y),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(zero(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)
               zero(div(x,x)) -> x
               zero(divides(x,x)) -> x
               zero(times(x,x)) -> x
               zero(quot(x,x,x)) -> x
               zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
              The DP: if#(false(),x,y) -> pr#(x,y) has the edges:
                1 >=  0
                2 >=  1
              The DP: pr#(x,s(s(y))) -> if#(divides(s(s(y)),x),x,s(y)) has the edges:
                0 >=  1
                1 >   2
              Qed
             
             DPs:
              div#(x,y) -> quot#(x,y,y)
              quot#(x,0(),s(z)) -> div#(x,s(z))
              quot#(s(x),s(y),z) -> quot#(x,y,z)
             TRS:
              p(0()) -> 0()
              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(zero(y),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(zero(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)
              zero(div(x,x)) -> x
              zero(divides(x,x)) -> x
              zero(times(x,x)) -> x
              zero(quot(x,x,x)) -> x
              zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
             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(0()) -> 0()
                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(zero(y),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(zero(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)
                zero(div(x,x)) -> x
                zero(divides(x,x)) -> x
                zero(times(x,x)) -> x
                zero(quot(x,x,x)) -> x
                zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
              Usable Rule Processor:
               DPs:
                div#(x,y) -> quot#(x,y,y)
                quot#(x,0(),s(z)) -> div#(x,s(z))
               TRS:
                
               Arctic Interpretation Processor:
                dimension: 1
                usable rules:
                 
                interpretation:
                 [quot#](x0, x1, x2) = x1 + x2,
                 
                 [div#](x0, x1) = x1 + 0,
                 
                 [s](x0) = 2,
                 
                 [0] = 7
                orientation:
                 div#(x,y) = y + 0 >= y = quot#(x,y,y)
                 
                 quot#(x,0(),s(z)) = 7 >= 2 = div#(x,s(z))
                problem:
                 DPs:
                  div#(x,y) -> quot#(x,y,y)
                 TRS:
                  
                Restore Modifier:
                 DPs:
                  div#(x,y) -> quot#(x,y,y)
                 TRS:
                  p(0()) -> 0()
                  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(zero(y),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(zero(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)
                  zero(div(x,x)) -> x
                  zero(divides(x,x)) -> x
                  zero(times(x,x)) -> x
                  zero(quot(x,x,x)) -> x
                  zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
                 SCC Processor:
                  #sccs: 0
                  #rules: 0
                  #arcs: 5/1
                  
    
    DPs:
     times#(s(x),y) -> times#(x,y)
    TRS:
     p(0()) -> 0()
     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(zero(y),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(zero(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)
     zero(div(x,x)) -> x
     zero(divides(x,x)) -> x
     zero(times(x,x)) -> x
     zero(quot(x,x,x)) -> x
     zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
    Subterm Criterion Processor:
     simple projection:
      pi(times#) = 0
     problem:
      DPs:
       
      TRS:
       p(0()) -> 0()
       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(zero(y),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(zero(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)
       zero(div(x,x)) -> x
       zero(divides(x,x)) -> x
       zero(times(x,x)) -> x
       zero(quot(x,x,x)) -> x
       zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
     Qed
    
    DPs:
     eq#(s(x),s(y)) -> eq#(x,y)
    TRS:
     p(0()) -> 0()
     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(zero(y),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(zero(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)
     zero(div(x,x)) -> x
     zero(divides(x,x)) -> x
     zero(times(x,x)) -> x
     zero(quot(x,x,x)) -> x
     zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
    Subterm Criterion Processor:
     simple projection:
      pi(eq#) = 0
     problem:
      DPs:
       
      TRS:
       p(0()) -> 0()
       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(zero(y),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(zero(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)
       zero(div(x,x)) -> x
       zero(divides(x,x)) -> x
       zero(times(x,x)) -> x
       zero(quot(x,x,x)) -> x
       zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
     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(0()) -> 0()
     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(zero(y),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(zero(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)
     zero(div(x,x)) -> x
     zero(divides(x,x)) -> x
     zero(times(x,x)) -> x
     zero(quot(x,x,x)) -> x
     zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
    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(0()) -> 0()
       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(zero(y),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(zero(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)
       zero(div(x,x)) -> x
       zero(divides(x,x)) -> x
       zero(times(x,x)) -> x
       zero(quot(x,x,x)) -> x
       zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
     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:
                  [1 1]     [0]
        [s](x0) = [1 1]x0 + [1],
        
                  [0 1]  
        [p](x0) = [1 0]x0
        labeled:
         plus#
        usable (for model):
         plus#
         s
         p
        argument filtering:
         pi(p) = []
         pi(s) = 0
         pi(plus#) = []
       precedence:
        plus# ~ s ~ p
       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(0()) -> 0()
         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(zero(y),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(zero(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)
         zero(div(x,x)) -> x
         zero(divides(x,x)) -> x
         zero(times(x,x)) -> x
         zero(quot(x,x,x)) -> x
         zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
        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]     [0]
           [s](x0) = [1 1]x0 + [1],
           
                     [0 1]     [1]
           [p](x0) = [1 0]x0 + [0]
           labeled:
            plus#
           usable (for model):
            plus#
            s
            p
           argument filtering:
            pi(p) = []
            pi(s) = 0
            pi(plus#) = []
          precedence:
           plus# ~ s ~ p
          problem:
           DPs:
            
           TRS:
            p(s(x)) -> x
          Qed