YES

Problem:
 nonZero(0()) -> false()
 nonZero(s(x)) -> true()
 p(s(0())) -> 0()
 p(s(s(x))) -> s(p(s(x)))
 id_inc(x) -> x
 id_inc(x) -> s(x)
 random(x) -> rand(x,0())
 rand(x,y) -> if(nonZero(x),x,y)
 if(false(),x,y) -> y
 if(true(),x,y) -> rand(p(x),id_inc(y))

Proof:
 DP Processor:
  DPs:
   p#(s(s(x))) -> p#(s(x))
   random#(x) -> rand#(x,0())
   rand#(x,y) -> nonZero#(x)
   rand#(x,y) -> if#(nonZero(x),x,y)
   if#(true(),x,y) -> id_inc#(y)
   if#(true(),x,y) -> p#(x)
   if#(true(),x,y) -> rand#(p(x),id_inc(y))
  TRS:
   nonZero(0()) -> false()
   nonZero(s(x)) -> true()
   p(s(0())) -> 0()
   p(s(s(x))) -> s(p(s(x)))
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x,0())
   rand(x,y) -> if(nonZero(x),x,y)
   if(false(),x,y) -> y
   if(true(),x,y) -> rand(p(x),id_inc(y))
  TDG Processor:
   DPs:
    p#(s(s(x))) -> p#(s(x))
    random#(x) -> rand#(x,0())
    rand#(x,y) -> nonZero#(x)
    rand#(x,y) -> if#(nonZero(x),x,y)
    if#(true(),x,y) -> id_inc#(y)
    if#(true(),x,y) -> p#(x)
    if#(true(),x,y) -> rand#(p(x),id_inc(y))
   TRS:
    nonZero(0()) -> false()
    nonZero(s(x)) -> true()
    p(s(0())) -> 0()
    p(s(s(x))) -> s(p(s(x)))
    id_inc(x) -> x
    id_inc(x) -> s(x)
    random(x) -> rand(x,0())
    rand(x,y) -> if(nonZero(x),x,y)
    if(false(),x,y) -> y
    if(true(),x,y) -> rand(p(x),id_inc(y))
   graph:
    if#(true(),x,y) -> rand#(p(x),id_inc(y)) ->
    rand#(x,y) -> if#(nonZero(x),x,y)
    if#(true(),x,y) -> rand#(p(x),id_inc(y)) -> rand#(x,y) -> nonZero#(x)
    if#(true(),x,y) -> p#(x) -> p#(s(s(x))) -> p#(s(x))
    rand#(x,y) -> if#(nonZero(x),x,y) ->
    if#(true(),x,y) -> rand#(p(x),id_inc(y))
    rand#(x,y) -> if#(nonZero(x),x,y) -> if#(true(),x,y) -> p#(x)
    rand#(x,y) -> if#(nonZero(x),x,y) -> if#(true(),x,y) -> id_inc#(y)
    random#(x) -> rand#(x,0()) -> rand#(x,y) -> if#(nonZero(x),x,y)
    random#(x) -> rand#(x,0()) -> rand#(x,y) -> nonZero#(x)
    p#(s(s(x))) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 9/49
    DPs:
     if#(true(),x,y) -> rand#(p(x),id_inc(y))
     rand#(x,y) -> if#(nonZero(x),x,y)
    TRS:
     nonZero(0()) -> false()
     nonZero(s(x)) -> true()
     p(s(0())) -> 0()
     p(s(s(x))) -> s(p(s(x)))
     id_inc(x) -> x
     id_inc(x) -> s(x)
     random(x) -> rand(x,0())
     rand(x,y) -> if(nonZero(x),x,y)
     if(false(),x,y) -> y
     if(true(),x,y) -> rand(p(x),id_inc(y))
    Usable Rule Processor:
     DPs:
      if#(true(),x,y) -> rand#(p(x),id_inc(y))
      rand#(x,y) -> if#(nonZero(x),x,y)
     TRS:
      id_inc(x) -> x
      id_inc(x) -> s(x)
      p(s(0())) -> 0()
      p(s(s(x))) -> s(p(s(x)))
      nonZero(0()) -> false()
      nonZero(s(x)) -> true()
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
       nonZero(0()) -> false()
       nonZero(s(x)) -> true()
      interpretation:
       [if#](x0, x1, x2) = x0 + x1 + 0,
       
       [rand#](x0, x1) = 1x0 + 2,
       
       [id_inc](x0) = x0 + 2,
       
       [p](x0) = -2x0 + 0,
       
       [true] = 2,
       
       [s](x0) = 2x0 + 4,
       
       [false] = 1,
       
       [nonZero](x0) = x0 + 0,
       
       [0] = 2
      orientation:
       if#(true(),x,y) = x + 2 >= -1x + 2 = rand#(p(x),id_inc(y))
       
       rand#(x,y) = 1x + 2 >= x + 0 = if#(nonZero(x),x,y)
       
       id_inc(x) = x + 2 >= x = x
       
       id_inc(x) = x + 2 >= 2x + 4 = s(x)
       
       p(s(0())) = 2 >= 2 = 0()
       
       p(s(s(x))) = 2x + 4 >= 2x + 4 = s(p(s(x)))
       
       nonZero(0()) = 2 >= 1 = false()
       
       nonZero(s(x)) = 2x + 4 >= 2 = true()
      problem:
       DPs:
        if#(true(),x,y) -> rand#(p(x),id_inc(y))
       TRS:
        id_inc(x) -> x
        id_inc(x) -> s(x)
        p(s(0())) -> 0()
        p(s(s(x))) -> s(p(s(x)))
        nonZero(0()) -> false()
        nonZero(s(x)) -> true()
      Restore Modifier:
       DPs:
        if#(true(),x,y) -> rand#(p(x),id_inc(y))
       TRS:
        nonZero(0()) -> false()
        nonZero(s(x)) -> true()
        p(s(0())) -> 0()
        p(s(s(x))) -> s(p(s(x)))
        id_inc(x) -> x
        id_inc(x) -> s(x)
        random(x) -> rand(x,0())
        rand(x,y) -> if(nonZero(x),x,y)
        if(false(),x,y) -> y
        if(true(),x,y) -> rand(p(x),id_inc(y))
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 2/1
        
    
    DPs:
     p#(s(s(x))) -> p#(s(x))
    TRS:
     nonZero(0()) -> false()
     nonZero(s(x)) -> true()
     p(s(0())) -> 0()
     p(s(s(x))) -> s(p(s(x)))
     id_inc(x) -> x
     id_inc(x) -> s(x)
     random(x) -> rand(x,0())
     rand(x,y) -> if(nonZero(x),x,y)
     if(false(),x,y) -> y
     if(true(),x,y) -> rand(p(x),id_inc(y))
    Subterm Criterion Processor:
     simple projection:
      pi(p#) = 0
     problem:
      DPs:
       
      TRS:
       nonZero(0()) -> false()
       nonZero(s(x)) -> true()
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
       id_inc(x) -> x
       id_inc(x) -> s(x)
       random(x) -> rand(x,0())
       rand(x,y) -> if(nonZero(x),x,y)
       if(false(),x,y) -> y
       if(true(),x,y) -> rand(p(x),id_inc(y))
     Qed