YES

Problem:
 nonZero(0()) -> false()
 nonZero(s(x)) -> true()
 p(0()) -> 0()
 p(s(x)) -> 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:
   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(0()) -> 0()
   p(s(x)) -> 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:
    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(0()) -> 0()
    p(s(x)) -> 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)
    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)
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 7/36
    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(0()) -> 0()
     p(s(x)) -> 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(0()) -> 0()
      p(s(x)) -> x
      nonZero(0()) -> false()
      nonZero(s(x)) -> true()
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       p(0()) -> 0()
       p(s(x)) -> x
       nonZero(0()) -> false()
       nonZero(s(x)) -> true()
      interpretation:
       [true] = 1,
       
       [if#](x0, x1, x2) = x0 + -7x1 + -16,
       
       [nonZero](x0) = x0 + -4,
       
       [s](x0) = 9x0 + 1,
       
       [p](x0) = -8x0 + 0,
       
       [0] = 0,
       
       [rand#](x0, x1) = x0 + 0,
       
       [id_inc](x0) = 6x0 + 1,
       
       [false] = 0
      orientation:
       if#(true(),x,y) = -7x + 1 >= -8x + 0 = rand#(p(x),id_inc(y))
       
       rand#(x,y) = x + 0 >= x + -4 = if#(nonZero(x),x,y)
       
       id_inc(x) = 6x + 1 >= x = x
       
       id_inc(x) = 6x + 1 >= 9x + 1 = s(x)
       
       p(0()) = 0 >= 0 = 0()
       
       p(s(x)) = 1x + 0 >= x = x
       
       nonZero(0()) = 0 >= 0 = false()
       
       nonZero(s(x)) = 9x + 1 >= 1 = true()
      problem:
       DPs:
        rand#(x,y) -> if#(nonZero(x),x,y)
       TRS:
        id_inc(x) -> x
        id_inc(x) -> s(x)
        p(0()) -> 0()
        p(s(x)) -> x
        nonZero(0()) -> false()
        nonZero(s(x)) -> true()
      Restore Modifier:
       DPs:
        rand#(x,y) -> if#(nonZero(x),x,y)
       TRS:
        nonZero(0()) -> false()
        nonZero(s(x)) -> true()
        p(0()) -> 0()
        p(s(x)) -> 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