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))
  Usable Rule 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()
    id_inc(x) -> x
    id_inc(x) -> s(x)
    p(0()) -> 0()
    p(s(x)) -> x
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     nonZero(0()) -> false()
     nonZero(s(x)) -> true()
     p(0()) -> 0()
     p(s(x)) -> x
    interpretation:
     [if#](x0, x1, x2) = -3x0 + x1,
     
     [rand#](x0, x1) = 1x0 + 0,
     
     [random#](x0) = 6x0 + 10,
     
     [id_inc#](x0) = 2,
     
     [p#](x0) = -14x0 + 1,
     
     [nonZero#](x0) = x0,
     
     [id_inc](x0) = 2x0 + 2,
     
     [p](x0) = -6x0 + 1,
     
     [true] = 6,
     
     [s](x0) = 10x0 + 4,
     
     [false] = 2,
     
     [nonZero](x0) = 3x0,
     
     [0] = 0
    orientation:
     random#(x) = 6x + 10 >= 1x + 0 = rand#(x,0())
     
     rand#(x,y) = 1x + 0 >= x = nonZero#(x)
     
     rand#(x,y) = 1x + 0 >= x = if#(nonZero(x),x,y)
     
     if#(true(),x,y) = x + 3 >= 2 = id_inc#(y)
     
     if#(true(),x,y) = x + 3 >= -14x + 1 = p#(x)
     
     if#(true(),x,y) = x + 3 >= -5x + 2 = rand#(p(x),id_inc(y))
     
     nonZero(0()) = 3 >= 2 = false()
     
     nonZero(s(x)) = 13x + 7 >= 6 = true()
     
     id_inc(x) = 2x + 2 >= x = x
     
     id_inc(x) = 2x + 2 >= 10x + 4 = s(x)
     
     p(0()) = 1 >= 0 = 0()
     
     p(s(x)) = 4x + 1 >= x = x
    problem:
     DPs:
      
     TRS:
      nonZero(0()) -> false()
      nonZero(s(x)) -> true()
      id_inc(x) -> x
      id_inc(x) -> s(x)
      p(0()) -> 0()
      p(s(x)) -> x
    Qed