YES

We show the termination of the relative TRS R/S:

  R:
  g(s(x)) -> f(x)
  f(|0|()) -> s(|0|())
  f(s(x)) -> s(s(g(x)))
  g(|0|()) -> |0|()

  S:
  rand(x) -> x
  rand(x) -> rand(s(x))

-- SCC decomposition.

Consider the non-minimal dependency pair problem (P, R), where P consists of

p1: g#(s(x)) -> f#(x)
p2: f#(s(x)) -> g#(x)

and R consists of:

r1: g(s(x)) -> f(x)
r2: f(|0|()) -> s(|0|())
r3: f(s(x)) -> s(s(g(x)))
r4: g(|0|()) -> |0|()
r5: rand(x) -> x
r6: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the non-minimal dependency pair problem (P, R), where P consists of

p1: g#(s(x)) -> f#(x)
p2: f#(s(x)) -> g#(x)

and R consists of:

r1: g(s(x)) -> f(x)
r2: f(|0|()) -> s(|0|())
r3: f(s(x)) -> s(s(g(x)))
r4: g(|0|()) -> |0|()
r5: rand(x) -> x
r6: rand(x) -> rand(s(x))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        g#_A(x1) = ((1,0),(1,0)) x1
        s_A(x1) = x1
        f#_A(x1) = ((1,0),(1,0)) x1
        g_A(x1) = x1 + (1,1)
        f_A(x1) = x1 + (1,0)
        |0|_A() = (1,1)
        rand_A(x1) = x1 + (1,0)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        g#_A(x1) = x1
        s_A(x1) = x1 + (2,1)
        f#_A(x1) = x1 + (1,0)
        g_A(x1) = ((1,1),(1,0)) x1 + (1,1)
        f_A(x1) = ((1,1),(1,0)) x1 + (4,1)
        |0|_A() = (1,2)
        rand_A(x1) = (1,0)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        g#_A(x1) = ((1,1),(0,0)) x1
        s_A(x1) = ((0,1),(1,0)) x1 + (1,2)
        f#_A(x1) = ((1,1),(0,1)) x1 + (4,1)
        g_A(x1) = ((1,1),(1,1)) x1 + (1,0)
        f_A(x1) = ((0,1),(1,1)) x1 + (3,1)
        |0|_A() = (1,0)
        rand_A(x1) = (0,0)
    

The next rules are strictly ordered:

  p1, p2

We remove them from the problem.  Then no dependency pair remains.