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. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          g#_A(x1) = x1
          s_A(x1) = max{3, x1}
          f#_A(x1) = max{3, x1}
          g_A(x1) = max{5, x1 + 2}
          f_A(x1) = x1 + 2
          |0|_A = 6
          rand_A(x1) = max{8, x1 + 4}
    
      precedence:
      
        |0| > g = f > rand > g# = s = f#
      
      partial status:
    
        pi(g#) = [1]
        pi(s) = [1]
        pi(f#) = [1]
        pi(g) = [1]
        pi(f) = [1]
        pi(|0|) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          g#_A(x1) = max{6, x1}
          s_A(x1) = 6
          f#_A(x1) = 6
          g_A(x1) = 0
          f_A(x1) = max{1, x1}
          |0|_A = 5
          rand_A(x1) = 5
    
      precedence:
      
        g# = s = f# > g = f = |0| = rand
      
      partial status:
    
        pi(g#) = [1]
        pi(s) = []
        pi(f#) = []
        pi(g) = []
        pi(f) = [1]
        pi(|0|) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

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

  (no SCCs)