YES

Problem:
 strict:
  g(s(x)) -> f(x)
  f(0()) -> s(0())
  f(s(x)) -> s(s(g(x)))
  g(0()) -> 0()
 weak:
  rand(x) -> x
  rand(x) -> rand(s(x))

Proof:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [rand](x0) = 8x0,
   
   [g](x0) = x0,
   
   [0] = 12,
   
   [s](x0) = x0,
   
   [f](x0) = x0
  orientation:
   g(s(x)) = x >= x = f(x)
   
   f(0()) = 12 >= 12 = s(0())
   
   f(s(x)) = x >= x = s(s(g(x)))
   
   g(0()) = 12 >= 12 = 0()
   
   rand(x) = 8x >= x = x
   
   rand(x) = 8x >= 8x = rand(s(x))
  problem:
   strict:
    g(s(x)) -> f(x)
    f(0()) -> s(0())
    f(s(x)) -> s(s(g(x)))
    g(0()) -> 0()
   weak:
    rand(x) -> rand(s(x))
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [rand](x0) = 8x0,
    
    [g](x0) = 8x0,
    
    [0] = 8,
    
    [s](x0) = x0,
    
    [f](x0) = 8x0
   orientation:
    g(s(x)) = 8x >= 8x = f(x)
    
    f(0()) = 16 >= 8 = s(0())
    
    f(s(x)) = 8x >= 8x = s(s(g(x)))
    
    g(0()) = 16 >= 8 = 0()
    
    rand(x) = 8x >= 8x = rand(s(x))
   problem:
    strict:
     g(s(x)) -> f(x)
     f(s(x)) -> s(s(g(x)))
    weak:
     rand(x) -> rand(s(x))
   Matrix Interpretation Processor: dim=2
    
    interpretation:
                  [2 0]     [0]
     [rand](x0) = [0 0]x0 + [2],
     
               [1 1]     [2]
     [g](x0) = [0 1]x0 + [0],
     
                    [0]
     [s](x0) = x0 + [1],
     
               [1 1]     [2]
     [f](x0) = [0 1]x0 + [1]
    orientation:
               [1 1]    [3]    [1 1]    [2]       
     g(s(x)) = [0 1]x + [1] >= [0 1]x + [1] = f(x)
     
               [1 1]    [3]    [1 1]    [2]             
     f(s(x)) = [0 1]x + [2] >= [0 1]x + [2] = s(s(g(x)))
     
               [2 0]    [0]    [2 0]    [0]             
     rand(x) = [0 0]x + [2] >= [0 0]x + [2] = rand(s(x))
    problem:
     strict:
      
     weak:
      rand(x) -> rand(s(x))
    Qed