YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

p1: f#(s(x)) -> f#(x)
p2: g#(s(|0|())) -> g#(f(s(|0|())))
p3: g#(s(|0|())) -> f#(s(|0|()))

and R consists of:

r1: f(f(x)) -> f(x)
r2: f(s(x)) -> f(x)
r3: g(s(|0|())) -> g(f(s(|0|())))

The estimated dependency graph contains the following SCCs:

  {p2}
  {p1}


-- Reduction pair.

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

p1: g#(s(|0|())) -> g#(f(s(|0|())))

and R consists of:

r1: f(f(x)) -> f(x)
r2: f(s(x)) -> f(x)
r3: g(s(|0|())) -> g(f(s(|0|())))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            g#_A(x1) = ((0,0),(1,0)) x1 + (6,1)
            s_A(x1) = (4,3)
            |0|_A() = (5,4)
            f_A(x1) = (1,2)
    
      precedence:
      
        g# = f > s = |0|
      
      partial status:
    
        pi(g#) = []
        pi(s) = []
        pi(|0|) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            g#_A(x1) = (3,3)
            s_A(x1) = (3,3)
            |0|_A() = (2,2)
            f_A(x1) = (1,1)
    
      precedence:
      
        g# = s > |0| > f
      
      partial status:
    
        pi(g#) = []
        pi(s) = []
        pi(|0|) = []
        pi(f) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

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

and R consists of:

r1: f(f(x)) -> f(x)
r2: f(s(x)) -> f(x)
r3: g(s(|0|())) -> g(f(s(|0|())))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        f# = s
      
      partial status:
    
        pi(f#) = []
        pi(s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1) = ((1,0),(0,0)) x1
            s_A(x1) = (1,1)
    
      precedence:
      
        s > f#
      
      partial status:
    
        pi(f#) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1

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