YES

We show the termination of the TRS R:

  f(a(),f(x,a())) -> f(a(),f(x,f(f(a(),a()),a())))

-- SCC decomposition.

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

p1: f#(a(),f(x,a())) -> f#(a(),f(x,f(f(a(),a()),a())))
p2: f#(a(),f(x,a())) -> f#(x,f(f(a(),a()),a()))
p3: f#(a(),f(x,a())) -> f#(f(a(),a()),a())
p4: f#(a(),f(x,a())) -> f#(a(),a())

and R consists of:

r1: f(a(),f(x,a())) -> f(a(),f(x,f(f(a(),a()),a())))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: f#(a(),f(x,a())) -> f#(a(),f(x,f(f(a(),a()),a())))
p2: f#(a(),f(x,a())) -> f#(x,f(f(a(),a()),a()))

and R consists of:

r1: f(a(),f(x,a())) -> f(a(),f(x,f(f(a(),a()),a())))

The set of usable rules consists of

  r1

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          f#_A(x1,x2) = max{2, x2}
          a_A = 9
          f_A(x1,x2) = max{1, x2 - 6}
    
      precedence:
      
        a > f# = f
      
      partial status:
    
        pi(f#) = [2]
        pi(a) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          f#_A(x1,x2) = max{0, x2 - 1}
          a_A = 9
          f_A(x1,x2) = 1
    
      precedence:
      
        a > f > f#
      
      partial status:
    
        pi(f#) = []
        pi(a) = []
        pi(f) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: f#(a(),f(x,a())) -> f#(x,f(f(a(),a()),a()))

and R consists of:

r1: f(a(),f(x,a())) -> f(a(),f(x,f(f(a(),a()),a())))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: f#(a(),f(x,a())) -> f#(x,f(f(a(),a()),a()))

and R consists of:

r1: f(a(),f(x,a())) -> f(a(),f(x,f(f(a(),a()),a())))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          f#_A(x1,x2) = max{x1 + 15, x2 + 21}
          a_A = 15
          f_A(x1,x2) = max{8, x1 - 5, x2 - 30}
    
      precedence:
      
        a > f# = f
      
      partial status:
    
        pi(f#) = [1, 2]
        pi(a) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          f#_A(x1,x2) = max{x1, x2 + 9}
          a_A = 1
          f_A(x1,x2) = 0
    
      precedence:
      
        f# = f > a
      
      partial status:
    
        pi(f#) = [1, 2]
        pi(a) = []
        pi(f) = []
    

The next rules are strictly ordered:

  p1

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