YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

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

and R consists of:

r1: f(a(),f(x,a())) -> f(f(a(),f(f(a(),a()),x)),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 + 1, x2 + 4}
          a_A = 11
          f_A(x1,x2) = max{0, x1 - 8, x2 - 10}
    
      precedence:
      
        f# = a > 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{13, x2 + 5}
          a_A = 4
          f_A(x1,x2) = 12
    
      precedence:
      
        f# = f > a
      
      partial status:
    
        pi(f#) = []
        pi(a) = []
        pi(f) = []
    

The next rules are strictly ordered:

  p1

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