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:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        f#_A(x1,x2) = max{1, x2 - 12}
        a_A = 21
        f_A(x1,x2) = max{14, x2 - 5}
  
    precedence:
    
      f# = a = 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:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        f#_A(x1,x2) = max{7, x1 - 19, x2 - 7}
        a_A = 31
        f_A(x1,x2) = max{18, x1 - 9, x2 - 14}
  
    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.  Then no dependency pair remains.