YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

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

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,x2) = x1 + x2 + (1,3)
            f_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (4,0)
            a_A() = (6,13)
    
      precedence:
      
        f# = f = a
      
      partial status:
    
        pi(f#) = []
        pi(f) = []
        pi(a) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1,x2) = ((1,0),(0,0)) x2 + (0,1)
            f_A(x1,x2) = (1,2)
            a_A() = (2,3)
    
      precedence:
      
        f# = f = a
      
      partial status:
    
        pi(f#) = []
        pi(f) = []
        pi(a) = []
    

The next rules are strictly ordered:

  p1

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