YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1, p4}


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

and R consists of:

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

The set of usable rules consists of

  r1

Take the reduction pair:

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

The next rules are strictly ordered:

  p2

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#(a(),f(f(a(),a()),f(a(),x)))

and R consists of:

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

and R consists of:

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

The set of usable rules consists of

  r1

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          f#_A(x1,x2) = ((1,0),(0,0)) x2 + (3,2)
          a_A() = (8,3)
          f_A(x1,x2) = ((0,1),(0,0)) x2 + (1,1)
  
    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.  Then no dependency pair remains.