YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

r1: f(f(a(),x),a()) -> f(f(f(x,f(a(),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#(f(a(),x),a()) -> f#(f(f(x,f(a(),a())),a()),a())
p2: f#(f(a(),x),a()) -> f#(f(x,f(a(),a())),a())

and R consists of:

r1: f(f(a(),x),a()) -> f(f(f(x,f(a(),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:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1,x2) = x1
            f_A(x1,x2) = x1 + ((1,0),(1,1)) x2
            a_A() = (0,0)
    
      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) = x1 + (0,1)
            f_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,1)
            a_A() = (2,6)
    
      precedence:
      
        a > f > f#
      
      partial status:
    
        pi(f#) = [1]
        pi(f) = []
        pi(a) = []
    

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

and R consists of:

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

and R consists of:

r1: f(f(a(),x),a()) -> f(f(f(x,f(a(),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{1, x1, x2 - 3}
          f_A(x1,x2) = max{0, x1 - 3, x2 - 5}
          a_A = 4
    
      precedence:
      
        f = a > f#
      
      partial status:
    
        pi(f#) = [1]
        pi(f) = []
        pi(a) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          f#_A(x1,x2) = 63
          f_A(x1,x2) = 64
          a_A = 16
    
      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.