YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

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

and R consists of:

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

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),(0,0)) x1 + (1,4)
          a_A() = (3,3)
          f_A(x1,x2) = (3,1)
  
    precedence:
    
      f# > a = f
    
    partial status:
  
      pi(f#) = []
      pi(a) = []
      pi(f) = []

The next rules are strictly ordered:

  p3

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

and R consists of:

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

and R consists of:

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

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
          a_A() = (0,1)
          f_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(0,0)) x2
  
    precedence:
    
      a > f# = 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(),x),f(a(),a())))

and R consists of:

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

and R consists of:

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

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) = x1 + ((1,1),(0,1)) x2 + (1,0)
          a_A() = (3,7)
          f_A(x1,x2) = ((0,1),(0,0)) x2 + (1,1)
  
    precedence:
    
      a > f > f#
    
    partial status:
  
      pi(f#) = [1, 2]
      pi(a) = []
      pi(f) = []

The next rules are strictly ordered:

  p1

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