YES

We show the termination of the TRS R:

  a(b(a(x))) -> b(a(b(x)))

-- SCC decomposition.

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

p1: a#(b(a(x))) -> a#(b(x))

and R consists of:

r1: a(b(a(x))) -> b(a(b(x)))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: a#(b(a(x))) -> a#(b(x))

and R consists of:

r1: a(b(a(x))) -> b(a(b(x)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a#_A(x1) = x1
        b_A(x1) = ((1,0),(1,1)) x1 + (1,1)
        a_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a#_A(x1) = ((0,1),(1,0)) x1
        b_A(x1) = ((0,0),(1,1)) x1 + (1,1)
        a_A(x1) = ((0,1),(0,1)) x1 + (1,0)
    

The next rules are strictly ordered:

  p1

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