YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

r1: a(b(x)) -> b(a(x))
r2: a(c(x)) -> 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(x)) -> a#(x)

and R consists of:

r1: a(b(x)) -> b(a(x))
r2: a(c(x)) -> x

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        a#_A(x1) = x1
        b_A(x1) = x1 + 1
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        b > a#
      
      argument filter:
    
        pi(a#) = 1
        pi(b) = [1]
    

The next rules are strictly ordered:

  p1
  r1, r2

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