YES

We show the termination of the TRS R:

  p(a(x0),p(a(b(x1)),x2)) -> p(a(b(a(x2))),p(a(a(x1)),x2))

-- SCC decomposition.

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

p1: p#(a(x0),p(a(b(x1)),x2)) -> p#(a(b(a(x2))),p(a(a(x1)),x2))
p2: p#(a(x0),p(a(b(x1)),x2)) -> p#(a(a(x1)),x2)

and R consists of:

r1: p(a(x0),p(a(b(x1)),x2)) -> p(a(b(a(x2))),p(a(a(x1)),x2))

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

p1: p#(a(x0),p(a(b(x1)),x2)) -> p#(a(a(x1)),x2)

and R consists of:

r1: p(a(x0),p(a(b(x1)),x2)) -> p(a(b(a(x2))),p(a(a(x1)),x2))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        a > p# > b > p
      
      argument filter:
    
        pi(p#) = 2
        pi(a) = [1]
        pi(p) = [1, 2]
        pi(b) = 1
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        p# > a > b > p
      
      argument filter:
    
        pi(p#) = 2
        pi(a) = 1
        pi(p) = 2
        pi(b) = [1]
    

The next rules are strictly ordered:

  p1

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