YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

and R consists of:

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

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:
        p#_A(x1,x2) = x1 + ((1,1),(1,0)) x2
        a_A(x1) = (0,1)
        p_A(x1,x2) = ((1,1),(1,1)) x2 + (1,0)
        b_A(x1) = (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        p#_A(x1,x2) = ((1,1),(0,0)) x1
        a_A(x1) = (1,1)
        p_A(x1,x2) = (1,1)
        b_A(x1) = (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        p#_A(x1,x2) = (0,0)
        a_A(x1) = (1,1)
        p_A(x1,x2) = (1,1)
        b_A(x1) = (1,1)
    

The next rules are strictly ordered:

  p1

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