YES

We show the termination of the TRS R:

  p(a(x0),p(b(x1),p(a(x2),x3))) -> p(x2,p(a(a(x0)),p(b(x1),x3)))

-- SCC decomposition.

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

p1: p#(a(x0),p(b(x1),p(a(x2),x3))) -> p#(x2,p(a(a(x0)),p(b(x1),x3)))
p2: p#(a(x0),p(b(x1),p(a(x2),x3))) -> p#(a(a(x0)),p(b(x1),x3))
p3: p#(a(x0),p(b(x1),p(a(x2),x3))) -> p#(b(x1),x3)

and R consists of:

r1: p(a(x0),p(b(x1),p(a(x2),x3))) -> p(x2,p(a(a(x0)),p(b(x1),x3)))

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(b(x1),p(a(x2),x3))) -> p#(a(a(x0)),p(b(x1),x3))

and R consists of:

r1: p(a(x0),p(b(x1),p(a(x2),x3))) -> p(x2,p(a(a(x0)),p(b(x1),x3)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          p#_A(x1,x2) = max{x1 + 5, x2 + 10}
          a_A(x1) = max{4, x1}
          p_A(x1,x2) = max{x1 + 3, x2 + 5}
          b_A(x1) = 1
    
      precedence:
      
        p# = a = p = b
      
      partial status:
    
        pi(p#) = [2]
        pi(a) = [1]
        pi(p) = [1, 2]
        pi(b) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          p#_A(x1,x2) = max{4, x2 - 7}
          a_A(x1) = x1 + 6
          p_A(x1,x2) = max{14, x1 + 8, x2 + 3}
          b_A(x1) = 5
    
      precedence:
      
        p# = a = p = b
      
      partial status:
    
        pi(p#) = []
        pi(a) = [1]
        pi(p) = []
        pi(b) = []
    

The next rules are strictly ordered:

  p1

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