YES

We show the termination of the TRS R:

  f(g(x,y),f(y,y)) -> f(g(y,x),y)

-- SCC decomposition.

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

p1: f#(g(x,y),f(y,y)) -> f#(g(y,x),y)

and R consists of:

r1: f(g(x,y),f(y,y)) -> f(g(y,x),y)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: f#(g(x,y),f(y,y)) -> f#(g(y,x),y)

and R consists of:

r1: f(g(x,y),f(y,y)) -> f(g(y,x),y)

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:
      
        f > f# > g
      
      argument filter:
    
        pi(f#) = 2
        pi(g) = 2
        pi(f) = [1, 2]
    
    2. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        f#_A(x1,x2) = 0
        g_A(x1,x2) = x2
        f_A(x1,x2) = x1 + 1
    

The next rules are strictly ordered:

  p1

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