YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

p1: f#(x,y) -> g#(x,y)
p2: g#(h(x),y) -> f#(x,y)
p3: g#(h(x),y) -> g#(x,y)

and R consists of:

r1: f(x,y) -> g(x,y)
r2: g(h(x),y) -> h(f(x,y))
r3: g(h(x),y) -> h(g(x,y))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: f#(x,y) -> g#(x,y)
p2: g#(h(x),y) -> g#(x,y)
p3: g#(h(x),y) -> f#(x,y)

and R consists of:

r1: f(x,y) -> g(x,y)
r2: g(h(x),y) -> h(f(x,y))
r3: g(h(x),y) -> h(g(x,y))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (4,3)
            g#_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (3,2)
            h_A(x1) = ((1,0),(0,0)) x1 + (2,1)
    
      precedence:
      
        f# = g# = h
      
      partial status:
    
        pi(f#) = []
        pi(g#) = [2]
        pi(h) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1,x2) = (0,0)
            g#_A(x1,x2) = (0,0)
            h_A(x1) = (1,1)
    
      precedence:
      
        h > f# = g#
      
      partial status:
    
        pi(f#) = []
        pi(g#) = []
        pi(h) = []
    

The next rules are strictly ordered:

  p1, p2

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: f(x,y) -> g(x,y)
r2: g(h(x),y) -> h(f(x,y))
r3: g(h(x),y) -> h(g(x,y))

The estimated dependency graph contains the following SCCs:

  (no SCCs)