YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

r1: f(a()) -> g(h(a()))
r2: h(g(x)) -> g(h(f(x)))
r3: k(x,h(x),a()) -> h(x)
r4: k(f(x),y,x) -> f(x)

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

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

and R consists of:

r1: f(a()) -> g(h(a()))
r2: h(g(x)) -> g(h(f(x)))
r3: k(x,h(x),a()) -> h(x)
r4: k(f(x),y,x) -> f(x)

The set of usable rules consists of

  r1

Take the reduction pair:

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

The next rules are strictly ordered:

  p1

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