YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

p1: f#(a(),h(x)) -> f#(g(x),h(x))
p2: f#(a(),h(x)) -> g#(x)
p3: h#(g(x)) -> h#(a())
p4: g#(h(x)) -> g#(x)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1}
  {p4}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  r2, r3, r4

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          f#_A(x1,x2) = max{0, x1 - 4}
          a_A = 6
          h_A(x1) = max{10, x1 + 4}
          g_A(x1) = 3
    
      precedence:
      
        f# = a = h = g
      
      partial status:
    
        pi(f#) = []
        pi(a) = []
        pi(h) = []
        pi(g) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          f#_A(x1,x2) = 13
          a_A = 13
          h_A(x1) = 12
          g_A(x1) = 11
    
      precedence:
      
        f# = h = g > a
      
      partial status:
    
        pi(f#) = []
        pi(a) = []
        pi(h) = []
        pi(g) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

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

and R consists of:

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

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:
        
          g#_A(x1) = max{4, x1 + 3}
          h_A(x1) = max{3, x1 + 2}
    
      precedence:
      
        g# = h
      
      partial status:
    
        pi(g#) = [1]
        pi(h) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          g#_A(x1) = max{1, x1 - 1}
          h_A(x1) = x1
    
      precedence:
      
        g# = h
      
      partial status:
    
        pi(g#) = []
        pi(h) = [1]
    

The next rules are strictly ordered:

  p1

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