YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  (no SCCs)