YES

We show the termination of the TRS R:

  f(f(x)) -> f(c(f(x)))
  f(f(x)) -> f(d(f(x)))
  g(c(x)) -> x
  g(d(x)) -> x
  g(c(|0|())) -> g(d(|1|()))
  g(c(|1|())) -> g(d(|0|()))

-- SCC decomposition.

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

p1: f#(f(x)) -> f#(c(f(x)))
p2: f#(f(x)) -> f#(d(f(x)))
p3: g#(c(|0|())) -> g#(d(|1|()))
p4: g#(c(|1|())) -> g#(d(|0|()))

and R consists of:

r1: f(f(x)) -> f(c(f(x)))
r2: f(f(x)) -> f(d(f(x)))
r3: g(c(x)) -> x
r4: g(d(x)) -> x
r5: g(c(|0|())) -> g(d(|1|()))
r6: g(c(|1|())) -> g(d(|0|()))

The estimated dependency graph contains the following SCCs:

  (no SCCs)