YES

We show the termination of the TRS R:

  f(f(X)) -> c(n__f(n__g(n__f(X))))
  c(X) -> d(activate(X))
  h(X) -> c(n__d(X))
  f(X) -> n__f(X)
  g(X) -> n__g(X)
  d(X) -> n__d(X)
  activate(n__f(X)) -> f(activate(X))
  activate(n__g(X)) -> g(X)
  activate(n__d(X)) -> d(X)
  activate(X) -> X

-- SCC decomposition.

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

p1: f#(f(X)) -> c#(n__f(n__g(n__f(X))))
p2: c#(X) -> d#(activate(X))
p3: c#(X) -> activate#(X)
p4: h#(X) -> c#(n__d(X))
p5: activate#(n__f(X)) -> f#(activate(X))
p6: activate#(n__f(X)) -> activate#(X)
p7: activate#(n__g(X)) -> g#(X)
p8: activate#(n__d(X)) -> d#(X)

and R consists of:

r1: f(f(X)) -> c(n__f(n__g(n__f(X))))
r2: c(X) -> d(activate(X))
r3: h(X) -> c(n__d(X))
r4: f(X) -> n__f(X)
r5: g(X) -> n__g(X)
r6: d(X) -> n__d(X)
r7: activate(n__f(X)) -> f(activate(X))
r8: activate(n__g(X)) -> g(X)
r9: activate(n__d(X)) -> d(X)
r10: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p3, p5, p6}


-- Reduction pair.

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

p1: f#(f(X)) -> c#(n__f(n__g(n__f(X))))
p2: c#(X) -> activate#(X)
p3: activate#(n__f(X)) -> activate#(X)
p4: activate#(n__f(X)) -> f#(activate(X))

and R consists of:

r1: f(f(X)) -> c(n__f(n__g(n__f(X))))
r2: c(X) -> d(activate(X))
r3: h(X) -> c(n__d(X))
r4: f(X) -> n__f(X)
r5: g(X) -> n__g(X)
r6: d(X) -> n__d(X)
r7: activate(n__f(X)) -> f(activate(X))
r8: activate(n__g(X)) -> g(X)
r9: activate(n__d(X)) -> d(X)
r10: activate(X) -> X

The set of usable rules consists of

  r1, r2, r4, r5, r6, r7, r8, r9, r10

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    f#_A(x1) = max{5, x1 + 4}
    f_A(x1) = x1 + 5
    c#_A(x1) = max{6, x1 + 3}
    n__f_A(x1) = x1 + 5
    n__g_A(x1) = max{0, x1 - 4}
    activate#_A(x1) = x1 + 1
    activate_A(x1) = x1 + 2
    c_A(x1) = 3
    d_A(x1) = 2
    g_A(x1) = max{2, x1 - 2}
    n__d_A(x1) = 0

The next rules are strictly ordered:

  p2, p3

We remove them from the problem.

-- SCC decomposition.

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

p1: f#(f(X)) -> c#(n__f(n__g(n__f(X))))
p2: activate#(n__f(X)) -> f#(activate(X))

and R consists of:

r1: f(f(X)) -> c(n__f(n__g(n__f(X))))
r2: c(X) -> d(activate(X))
r3: h(X) -> c(n__d(X))
r4: f(X) -> n__f(X)
r5: g(X) -> n__g(X)
r6: d(X) -> n__d(X)
r7: activate(n__f(X)) -> f(activate(X))
r8: activate(n__g(X)) -> g(X)
r9: activate(n__d(X)) -> d(X)
r10: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)