The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(f(a)) → c(n__f(n__g(n__f(n__a))))
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
activate(n__f(X)) →⁺ f(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__f(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

f(f(a)) → c(n__f(n__g(n__f(n__a))))
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
c/0

(6) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

f(f(a)) → c
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X

S is empty.
Rewrite Strategy: FULL

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(8) Obligation:

TRS:
Rules:
f(f(a)) → c
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X

Types:
f :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
a :: c:n__f:n__g:n__a
c :: c:n__f:n__g:n__a
n__f :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
g :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
n__g :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
n__a :: c:n__f:n__g:n__a
activate :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
hole_c:n__f:n__g:n__a1_0 :: c:n__f:n__g:n__a
gen_c:n__f:n__g:n__a2_0 :: Nat → c:n__f:n__g:n__a

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
activate

(10) Obligation:

TRS:
Rules:
f(f(a)) → c
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X

Types:
f :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
a :: c:n__f:n__g:n__a
c :: c:n__f:n__g:n__a
n__f :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
g :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
n__g :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
n__a :: c:n__f:n__g:n__a
activate :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
hole_c:n__f:n__g:n__a1_0 :: c:n__f:n__g:n__a
gen_c:n__f:n__g:n__a2_0 :: Nat → c:n__f:n__g:n__a

Generator Equations:
gen_c:n__f:n__g:n__a2_0(0) ⇔ n__a
gen_c:n__f:n__g:n__a2_0(+(x, 1)) ⇔ n__f(gen_c:n__f:n__g:n__a2_0(x))

The following defined symbols remain to be analysed:
activate

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_c:n__f:n__g:n__a2_0(n4_0)) → gen_c:n__f:n__g:n__a2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
activate(gen_c:n__f:n__g:n__a2_0(0)) →_R^Ω(1)
gen_c:n__f:n__g:n__a2_0(0)

Induction Step:
activate(gen_c:n__f:n__g:n__a2_0(+(n4_0, 1))) →_R^Ω(1)
f(activate(gen_c:n__f:n__g:n__a2_0(n4_0))) →_IH
f(gen_c:n__f:n__g:n__a2_0(c5_0)) →_R^Ω(1)
n__f(gen_c:n__f:n__g:n__a2_0(n4_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
f(f(a)) → c
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X

Types:
f :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
a :: c:n__f:n__g:n__a
c :: c:n__f:n__g:n__a
n__f :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
g :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
n__g :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
n__a :: c:n__f:n__g:n__a
activate :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
hole_c:n__f:n__g:n__a1_0 :: c:n__f:n__g:n__a
gen_c:n__f:n__g:n__a2_0 :: Nat → c:n__f:n__g:n__a

Lemmas:
activate(gen_c:n__f:n__g:n__a2_0(n4_0)) → gen_c:n__f:n__g:n__a2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_c:n__f:n__g:n__a2_0(0) ⇔ n__a
gen_c:n__f:n__g:n__a2_0(+(x, 1)) ⇔ n__f(gen_c:n__f:n__g:n__a2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_c:n__f:n__g:n__a2_0(n4_0)) → gen_c:n__f:n__g:n__a2_0(n4_0), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
f(f(a)) → c
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X

Types:
f :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
a :: c:n__f:n__g:n__a
c :: c:n__f:n__g:n__a
n__f :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
g :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
n__g :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
n__a :: c:n__f:n__g:n__a
activate :: c:n__f:n__g:n__a → c:n__f:n__g:n__a
hole_c:n__f:n__g:n__a1_0 :: c:n__f:n__g:n__a
gen_c:n__f:n__g:n__a2_0 :: Nat → c:n__f:n__g:n__a

Lemmas:
activate(gen_c:n__f:n__g:n__a2_0(n4_0)) → gen_c:n__f:n__g:n__a2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_c:n__f:n__g:n__a2_0(0) ⇔ n__a
gen_c:n__f:n__g:n__a2_0(+(x, 1)) ⇔ n__f(gen_c:n__f:n__g:n__a2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_c:n__f:n__g:n__a2_0(n4_0)) → gen_c:n__f:n__g:n__a2_0(n4_0), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)