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

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 94 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 typed CpxTrs
    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 48 ms)
    ↳10 typed CpxTrs
    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 198 ms)
    ↳12 BEST
        ↳13 typed CpxTrs
            ↳14 LowerBoundsProof (⇔, 0 ms)
            ↳15 BOUNDS(n^1, INF)
        ↳16 typed CpxTrs
            ↳17 LowerBoundsProof (⇔, 0 ms)
            ↳18 BOUNDS(n^1, INF)

(0) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
f :: n__a:n__f:g → n__a:n__f:g
a :: n__a:n__f:g
g :: n__a:n__f:g → n__a:n__f:g
n__f :: n__a:n__f:g → n__a:n__f:g
n__a :: n__a:n__f:g
activate :: n__a:n__f:g → n__a:n__f:g
hole_n__a:n__f:g1_0 :: n__a:n__f:g
gen_n__a:n__f:g2_0 :: Nat → n__a:n__f:g

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f, activate

They will be analysed ascendingly in the following order:
f < activate

(8) Obligation:

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

Types:
f :: n__a:n__f:g → n__a:n__f:g
a :: n__a:n__f:g
g :: n__a:n__f:g → n__a:n__f:g
n__f :: n__a:n__f:g → n__a:n__f:g
n__a :: n__a:n__f:g
activate :: n__a:n__f:g → n__a:n__f:g
hole_n__a:n__f:g1_0 :: n__a:n__f:g
gen_n__a:n__f:g2_0 :: Nat → n__a:n__f:g

Generator Equations:
gen_n__a:n__f:g2_0(0) ⇔ n__a
gen_n__a:n__f:g2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:g2_0(x))

The following defined symbols remain to be analysed:
f, activate

They will be analysed ascendingly in the following order:
f < activate

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(10) Obligation:

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

Types:
f :: n__a:n__f:g → n__a:n__f:g
a :: n__a:n__f:g
g :: n__a:n__f:g → n__a:n__f:g
n__f :: n__a:n__f:g → n__a:n__f:g
n__a :: n__a:n__f:g
activate :: n__a:n__f:g → n__a:n__f:g
hole_n__a:n__f:g1_0 :: n__a:n__f:g
gen_n__a:n__f:g2_0 :: Nat → n__a:n__f:g

Generator Equations:
gen_n__a:n__f:g2_0(0) ⇔ n__a
gen_n__a:n__f:g2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:g2_0(x))

The following defined symbols remain to be analysed:
activate

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

Proved the following rewrite lemma:
activate(gen_n__a:n__f:g2_0(n9_0)) → gen_n__a:n__f:g2_0(n9_0), rt ∈ Ω(1 + n90)

Induction Base:
activate(gen_n__a:n__f:g2_0(0)) →RΩ(1)
gen_n__a:n__f:g2_0(0)

Induction Step:
activate(gen_n__a:n__f:g2_0(+(n9_0, 1))) →RΩ(1)
f(activate(gen_n__a:n__f:g2_0(n9_0))) →IH
f(gen_n__a:n__f:g2_0(c10_0)) →RΩ(1)
n__f(gen_n__a:n__f:g2_0(n9_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
f :: n__a:n__f:g → n__a:n__f:g
a :: n__a:n__f:g
g :: n__a:n__f:g → n__a:n__f:g
n__f :: n__a:n__f:g → n__a:n__f:g
n__a :: n__a:n__f:g
activate :: n__a:n__f:g → n__a:n__f:g
hole_n__a:n__f:g1_0 :: n__a:n__f:g
gen_n__a:n__f:g2_0 :: Nat → n__a:n__f:g

Lemmas:
activate(gen_n__a:n__f:g2_0(n9_0)) → gen_n__a:n__f:g2_0(n9_0), rt ∈ Ω(1 + n90)

Generator Equations:
gen_n__a:n__f:g2_0(0) ⇔ n__a
gen_n__a:n__f:g2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:g2_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_n__a:n__f:g2_0(n9_0)) → gen_n__a:n__f:g2_0(n9_0), rt ∈ Ω(1 + n90)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
f :: n__a:n__f:g → n__a:n__f:g
a :: n__a:n__f:g
g :: n__a:n__f:g → n__a:n__f:g
n__f :: n__a:n__f:g → n__a:n__f:g
n__a :: n__a:n__f:g
activate :: n__a:n__f:g → n__a:n__f:g
hole_n__a:n__f:g1_0 :: n__a:n__f:g
gen_n__a:n__f:g2_0 :: Nat → n__a:n__f:g

Lemmas:
activate(gen_n__a:n__f:g2_0(n9_0)) → gen_n__a:n__f:g2_0(n9_0), rt ∈ Ω(1 + n90)

Generator Equations:
gen_n__a:n__f:g2_0(0) ⇔ n__a
gen_n__a:n__f:g2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:g2_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_n__a:n__f:g2_0(n9_0)) → gen_n__a:n__f:g2_0(n9_0), rt ∈ Ω(1 + n90)

(18) BOUNDS(n^1, INF)