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

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 SlicingProof (LOWER BOUND(ID), 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), 4 ms)
↳10 typed CpxTrs
↳11 RewriteLemmaProof (LOWER BOUND(ID), 243 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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
g/0

(4) Obligation:

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

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

Types:
f :: g:n__f:n__a → g:n__f:n__a
a :: g:n__f:n__a
g :: g:n__f:n__a
n__f :: g:n__f:n__a → g:n__f:n__a
n__a :: g:n__f:n__a
activate :: g:n__f:n__a → g:n__f:n__a
hole_g:n__f:n__a1_0 :: g:n__f:n__a
gen_g:n__f:n__a2_0 :: Nat → g:n__f:n__a

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

Types:
f :: g:n__f:n__a → g:n__f:n__a
a :: g:n__f:n__a
g :: g:n__f:n__a
n__f :: g:n__f:n__a → g:n__f:n__a
n__a :: g:n__f:n__a
activate :: g:n__f:n__a → g:n__f:n__a
hole_g:n__f:n__a1_0 :: g:n__f:n__a
gen_g:n__f:n__a2_0 :: Nat → g:n__f:n__a

Generator Equations:
gen_g:n__f:n__a2_0(0) ⇔ n__a
gen_g:n__f:n__a2_0(+(x, 1)) ⇔ n__f(gen_g:n__f:n__a2_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)
f(X) → n__f(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__a) → a
activate(X) → X

Types:
f :: g:n__f:n__a → g:n__f:n__a
a :: g:n__f:n__a
g :: g:n__f:n__a
n__f :: g:n__f:n__a → g:n__f:n__a
n__a :: g:n__f:n__a
activate :: g:n__f:n__a → g:n__f:n__a
hole_g:n__f:n__a1_0 :: g:n__f:n__a
gen_g:n__f:n__a2_0 :: Nat → g:n__f:n__a

Generator Equations:
gen_g:n__f:n__a2_0(0) ⇔ n__a
gen_g:n__f:n__a2_0(+(x, 1)) ⇔ n__f(gen_g:n__f: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_g:n__f:n__a2_0(n9_0)) → gen_g:n__f:n__a2_0(n9_0), rt ∈ Ω(1 + n90)

Induction Base:
activate(gen_g:n__f:n__a2_0(0)) →RΩ(1)
gen_g:n__f:n__a2_0(0)

Induction Step:
activate(gen_g:n__f:n__a2_0(+(n9_0, 1))) →RΩ(1)
f(activate(gen_g:n__f:n__a2_0(n9_0))) →IH
f(gen_g:n__f:n__a2_0(c10_0)) →RΩ(1)
n__f(gen_g:n__f:n__a2_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)
f(X) → n__f(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__a) → a
activate(X) → X

Types:
f :: g:n__f:n__a → g:n__f:n__a
a :: g:n__f:n__a
g :: g:n__f:n__a
n__f :: g:n__f:n__a → g:n__f:n__a
n__a :: g:n__f:n__a
activate :: g:n__f:n__a → g:n__f:n__a
hole_g:n__f:n__a1_0 :: g:n__f:n__a
gen_g:n__f:n__a2_0 :: Nat → g:n__f:n__a

Lemmas:
activate(gen_g:n__f:n__a2_0(n9_0)) → gen_g:n__f:n__a2_0(n9_0), rt ∈ Ω(1 + n90)

Generator Equations:
gen_g:n__f:n__a2_0(0) ⇔ n__a
gen_g:n__f:n__a2_0(+(x, 1)) ⇔ n__f(gen_g:n__f: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_g:n__f:n__a2_0(n9_0)) → gen_g:n__f:n__a2_0(n9_0), rt ∈ Ω(1 + n90)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
f :: g:n__f:n__a → g:n__f:n__a
a :: g:n__f:n__a
g :: g:n__f:n__a
n__f :: g:n__f:n__a → g:n__f:n__a
n__a :: g:n__f:n__a
activate :: g:n__f:n__a → g:n__f:n__a
hole_g:n__f:n__a1_0 :: g:n__f:n__a
gen_g:n__f:n__a2_0 :: Nat → g:n__f:n__a

Lemmas:
activate(gen_g:n__f:n__a2_0(n9_0)) → gen_g:n__f:n__a2_0(n9_0), rt ∈ Ω(1 + n90)

Generator Equations:
gen_g:n__f:n__a2_0(0) ⇔ n__a
gen_g:n__f:n__a2_0(+(x, 1)) ⇔ n__f(gen_g:n__f: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_g:n__f:n__a2_0(n9_0)) → gen_g:n__f:n__a2_0(n9_0), rt ∈ Ω(1 + n90)

(18) BOUNDS(n^1, INF)