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

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 typed CpxTrs
↳5 OrderProof (LOWER BOUND(ID), 0 ms)
↳6 typed CpxTrs
↳7 NoRewriteLemmaProof (LOWER BOUND(ID), 59 ms)
↳8 typed CpxTrs
↳9 RewriteLemmaProof (LOWER BOUND(ID), 252 ms)
↳10 BEST
    ↳11 typed CpxTrs
        ↳12 LowerBoundsProof (⇔, 0 ms)
        ↳13 BOUNDS(n^1, INF)
    ↳14 typed CpxTrs
        ↳15 LowerBoundsProof (⇔, 0 ms)
        ↳16 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: INNERMOST

(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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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

(5) 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

(6) Obligation:

Innermost 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

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

Innermost 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

(9) 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).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost 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.

(12) 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)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost 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.

(15) 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)

(16) BOUNDS(n^1, INF)