The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 93 ms)
↳2 BOUNDS(2^n, 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 RewriteLemmaProof (LOWER BOUND(ID), 392 ms)
    ↳10 BEST
        ↳11 typed CpxTrs
            ↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
            ↳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(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(activate(X))
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
activate(n__f(n__g(X26_0), X2)) →+ f(activate(X26_0), n__f(n__g(activate(X26_0)), activate(X2)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X26_0 / n__f(n__g(X26_0), X2)].
The result substitution is [ ].

The rewrite sequence
activate(n__f(n__g(X26_0), X2)) →+ f(activate(X26_0), n__f(n__g(activate(X26_0)), activate(X2)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X26_0 / n__f(n__g(X26_0), X2)].
The result substitution is [ ].

(2) BOUNDS(2^n, 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(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f → n__g:n__f → n__g:n__f
g :: n__g:n__f → n__g:n__f
n__f :: n__g:n__f → n__g:n__f → n__g:n__f
n__g :: n__g:n__f → n__g:n__f
activate :: n__g:n__f → n__g:n__f
hole_n__g:n__f1_0 :: n__g:n__f
gen_n__g:n__f2_0 :: Nat → n__g:n__f

(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(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f → n__g:n__f → n__g:n__f
g :: n__g:n__f → n__g:n__f
n__f :: n__g:n__f → n__g:n__f → n__g:n__f
n__g :: n__g:n__f → n__g:n__f
activate :: n__g:n__f → n__g:n__f
hole_n__g:n__f1_0 :: n__g:n__f
gen_n__g:n__f2_0 :: Nat → n__g:n__f

Generator Equations:
gen_n__g:n__f2_0(0) ⇔ hole_n__g:n__f1_0
gen_n__g:n__f2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f2_0(x), hole_n__g:n__f1_0)

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

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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__g:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Induction Base:
activate(gen_n__g:n__f2_0(+(1, 0)))

Induction Step:
activate(gen_n__g:n__f2_0(+(1, +(n4_0, 1)))) →RΩ(1)
f(activate(gen_n__g:n__f2_0(+(1, n4_0))), hole_n__g:n__f1_0) →IH
f(*3_0, hole_n__g:n__f1_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
f(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f → n__g:n__f → n__g:n__f
g :: n__g:n__f → n__g:n__f
n__f :: n__g:n__f → n__g:n__f → n__g:n__f
n__g :: n__g:n__f → n__g:n__f
activate :: n__g:n__f → n__g:n__f
hole_n__g:n__f1_0 :: n__g:n__f
gen_n__g:n__f2_0 :: Nat → n__g:n__f

Lemmas:
activate(gen_n__g:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_n__g:n__f2_0(0) ⇔ hole_n__g:n__f1_0
gen_n__g:n__f2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f2_0(x), hole_n__g:n__f1_0)

The following defined symbols remain to be analysed:
f

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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
f(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f → n__g:n__f → n__g:n__f
g :: n__g:n__f → n__g:n__f
n__f :: n__g:n__f → n__g:n__f → n__g:n__f
n__g :: n__g:n__f → n__g:n__f
activate :: n__g:n__f → n__g:n__f
hole_n__g:n__f1_0 :: n__g:n__f
gen_n__g:n__f2_0 :: Nat → n__g:n__f

Lemmas:
activate(gen_n__g:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_n__g:n__f2_0(0) ⇔ hole_n__g:n__f1_0
gen_n__g:n__f2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f2_0(x), hole_n__g:n__f1_0)

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_n__g:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
f(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f → n__g:n__f → n__g:n__f
g :: n__g:n__f → n__g:n__f
n__f :: n__g:n__f → n__g:n__f → n__g:n__f
n__g :: n__g:n__f → n__g:n__f
activate :: n__g:n__f → n__g:n__f
hole_n__g:n__f1_0 :: n__g:n__f
gen_n__g:n__f2_0 :: Nat → n__g:n__f

Lemmas:
activate(gen_n__g:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_n__g:n__f2_0(0) ⇔ hole_n__g:n__f1_0
gen_n__g:n__f2_0(+(x, 1)) ⇔ n__f(gen_n__g:n__f2_0(x), hole_n__g:n__f1_0)

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_n__g:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

(18) BOUNDS(n^1, INF)