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

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 190 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 SlicingProof (LOWER BOUND(ID), 0 ms)
    ↳6 CpxRelTRS
    ↳7 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳10 typed CpxTrs
    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 354 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(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
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__h(X)) →+ h(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__h(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(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
g/0

(6) Obligation:

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

f(X) → g
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
f(X) → g
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Types:
f :: g:n__h:n__f → g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f → g:n__h:n__f
n__h :: g:n__h:n__f → g:n__h:n__f
n__f :: g:n__h:n__f → g:n__h:n__f
activate :: g:n__h:n__f → g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat → g:n__h:n__f

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

Heuristically decided to analyse the following defined symbols:
activate

(10) Obligation:

TRS:
Rules:
f(X) → g
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Types:
f :: g:n__h:n__f → g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f → g:n__h:n__f
n__h :: g:n__h:n__f → g:n__h:n__f
n__f :: g:n__h:n__f → g:n__h:n__f
activate :: g:n__h:n__f → g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat → g:n__h:n__f

Generator Equations:
gen_g:n__h:n__f2_0(0) ⇔ g
gen_g:n__h:n__f2_0(+(x, 1)) ⇔ n__h(gen_g:n__h:n__f2_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__h:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Induction Base:
activate(gen_g:n__h:n__f2_0(+(1, 0)))

Induction Step:
activate(gen_g:n__h:n__f2_0(+(1, +(n4_0, 1)))) →RΩ(1)
h(activate(gen_g:n__h:n__f2_0(+(1, n4_0)))) →IH
h(*3_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(X) → g
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Types:
f :: g:n__h:n__f → g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f → g:n__h:n__f
n__h :: g:n__h:n__f → g:n__h:n__f
n__f :: g:n__h:n__f → g:n__h:n__f
activate :: g:n__h:n__f → g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat → g:n__h:n__f

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

Generator Equations:
gen_g:n__h:n__f2_0(0) ⇔ g
gen_g:n__h:n__f2_0(+(x, 1)) ⇔ n__h(gen_g:n__h:n__f2_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__h:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
f(X) → g
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Types:
f :: g:n__h:n__f → g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f → g:n__h:n__f
n__h :: g:n__h:n__f → g:n__h:n__f
n__f :: g:n__h:n__f → g:n__h:n__f
activate :: g:n__h:n__f → g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat → g:n__h:n__f

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

Generator Equations:
gen_g:n__h:n__f2_0(0) ⇔ g
gen_g:n__h:n__f2_0(+(x, 1)) ⇔ n__h(gen_g:n__h:n__f2_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__h:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

(18) BOUNDS(n^1, INF)