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 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 typed CpxTrs
↳5 OrderProof (LOWER BOUND(ID), 0 ms)
↳6 typed CpxTrs
↳7 RewriteLemmaProof (LOWER BOUND(ID), 357 ms)
↳8 BEST
    ↳9 typed CpxTrs
        ↳10 LowerBoundsProof (⇔, 0 ms)
        ↳11 BOUNDS(n^1, INF)
    ↳12 typed CpxTrs
        ↳13 LowerBoundsProof (⇔, 0 ms)
        ↳14 BOUNDS(n^1, INF)

(0) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

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

Types:
f :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
a :: n__a:n__f:n__g:c
c :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__f :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__g :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__a :: n__a:n__f:n__g:c
g :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
activate :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
hole_n__a:n__f:n__g:c1_0 :: n__a:n__f:n__g:c
gen_n__a:n__f:n__g:c2_0 :: Nat → n__a:n__f:n__g:c

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
activate

(6) Obligation:

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

Types:
f :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
a :: n__a:n__f:n__g:c
c :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__f :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__g :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__a :: n__a:n__f:n__g:c
g :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
activate :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
hole_n__a:n__f:n__g:c1_0 :: n__a:n__f:n__g:c
gen_n__a:n__f:n__g:c2_0 :: Nat → n__a:n__f:n__g:c

Generator Equations:
gen_n__a:n__f:n__g:c2_0(0) ⇔ n__a
gen_n__a:n__f:n__g:c2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:n__g:c2_0(x))

The following defined symbols remain to be analysed:
activate

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__a:n__f:n__g:c2_0(n4_0)) → gen_n__a:n__f:n__g:c2_0(n4_0), rt ∈ Ω(1 + n40)

Induction Base:
activate(gen_n__a:n__f:n__g:c2_0(0)) →RΩ(1)
gen_n__a:n__f:n__g:c2_0(0)

Induction Step:
activate(gen_n__a:n__f:n__g:c2_0(+(n4_0, 1))) →RΩ(1)
f(activate(gen_n__a:n__f:n__g:c2_0(n4_0))) →IH
f(gen_n__a:n__f:n__g:c2_0(c5_0)) →RΩ(1)
n__f(gen_n__a:n__f:n__g:c2_0(n4_0))

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

(8) Complex Obligation (BEST)

(9) Obligation:

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

Types:
f :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
a :: n__a:n__f:n__g:c
c :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__f :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__g :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__a :: n__a:n__f:n__g:c
g :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
activate :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
hole_n__a:n__f:n__g:c1_0 :: n__a:n__f:n__g:c
gen_n__a:n__f:n__g:c2_0 :: Nat → n__a:n__f:n__g:c

Lemmas:
activate(gen_n__a:n__f:n__g:c2_0(n4_0)) → gen_n__a:n__f:n__g:c2_0(n4_0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_n__a:n__f:n__g:c2_0(0) ⇔ n__a
gen_n__a:n__f:n__g:c2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:n__g:c2_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_n__a:n__f:n__g:c2_0(n4_0)) → gen_n__a:n__f:n__g:c2_0(n4_0), rt ∈ Ω(1 + n40)

(11) BOUNDS(n^1, INF)

(12) Obligation:

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

Types:
f :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
a :: n__a:n__f:n__g:c
c :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__f :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__g :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
n__a :: n__a:n__f:n__g:c
g :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
activate :: n__a:n__f:n__g:c → n__a:n__f:n__g:c
hole_n__a:n__f:n__g:c1_0 :: n__a:n__f:n__g:c
gen_n__a:n__f:n__g:c2_0 :: Nat → n__a:n__f:n__g:c

Lemmas:
activate(gen_n__a:n__f:n__g:c2_0(n4_0)) → gen_n__a:n__f:n__g:c2_0(n4_0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_n__a:n__f:n__g:c2_0(0) ⇔ n__a
gen_n__a:n__f:n__g:c2_0(+(x, 1)) ⇔ n__f(gen_n__a:n__f:n__g:c2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_n__a:n__f:n__g:c2_0(n4_0)) → gen_n__a:n__f:n__g:c2_0(n4_0), rt ∈ Ω(1 + n40)

(14) BOUNDS(n^1, INF)