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

(0) Obligation:

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

f(X) → cons(X, n__f(n__g(X)))
g(0) → s(0)
g(s(X)) → s(s(g(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
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(X)) →⁺ cons(activate(X), n__f(n__g(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__f(X)].
The result substitution is [ ].

The rewrite sequence
activate(n__f(X)) →⁺ cons(activate(X), n__f(n__g(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / n__f(X)].
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(X) → cons(X, n__f(n__g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
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(X) → cons(X, n__f(n__g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
cons :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
0' :: n__g:n__f:cons:0':s
s :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
sel :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
activate :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
hole_n__g:n__f:cons:0':s1_0 :: n__g:n__f:cons:0':s
gen_n__g:n__f:cons:0':s2_0 :: Nat → n__g:n__f:cons:0':s

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
g, sel, activate

They will be analysed ascendingly in the following order:
g < activate
activate < sel

(8) Obligation:

TRS:
Rules:
f(X) → cons(X, n__f(n__g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
cons :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
0' :: n__g:n__f:cons:0':s
s :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
sel :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
activate :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
hole_n__g:n__f:cons:0':s1_0 :: n__g:n__f:cons:0':s
gen_n__g:n__f:cons:0':s2_0 :: Nat → n__g:n__f:cons:0':s

Generator Equations:
gen_n__g:n__f:cons:0':s2_0(0) ⇔ 0'
gen_n__g:n__f:cons:0':s2_0(+(x, 1)) ⇔ cons(0', gen_n__g:n__f:cons:0':s2_0(x))

The following defined symbols remain to be analysed:
g, sel, activate

They will be analysed ascendingly in the following order:
g < activate
activate < sel

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
f(X) → cons(X, n__f(n__g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
cons :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
0' :: n__g:n__f:cons:0':s
s :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
sel :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
activate :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
hole_n__g:n__f:cons:0':s1_0 :: n__g:n__f:cons:0':s
gen_n__g:n__f:cons:0':s2_0 :: Nat → n__g:n__f:cons:0':s

Generator Equations:
gen_n__g:n__f:cons:0':s2_0(0) ⇔ 0'
gen_n__g:n__f:cons:0':s2_0(+(x, 1)) ⇔ cons(0', gen_n__g:n__f:cons:0':s2_0(x))

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

They will be analysed ascendingly in the following order:
activate < sel

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
f(X) → cons(X, n__f(n__g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
cons :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
0' :: n__g:n__f:cons:0':s
s :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
sel :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
activate :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
hole_n__g:n__f:cons:0':s1_0 :: n__g:n__f:cons:0':s
gen_n__g:n__f:cons:0':s2_0 :: Nat → n__g:n__f:cons:0':s

Generator Equations:
gen_n__g:n__f:cons:0':s2_0(0) ⇔ 0'
gen_n__g:n__f:cons:0':s2_0(+(x, 1)) ⇔ cons(0', gen_n__g:n__f:cons:0':s2_0(x))

The following defined symbols remain to be analysed:
sel

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

TRS:
Rules:
f(X) → cons(X, n__f(n__g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X

Types:
f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
cons :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__f :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
n__g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
g :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
0' :: n__g:n__f:cons:0':s
s :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
sel :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
activate :: n__g:n__f:cons:0':s → n__g:n__f:cons:0':s
hole_n__g:n__f:cons:0':s1_0 :: n__g:n__f:cons:0':s
gen_n__g:n__f:cons:0':s2_0 :: Nat → n__g:n__f:cons:0':s

Generator Equations:
gen_n__g:n__f:cons:0':s2_0(0) ⇔ 0'
gen_n__g:n__f:cons:0':s2_0(+(x, 1)) ⇔ cons(0', gen_n__g:n__f:cons:0':s2_0(x))

No more defined symbols left to analyse.