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

(0) Obligation:

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

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

Types:
f :: 0':s → n__f:cons
cons :: 0':s → n__f:cons → n__f:cons
n__f :: 0':s → n__f:cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → n__f:cons → 0':s
activate :: n__f:cons → n__f:cons
hole_n__f:cons1_0 :: n__f:cons
hole_0':s2_0 :: 0':s
gen_n__f:cons3_0 :: Nat → n__f:cons
gen_0':s4_0 :: Nat → 0':s

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

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

(8) Obligation:

TRS:
Rules:
f(X) → cons(X, n__f(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)
activate(n__f(X)) → f(X)
activate(X) → X

Types:
f :: 0':s → n__f:cons
cons :: 0':s → n__f:cons → n__f:cons
n__f :: 0':s → n__f:cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → n__f:cons → 0':s
activate :: n__f:cons → n__f:cons
hole_n__f:cons1_0 :: n__f:cons
hole_0':s2_0 :: 0':s
gen_n__f:cons3_0 :: Nat → n__f:cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_n__f:cons3_0(0) ⇔ n__f(0')
gen_n__f:cons3_0(+(x, 1)) ⇔ cons(0', gen_n__f:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

Proved the following rewrite lemma:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

Induction Base:
g(gen_0':s4_0(0)) →_R^Ω(1)
s(0')

Induction Step:
g(gen_0':s4_0(+(n6_0, 1))) →_R^Ω(1)
s(s(g(gen_0':s4_0(n6_0)))) →_IH
s(s(gen_0':s4_0(+(1, *(2, c7_0)))))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
f(X) → cons(X, n__f(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)
activate(n__f(X)) → f(X)
activate(X) → X

Types:
f :: 0':s → n__f:cons
cons :: 0':s → n__f:cons → n__f:cons
n__f :: 0':s → n__f:cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → n__f:cons → 0':s
activate :: n__f:cons → n__f:cons
hole_n__f:cons1_0 :: n__f:cons
hole_0':s2_0 :: 0':s
gen_n__f:cons3_0 :: Nat → n__f:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_n__f:cons3_0(0) ⇔ n__f(0')
gen_n__f:cons3_0(+(x, 1)) ⇔ cons(0', gen_n__f:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
sel

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

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

(13) Obligation:

TRS:
Rules:
f(X) → cons(X, n__f(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)
activate(n__f(X)) → f(X)
activate(X) → X

Types:
f :: 0':s → n__f:cons
cons :: 0':s → n__f:cons → n__f:cons
n__f :: 0':s → n__f:cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → n__f:cons → 0':s
activate :: n__f:cons → n__f:cons
hole_n__f:cons1_0 :: n__f:cons
hole_0':s2_0 :: 0':s
gen_n__f:cons3_0 :: Nat → n__f:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_n__f:cons3_0(0) ⇔ n__f(0')
gen_n__f:cons3_0(+(x, 1)) ⇔ cons(0', gen_n__f:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
f(X) → cons(X, n__f(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)
activate(n__f(X)) → f(X)
activate(X) → X

Types:
f :: 0':s → n__f:cons
cons :: 0':s → n__f:cons → n__f:cons
n__f :: 0':s → n__f:cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → n__f:cons → 0':s
activate :: n__f:cons → n__f:cons
hole_n__f:cons1_0 :: n__f:cons
hole_0':s2_0 :: 0':s
gen_n__f:cons3_0 :: Nat → n__f:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_n__f:cons3_0(0) ⇔ n__f(0')
gen_n__f:cons3_0(+(x, 1)) ⇔ cons(0', gen_n__f:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

(18) BOUNDS(n^1, INF)