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:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

The rewrite sequence
mark(f(X)) →⁺ cons(mark(mark(X)), f(g(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / f(X)].
The result substitution is [ ].

(2) BOUNDS(2^n, INF)