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

(0) Obligation:

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

map(Cons(x, xs)) → Cons(f(x), map(xs))
map(Nil) → Nil
goal(xs) → map(xs)
f(x) → *(x, x)
+Full(S(x), y) → +Full(x, S(y))
+Full(0, y) → y

The (relative) TRS S consists of the following rules:

*(x, S(S(y))) → +(x, *(x, S(y)))
*(x, S(0)) → x
*(x, 0) → 0
*(0, y) → 0

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
map(Cons(x, xs)) →⁺ Cons(f(x), map(xs))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs / Cons(x, xs)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)