The Runtime Complexity (innermost) 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:

mul0(C(x, y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(x, y), y') → add0(y, C(S, y'))
add0(Z, y) → y
second(C(x, y)) → y
isZero(C(x, y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

Rewrite Strategy: INNERMOST

(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) Obligation:

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

mul0(C(x, y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(x, y), y') → add0(y, C(S, y'))
add0(Z, y) → y
second(C(x, y)) → y
isZero(C(x, y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
C/0

(4) Obligation:

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

mul0(C(y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(y), y') → add0(y, C(y'))
add0(Z, y) → y
second(C(y)) → y
isZero(C(y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

S is empty.
Rewrite Strategy: INNERMOST

(5) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
mul0(C(y), y') →⁺ add0(mul0(y, y'), y')
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [y / C(y)].
The result substitution is [ ].

(6) BOUNDS(n^1, INF)