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:

f4(S(x''), S(x'), x3, x4, S(x)) → f2(S(x''), x', x3, x4, x)
f4(S(x'), 0, x3, x4, S(x)) → f3(x', 0, x3, x4, x)
f2(x1, x2, S(x''), S(x'), S(x)) → f5(x1, x2, S(x''), x', x)
f2(x1, x2, S(x'), 0, S(x)) → f3(x1, x2, x', 0, x)
f4(S(x'), S(x), x3, x4, 0) → 0
f4(S(x), 0, x3, x4, 0) → 0
f2(x1, x2, S(x'), S(x), 0) → 0
f2(x1, x2, S(x), 0, 0) → 0
f0(x1, S(x'), x3, S(x), x5) → f1(x', S(x'), x, S(x), S(x))
f0(x1, S(x), x3, 0, x5) → 0
f6(x1, x2, x3, x4, S(x)) → f0(x1, x2, x4, x3, x)
f5(x1, x2, x3, x4, S(x)) → f6(x2, x1, x3, x4, x)
f3(x1, x2, x3, x4, S(x)) → f4(x1, x2, x4, x3, x)
f1(x1, x2, x3, x4, S(x)) → f2(x2, x1, x3, x4, x)
f6(x1, x2, x3, x4, 0) → 0
f5(x1, x2, x3, x4, 0) → 0
f4(0, x2, x3, x4, x5) → 0
f3(x1, x2, x3, x4, 0) → 0
f2(x1, x2, 0, x4, x5) → 0
f1(x1, x2, x3, x4, 0) → 0
f0(x1, 0, x3, x4, x5) → 0

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f4(S(x'), 0, x3, x4, S(S(x20081_7))) →⁺ f4(x', 0, x4, x3, x20081_7)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x' / S(x'), x20081_7 / S(S(x20081_7))].
The result substitution is [x3 / x4, x4 / x3].

(2) BOUNDS(n^1, INF)