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:

dbl(S(0), S(0)) → S(S(S(S(0))))
unsafe(S(x)) → dbl(unsafe(x), 0)
unsafe(0) → 0
dbl(0, y) → y

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:

dbl(S(0), S(0)) → S(S(S(S(0))))
unsafe(S(x)) → dbl(unsafe(x), 0)
unsafe(0) → 0
dbl(0, y) → y

S is empty.
Rewrite Strategy: INNERMOST

(3) DecreasingLoopProof (EQUIVALENT transformation)

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

(4) BOUNDS(n^1, INF)