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:

compS_f#1(compS_f(x2), x1) → compS_f#1(x2, S(x1))
compS_f#1(id, x3) → S(x3)
iter#3(0) → id
iter#3(S(x6)) → compS_f(iter#3(x6))
main(0) → 0
main(S(x9)) → compS_f#1(iter#3(x9), 0)

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:

compS_f#1(compS_f(x2), x1) → compS_f#1(x2, S(x1))
compS_f#1(id, x3) → S(x3)
iter#3(0) → id
iter#3(S(x6)) → compS_f(iter#3(x6))
main(0) → 0
main(S(x9)) → compS_f#1(iter#3(x9), 0)

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
compS_f#1(compS_f(x2), x1) →⁺ compS_f#1(x2, S(x1))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x2 / compS_f(x2)].
The result substitution is [x1 / S(x1)].

(4) BOUNDS(n^1, INF)