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:

plus#2(0, x12) → x12
plus#2(S(x4), x2) → S(plus#2(x4, x2))
fold#3(Nil) → 0
fold#3(Cons(x4, x2)) → plus#2(x4, fold#3(x2))
main(x1) → fold#3(x1)

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:

plus#2(0, x12) → x12
plus#2(S(x4), x2) → S(plus#2(x4, x2))
fold#3(Nil) → 0
fold#3(Cons(x4, x2)) → plus#2(x4, fold#3(x2))
main(x1) → fold#3(x1)

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
plus#2(S(x4), x2) →⁺ S(plus#2(x4, x2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x4 / S(x4)].
The result substitution is [ ].

(4) BOUNDS(n^1, INF)