The Runtime Complexity (full) 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:

top(sent(x)) → top(check(rest(x)))
rest(nil) → sent(nil)
rest(cons(x, y)) → sent(y)
check(sent(x)) → sent(check(x))
check(rest(x)) → rest(check(x))
check(cons(x, y)) → cons(check(x), y)
check(cons(x, y)) → cons(x, check(y))
check(cons(x, y)) → cons(x, y)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
top(sent(cons(x2_0, cons(x193_0, y194_0)))) →⁺ top(sent(cons(check(x193_0), y194_0)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [y194_0 / cons(x193_0, y194_0)].
The result substitution is [x2_0 / check(x193_0)].

(2) BOUNDS(n^1, INF)