The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

bsort(S(x'), Cons(x, xs)) → bsort(x', bubble(x, xs))
len(Cons(x, xs)) → +(S(0), len(xs))
bubble(x', Cons(x, xs)) → bubble[Ite][False][Ite](<(x', x), x', Cons(x, xs))
len(Nil) → 0
bubble(x, Nil) → Cons(x, Nil)
bsort(0, xs) → xs
bubblesort(xs) → bsort(len(xs), xs)

The (relative) TRS S consists of the following rules:

+(x, S(0)) → S(x)
+(S(0), y) → S(y)
<(S(x), S(y)) → <(x, y)
<(0, S(y)) → True
<(x, 0) → False
bubble[Ite][False][Ite](False, x', Cons(x, xs)) → Cons(x, bubble(x', xs))
bubble[Ite][False][Ite](True, x', Cons(x, xs)) → Cons(x', bubble(x, xs))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)