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:

isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)

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

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

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)