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:

subsets(Cons(x, xs)) → subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
subsets(Nil) → Cons(Nil, Nil)
mapconsapp(x', Cons(x, xs), rest) → Cons(Cons(x', x), mapconsapp(x', xs, rest))
mapconsapp(x, Nil, rest) → rest
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)

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

subsets[Ite][True][Let](Cons(x, xs), subs) → mapconsapp(x, subs, subs)

Rewrite Strategy: INNERMOST

(1) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0
mapconsapp/0

(2) Obligation:

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

subsets(Cons(xs)) → subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) → Cons(Nil)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) → rest
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
goal(xs) → subsets(xs)

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

subsets[Ite][True][Let](Cons(xs), subs) → mapconsapp(subs, subs)

Rewrite Strategy: INNERMOST

(3) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
subsets(Cons(xs)) →⁺ subsets[Ite][True][Let](Cons(xs), subsets(xs))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs / Cons(xs)].
The result substitution is [ ].

(4) BOUNDS(n^1, INF)