The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

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

select(x', revprefix, Cons(x, xs)) → mapconsapp(x', permute(revapp(revprefix, Cons(x, xs))), select(x, Cons(x', revprefix), xs))
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
permute(Cons(x, xs)) → select(x, Nil, xs)
mapconsapp(x', Cons(x, xs), rest) → Cons(Cons(x', x), mapconsapp(x', xs, rest))
select(x, revprefix, Nil) → mapconsapp(x, permute(revapp(revprefix, Nil)), Nil)
revapp(Nil, rest) → rest
permute(Nil) → Cons(Nil, Nil)
mapconsapp(x, Nil, rest) → rest
goal(xs) → permute(xs)

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:

select(x', revprefix, Cons(x, xs)) → mapconsapp(x', permute(revapp(revprefix, Cons(x, xs))), select(x, Cons(x', revprefix), xs))
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
permute(Cons(x, xs)) → select(x, Nil, xs)
mapconsapp(x', Cons(x, xs), rest) → Cons(Cons(x', x), mapconsapp(x', xs, rest))
select(x, revprefix, Nil) → mapconsapp(x, permute(revapp(revprefix, Nil)), Nil)
revapp(Nil, rest) → rest
permute(Nil) → Cons(Nil, Nil)
mapconsapp(x, Nil, rest) → rest
goal(xs) → permute(xs)

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
select/0
Cons/0
mapconsapp/0

(4) Obligation:

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

select(revprefix, Cons(xs)) → mapconsapp(permute(revapp(revprefix, Cons(xs))), select(Cons(revprefix), xs))
revapp(Cons(xs), rest) → revapp(xs, Cons(rest))
permute(Cons(xs)) → select(Nil, xs)
mapconsapp(Cons(xs), rest) → Cons(mapconsapp(xs, rest))
select(revprefix, Nil) → mapconsapp(permute(revapp(revprefix, Nil)), Nil)
revapp(Nil, rest) → rest
permute(Nil) → Cons(Nil)
mapconsapp(Nil, rest) → rest
goal(xs) → permute(xs)

S is empty.
Rewrite Strategy: INNERMOST

(5) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
permute(Cons(Cons(Cons(xs13774_0)))) →⁺ mapconsapp(permute(Cons(Cons(xs13774_0))), mapconsapp(permute(Cons(Cons(xs13774_0))), select(Cons(Cons(Nil)), xs13774_0)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [xs13774_0 / Cons(xs13774_0)].
The result substitution is [ ].

The rewrite sequence
permute(Cons(Cons(Cons(xs13774_0)))) →⁺ mapconsapp(permute(Cons(Cons(xs13774_0))), mapconsapp(permute(Cons(Cons(xs13774_0))), select(Cons(Cons(Nil)), xs13774_0)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
The pumping substitution is [xs13774_0 / Cons(xs13774_0)].
The result substitution is [ ].

(6) BOUNDS(2^n, INF)