The Runtime Complexity (innermost) 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:

selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) → selects(x, Nil, xs)
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) → Nil
revapp(Nil, rest) → rest

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) → selects(x, Nil, xs)
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) → Nil
revapp(Nil, rest) → rest

S is empty.
Rewrite Strategy: INNERMOST

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) → selects(x, Nil, xs)
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) → Nil
revapp(Nil, rest) → rest

Types:
selects :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
select :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
selects, revapp

They will be analysed ascendingly in the following order:
revapp < selects

(8) Obligation:

Innermost TRS:
Rules:
selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) → selects(x, Nil, xs)
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) → Nil
revapp(Nil, rest) → rest

Types:
selects :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
select :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_0(x))

The following defined symbols remain to be analysed:
revapp, selects

They will be analysed ascendingly in the following order:
revapp < selects

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
revapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Induction Base:
revapp(gen_Cons:Nil2_0(0), gen_Cons:Nil2_0(b)) →_R^Ω(1)
gen_Cons:Nil2_0(b)

Induction Step:
revapp(gen_Cons:Nil2_0(+(n4_0, 1)), gen_Cons:Nil2_0(b)) →_R^Ω(1)
revapp(gen_Cons:Nil2_0(n4_0), Cons(Nil, gen_Cons:Nil2_0(b))) →_IH
gen_Cons:Nil2_0(+(+(b, 1), c5_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) → selects(x, Nil, xs)
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) → Nil
revapp(Nil, rest) → rest

Types:
selects :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
select :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
revapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_0(x))

The following defined symbols remain to be analysed:
selects

(12) Obligation:

Innermost TRS:
Rules:
selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) → selects(x, Nil, xs)
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) → Nil
revapp(Nil, rest) → rest

Types:
selects :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
select :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
revapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.