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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
permute(Cons(x, Cons(x23981_0, Cons(x24266_0, xs24267_0)))) →⁺ mapconsapp(x, permute(Cons(x23981_0, Cons(x24266_0, xs24267_0))), mapconsapp(x23981_0, permute(Cons(x, Cons(x24266_0, xs24267_0))), select(x24266_0, Cons(x23981_0, Cons(x, Nil)), xs24267_0)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs24267_0 / Cons(x24266_0, xs24267_0)].
The result substitution is [x / x23981_0, x23981_0 / x24266_0].

The rewrite sequence
permute(Cons(x, Cons(x23981_0, Cons(x24266_0, xs24267_0)))) →⁺ mapconsapp(x, permute(Cons(x23981_0, Cons(x24266_0, xs24267_0))), mapconsapp(x23981_0, permute(Cons(x, Cons(x24266_0, xs24267_0))), select(x24266_0, Cons(x23981_0, Cons(x, Nil)), xs24267_0)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
The pumping substitution is [xs24267_0 / Cons(x24266_0, xs24267_0)].
The result substitution is [x23981_0 / x24266_0].

(2) BOUNDS(2^n, 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:

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

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

Infered types.

(6) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
permute :: Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons: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:
select, mapconsapp, permute, revapp

They will be analysed ascendingly in the following order:
mapconsapp < select
select = permute
revapp < select

(8) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
permute :: Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons: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:
mapconsapp, select, permute, revapp

They will be analysed ascendingly in the following order:
mapconsapp < select
select = permute
revapp < select

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol mapconsapp.

(10) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
permute :: Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons: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, select, permute

They will be analysed ascendingly in the following order:
select = permute
revapp < select

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
revapp(gen_Cons:Nil2_0(n3616860_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n3616860_0, b)), rt ∈ Ω(1 + n3616860₀)

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(+(n3616860_0, 1)), gen_Cons:Nil2_0(b)) →_R^Ω(1)
revapp(gen_Cons:Nil2_0(n3616860_0), Cons(Nil, gen_Cons:Nil2_0(b))) →_IH
gen_Cons:Nil2_0(+(+(b, 1), c3616861_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
permute :: Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
revapp(gen_Cons:Nil2_0(n3616860_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n3616860_0, b)), rt ∈ Ω(1 + n3616860₀)

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:
permute, select

They will be analysed ascendingly in the following order:
select = permute

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol permute.

(15) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
permute :: Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
revapp(gen_Cons:Nil2_0(n3616860_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n3616860_0, b)), rt ∈ Ω(1 + n3616860₀)

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:
select

They will be analysed ascendingly in the following order:
select = permute

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol select.

(17) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
permute :: Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
revapp(gen_Cons:Nil2_0(n3616860_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n3616860_0, b)), rt ∈ Ω(1 + n3616860₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
revapp(gen_Cons:Nil2_0(n3616860_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n3616860_0, b)), rt ∈ Ω(1 + n3616860₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mapconsapp :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
permute :: Cons:Nil → Cons:Nil
revapp :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
revapp(gen_Cons:Nil2_0(n3616860_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n3616860_0, b)), rt ∈ Ω(1 + n3616860₀)

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.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
revapp(gen_Cons:Nil2_0(n3616860_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n3616860_0, b)), rt ∈ Ω(1 + n3616860₀)

(22) BOUNDS(n^1, INF)