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:

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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

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

Infered types.

(6) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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(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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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(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) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

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

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

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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:
mapconsapp(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(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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
revapp(gen_Cons:Nil2_0(n574_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n574_0, b)), rt ∈ Ω(1 + n574₀)

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(+(n574_0, 1)), gen_Cons:Nil2_0(b)) →_R^Ω(1)
revapp(gen_Cons:Nil2_0(n574_0), Cons(gen_Cons:Nil2_0(b))) →_IH
gen_Cons:Nil2_0(+(+(b, 1), c575_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
revapp(gen_Cons:Nil2_0(n574_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n574_0, b)), rt ∈ Ω(1 + n574₀)

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

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
permute(gen_Cons:Nil2_0(+(1, n1155_0))) → *3_0, rt ∈ Ω(n1155₀)

Induction Base:
permute(gen_Cons:Nil2_0(+(1, 0)))

Induction Step:
permute(gen_Cons:Nil2_0(+(1, +(n1155_0, 1)))) →_R^Ω(1)
select(Nil, gen_Cons:Nil2_0(+(1, n1155_0))) →_R^Ω(1)
mapconsapp(permute(revapp(Nil, Cons(gen_Cons:Nil2_0(n1155_0)))), select(Cons(Nil), gen_Cons:Nil2_0(n1155_0))) →_L^Ω(1)
mapconsapp(permute(gen_Cons:Nil2_0(+(0, +(n1155_0, 1)))), select(Cons(Nil), gen_Cons:Nil2_0(n1155_0))) →_IH
mapconsapp(*3_0, select(Cons(Nil), gen_Cons:Nil2_0(n1155_0)))

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
revapp(gen_Cons:Nil2_0(n574_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n574_0, b)), rt ∈ Ω(1 + n574₀)
permute(gen_Cons:Nil2_0(+(1, n1155_0))) → *3_0, rt ∈ Ω(n1155₀)

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

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
revapp(gen_Cons:Nil2_0(n574_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n574_0, b)), rt ∈ Ω(1 + n574₀)
permute(gen_Cons:Nil2_0(+(1, n1155_0))) → *3_0, rt ∈ Ω(n1155₀)

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

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
revapp(gen_Cons:Nil2_0(n574_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n574_0, b)), rt ∈ Ω(1 + n574₀)
permute(gen_Cons:Nil2_0(+(1, n1155_0))) → *3_0, rt ∈ Ω(n1155₀)

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

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(24) BOUNDS(n^1, INF)

(25) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
revapp(gen_Cons:Nil2_0(n574_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n574_0, b)), rt ∈ Ω(1 + n574₀)

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

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

Innermost TRS:
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)

Types:
select :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
mapconsapp :: 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:
mapconsapp(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(gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mapconsapp(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(30) BOUNDS(n^1, INF)