The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).

(0) Obligation:

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

shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(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:

shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
reverse :: Cons:Nil → Cons:Nil
append :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
shuffle, reverse, append

They will be analysed ascendingly in the following order:
reverse < shuffle
append < reverse

(6) Obligation:

Innermost TRS:
Rules:
shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
reverse :: Cons:Nil → Cons:Nil
append :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
append, shuffle, reverse

They will be analysed ascendingly in the following order:
reverse < shuffle
append < reverse

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
append(gen_Cons:Nil3_0(0), gen_Cons:Nil3_0(b)) →_R^Ω(1)
gen_Cons:Nil3_0(b)

Induction Step:
append(gen_Cons:Nil3_0(+(n5_0, 1)), gen_Cons:Nil3_0(b)) →_R^Ω(1)
Cons(hole_a2_0, append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b))) →_IH
Cons(hole_a2_0, gen_Cons:Nil3_0(+(b, c6_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
reverse :: Cons:Nil → Cons:Nil
append :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
reverse, shuffle

They will be analysed ascendingly in the following order:
reverse < shuffle

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
reverse(gen_Cons:Nil3_0(n528_0)) → gen_Cons:Nil3_0(n528_0), rt ∈ Ω(1 + n528₀ + n528₀²)

Induction Base:
reverse(gen_Cons:Nil3_0(0)) →_R^Ω(1)
Nil

Induction Step:
reverse(gen_Cons:Nil3_0(+(n528_0, 1))) →_R^Ω(1)
append(reverse(gen_Cons:Nil3_0(n528_0)), Cons(hole_a2_0, Nil)) →_IH
append(gen_Cons:Nil3_0(c529_0), Cons(hole_a2_0, Nil)) →_L^{Ω(1 + n528₀)}
gen_Cons:Nil3_0(+(n528_0, +(0, 1)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
reverse :: Cons:Nil → Cons:Nil
append :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
reverse(gen_Cons:Nil3_0(n528_0)) → gen_Cons:Nil3_0(n528_0), rt ∈ Ω(1 + n528₀ + n528₀²)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
shuffle

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
shuffle(gen_Cons:Nil3_0(n766_0)) → gen_Cons:Nil3_0(n766_0), rt ∈ Ω(1 + n766₀ + n766₀² + n766₀³)

Induction Base:
shuffle(gen_Cons:Nil3_0(0)) →_R^Ω(1)
Nil

Induction Step:
shuffle(gen_Cons:Nil3_0(+(n766_0, 1))) →_R^Ω(1)
Cons(hole_a2_0, shuffle(reverse(gen_Cons:Nil3_0(n766_0)))) →_L^{Ω(1 + n766₀ + n766₀2)}
Cons(hole_a2_0, shuffle(gen_Cons:Nil3_0(n766_0))) →_IH
Cons(hole_a2_0, gen_Cons:Nil3_0(c767_0))

We have rt ∈ Ω(n³) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n³).

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
reverse :: Cons:Nil → Cons:Nil
append :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
reverse(gen_Cons:Nil3_0(n528_0)) → gen_Cons:Nil3_0(n528_0), rt ∈ Ω(1 + n528₀ + n528₀²)
shuffle(gen_Cons:Nil3_0(n766_0)) → gen_Cons:Nil3_0(n766_0), rt ∈ Ω(1 + n766₀ + n766₀² + n766₀³)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
shuffle(gen_Cons:Nil3_0(n766_0)) → gen_Cons:Nil3_0(n766_0), rt ∈ Ω(1 + n766₀ + n766₀² + n766₀³)

(17) BOUNDS(n^3, INF)

(18) Obligation:

Innermost TRS:
Rules:
shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
reverse :: Cons:Nil → Cons:Nil
append :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
reverse(gen_Cons:Nil3_0(n528_0)) → gen_Cons:Nil3_0(n528_0), rt ∈ Ω(1 + n528₀ + n528₀²)
shuffle(gen_Cons:Nil3_0(n766_0)) → gen_Cons:Nil3_0(n766_0), rt ∈ Ω(1 + n766₀ + n766₀² + n766₀³)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
shuffle(gen_Cons:Nil3_0(n766_0)) → gen_Cons:Nil3_0(n766_0), rt ∈ Ω(1 + n766₀ + n766₀² + n766₀³)

(20) BOUNDS(n^3, INF)

(21) Obligation:

Innermost TRS:
Rules:
shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
reverse :: Cons:Nil → Cons:Nil
append :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
reverse(gen_Cons:Nil3_0(n528_0)) → gen_Cons:Nil3_0(n528_0), rt ∈ Ω(1 + n528₀ + n528₀²)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
reverse(gen_Cons:Nil3_0(n528_0)) → gen_Cons:Nil3_0(n528_0), rt ∈ Ω(1 + n528₀ + n528₀²)

(23) BOUNDS(n^2, INF)

(24) Obligation:

Innermost TRS:
Rules:
shuffle(Cons(x, xs)) → Cons(x, shuffle(reverse(xs)))
reverse(Cons(x, xs)) → append(reverse(xs), Cons(x, Nil))
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
shuffle(Nil) → Nil
reverse(Nil) → Nil
append(Nil, ys) → ys
goal(xs) → shuffle(xs)

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
reverse :: Cons:Nil → Cons:Nil
append :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(26) BOUNDS(n^1, INF)