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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
reverse(Cons(x, xs)) →⁺ append(reverse(xs), Cons(x, Nil))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [xs / Cons(x, xs)].
The result substitution is [ ].

(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:

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

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0

(6) Obligation:

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

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

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

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: 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
gen_Cons:Nil2_0 :: Nat → Cons:Nil

(9) 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

(10) Obligation:

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

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: 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
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:
append, shuffle, reverse

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

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

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

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

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: 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
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
append(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:
reverse, shuffle

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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
reverse(gen_Cons:Nil2_0(n518_0)) → gen_Cons:Nil2_0(n518_0), rt ∈ Ω(1 + n518₀ + n518₀²)

Induction Base:
reverse(gen_Cons:Nil2_0(0)) →_R^Ω(1)
Nil

Induction Step:
reverse(gen_Cons:Nil2_0(+(n518_0, 1))) →_R^Ω(1)
append(reverse(gen_Cons:Nil2_0(n518_0)), Cons(Nil)) →_IH
append(gen_Cons:Nil2_0(c519_0), Cons(Nil)) →_L^{Ω(1 + n518₀)}
gen_Cons:Nil2_0(+(n518_0, +(0, 1)))

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

(15) Complex Obligation (BEST)

(16) Obligation:

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

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: 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
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
reverse(gen_Cons:Nil2_0(n518_0)) → gen_Cons:Nil2_0(n518_0), rt ∈ Ω(1 + n518₀ + n518₀²)

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:
shuffle

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
shuffle(gen_Cons:Nil2_0(n747_0)) → gen_Cons:Nil2_0(n747_0), rt ∈ Ω(1 + n747₀ + n747₀² + n747₀³)

Induction Base:
shuffle(gen_Cons:Nil2_0(0)) →_R^Ω(1)
Nil

Induction Step:
shuffle(gen_Cons:Nil2_0(+(n747_0, 1))) →_R^Ω(1)
Cons(shuffle(reverse(gen_Cons:Nil2_0(n747_0)))) →_L^{Ω(1 + n747₀ + n747₀2)}
Cons(shuffle(gen_Cons:Nil2_0(n747_0))) →_IH
Cons(gen_Cons:Nil2_0(c748_0))

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

(18) Complex Obligation (BEST)

(19) Obligation:

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

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: 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
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
reverse(gen_Cons:Nil2_0(n518_0)) → gen_Cons:Nil2_0(n518_0), rt ∈ Ω(1 + n518₀ + n518₀²)
shuffle(gen_Cons:Nil2_0(n747_0)) → gen_Cons:Nil2_0(n747_0), rt ∈ Ω(1 + n747₀ + n747₀² + n747₀³)

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(n³) was proven with the following lemma:
shuffle(gen_Cons:Nil2_0(n747_0)) → gen_Cons:Nil2_0(n747_0), rt ∈ Ω(1 + n747₀ + n747₀² + n747₀³)

(21) BOUNDS(n^3, INF)

(22) Obligation:

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

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: 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
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
reverse(gen_Cons:Nil2_0(n518_0)) → gen_Cons:Nil2_0(n518_0), rt ∈ Ω(1 + n518₀ + n518₀²)
shuffle(gen_Cons:Nil2_0(n747_0)) → gen_Cons:Nil2_0(n747_0), rt ∈ Ω(1 + n747₀ + n747₀² + n747₀³)

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(n³) was proven with the following lemma:
shuffle(gen_Cons:Nil2_0(n747_0)) → gen_Cons:Nil2_0(n747_0), rt ∈ Ω(1 + n747₀ + n747₀² + n747₀³)

(24) BOUNDS(n^3, INF)

(25) Obligation:

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

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: 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
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
reverse(gen_Cons:Nil2_0(n518_0)) → gen_Cons:Nil2_0(n518_0), rt ∈ Ω(1 + n518₀ + n518₀²)

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(n²) was proven with the following lemma:
reverse(gen_Cons:Nil2_0(n518_0)) → gen_Cons:Nil2_0(n518_0), rt ∈ Ω(1 + n518₀ + n518₀²)

(27) BOUNDS(n^2, INF)

(28) Obligation:

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

Types:
shuffle :: Cons:Nil → Cons:Nil
Cons :: 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
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
append(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:
append(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)