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:

mul0(Cons(x, xs), y) → add0(mul0(xs, y), y)
add0(Cons(x, xs), y) → add0(xs, Cons(S, y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
mul0(Cons(x, xs), y) →⁺ add0(mul0(xs, y), y)
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:

mul0(Cons(x, xs), y) → add0(mul0(xs, y), y)
add0(Cons(x, xs), y) → add0(xs, Cons(S, y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

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:

mul0(Cons(xs), y) → add0(mul0(xs, y), y)
add0(Cons(xs), y) → add0(xs, Cons(y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Innermost TRS:
Rules:
mul0(Cons(xs), y) → add0(mul0(xs, y), y)
add0(Cons(xs), y) → add0(xs, Cons(y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat → Cons:Nil

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mul0, add0

They will be analysed ascendingly in the following order:
add0 < mul0

(10) Obligation:

Innermost TRS:
Rules:
mul0(Cons(xs), y) → add0(mul0(xs, y), y)
add0(Cons(xs), y) → add0(xs, Cons(y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil2_1(0) ⇔ Nil
gen_Cons:Nil2_1(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_1(x))

The following defined symbols remain to be analysed:
add0, mul0

They will be analysed ascendingly in the following order:
add0 < mul0

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

Proved the following rewrite lemma:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(+(n4_1, b)), rt ∈ Ω(1 + n4₁)

Induction Base:
add0(gen_Cons:Nil2_1(0), gen_Cons:Nil2_1(b)) →_R^Ω(1)
gen_Cons:Nil2_1(b)

Induction Step:
add0(gen_Cons:Nil2_1(+(n4_1, 1)), gen_Cons:Nil2_1(b)) →_R^Ω(1)
add0(gen_Cons:Nil2_1(n4_1), Cons(gen_Cons:Nil2_1(b))) →_IH
gen_Cons:Nil2_1(+(+(b, 1), c5_1))

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:
mul0(Cons(xs), y) → add0(mul0(xs, y), y)
add0(Cons(xs), y) → add0(xs, Cons(y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat → Cons:Nil

Lemmas:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(+(n4_1, b)), rt ∈ Ω(1 + n4₁)

Generator Equations:
gen_Cons:Nil2_1(0) ⇔ Nil
gen_Cons:Nil2_1(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_1(x))

The following defined symbols remain to be analysed:
mul0

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

Proved the following rewrite lemma:
mul0(gen_Cons:Nil2_1(n458_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(*(n458_1, b)), rt ∈ Ω(1 + b·n458₁² + n458₁)

Induction Base:
mul0(gen_Cons:Nil2_1(0), gen_Cons:Nil2_1(b)) →_R^Ω(1)
Nil

Induction Step:
mul0(gen_Cons:Nil2_1(+(n458_1, 1)), gen_Cons:Nil2_1(b)) →_R^Ω(1)
add0(mul0(gen_Cons:Nil2_1(n458_1), gen_Cons:Nil2_1(b)), gen_Cons:Nil2_1(b)) →_IH
add0(gen_Cons:Nil2_1(*(c459_1, b)), gen_Cons:Nil2_1(b)) →_L^{Ω(1 + b·n458₁)}
gen_Cons:Nil2_1(+(*(n458_1, b), b))

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

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
mul0(Cons(xs), y) → add0(mul0(xs, y), y)
add0(Cons(xs), y) → add0(xs, Cons(y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat → Cons:Nil

Lemmas:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(+(n4_1, b)), rt ∈ Ω(1 + n4₁)
mul0(gen_Cons:Nil2_1(n458_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(*(n458_1, b)), rt ∈ Ω(1 + b·n458₁² + n458₁)

Generator Equations:
gen_Cons:Nil2_1(0) ⇔ Nil
gen_Cons:Nil2_1(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_1(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
mul0(gen_Cons:Nil2_1(n458_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(*(n458_1, b)), rt ∈ Ω(1 + b·n458₁² + n458₁)

(18) BOUNDS(n^3, INF)

(19) Obligation:

Innermost TRS:
Rules:
mul0(Cons(xs), y) → add0(mul0(xs, y), y)
add0(Cons(xs), y) → add0(xs, Cons(y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat → Cons:Nil

Lemmas:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(+(n4_1, b)), rt ∈ Ω(1 + n4₁)
mul0(gen_Cons:Nil2_1(n458_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(*(n458_1, b)), rt ∈ Ω(1 + b·n458₁² + n458₁)

Generator Equations:
gen_Cons:Nil2_1(0) ⇔ Nil
gen_Cons:Nil2_1(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_1(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
mul0(gen_Cons:Nil2_1(n458_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(*(n458_1, b)), rt ∈ Ω(1 + b·n458₁² + n458₁)

(21) BOUNDS(n^3, INF)

(22) Obligation:

Innermost TRS:
Rules:
mul0(Cons(xs), y) → add0(mul0(xs, y), y)
add0(Cons(xs), y) → add0(xs, Cons(y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat → Cons:Nil

Lemmas:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(+(n4_1, b)), rt ∈ Ω(1 + n4₁)

Generator Equations:
gen_Cons:Nil2_1(0) ⇔ Nil
gen_Cons:Nil2_1(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_1(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) → gen_Cons:Nil2_1(+(n4_1, b)), rt ∈ Ω(1 + n4₁)

(24) BOUNDS(n^1, INF)