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(C(x, y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(x, y), y') → add0(y, C(S, y'))
add0(Z, y) → y
second(C(x, y)) → y
isZero(C(x, y)) → False
isZero(Z) → True
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(C(x, y), y') →⁺ add0(mul0(y, y'), y')
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [y / C(x, y)].
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(C(x, y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(x, y), y') → add0(y, C(S, y'))
add0(Z, y) → y
second(C(x, y)) → y
isZero(C(x, y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

S is empty.
Rewrite Strategy: INNERMOST

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

Sliced the following arguments:
C/0

(6) Obligation:

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

mul0(C(y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(y), y') → add0(y, C(y'))
add0(Z, y) → y
second(C(y)) → y
isZero(C(y)) → False
isZero(Z) → True
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(C(y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(y), y') → add0(y, C(y'))
add0(Z, y) → y
second(C(y)) → y
isZero(C(y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: C:Z → C:Z → C:Z
C :: C:Z → C:Z
add0 :: C:Z → C:Z → C:Z
Z :: C:Z
second :: C:Z → C:Z
isZero :: C:Z → False:True
False :: False:True
True :: False:True
goal :: C:Z → C:Z → C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat → C:Z

(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(C(y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(y), y') → add0(y, C(y'))
add0(Z, y) → y
second(C(y)) → y
isZero(C(y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: C:Z → C:Z → C:Z
C :: C:Z → C:Z
add0 :: C:Z → C:Z → C:Z
Z :: C:Z
second :: C:Z → C:Z
isZero :: C:Z → False:True
False :: False:True
True :: False:True
goal :: C:Z → C:Z → C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat → C:Z

Generator Equations:
gen_C:Z3_1(0) ⇔ Z
gen_C:Z3_1(+(x, 1)) ⇔ C(gen_C:Z3_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_C:Z3_1(n5_1), gen_C:Z3_1(b)) → gen_C:Z3_1(+(n5_1, b)), rt ∈ Ω(1 + n5₁)

Induction Base:
add0(gen_C:Z3_1(0), gen_C:Z3_1(b)) →_R^Ω(1)
gen_C:Z3_1(b)

Induction Step:
add0(gen_C:Z3_1(+(n5_1, 1)), gen_C:Z3_1(b)) →_R^Ω(1)
add0(gen_C:Z3_1(n5_1), C(gen_C:Z3_1(b))) →_IH
gen_C:Z3_1(+(+(b, 1), c6_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(C(y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(y), y') → add0(y, C(y'))
add0(Z, y) → y
second(C(y)) → y
isZero(C(y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: C:Z → C:Z → C:Z
C :: C:Z → C:Z
add0 :: C:Z → C:Z → C:Z
Z :: C:Z
second :: C:Z → C:Z
isZero :: C:Z → False:True
False :: False:True
True :: False:True
goal :: C:Z → C:Z → C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat → C:Z

Lemmas:
add0(gen_C:Z3_1(n5_1), gen_C:Z3_1(b)) → gen_C:Z3_1(+(n5_1, b)), rt ∈ Ω(1 + n5₁)

Generator Equations:
gen_C:Z3_1(0) ⇔ Z
gen_C:Z3_1(+(x, 1)) ⇔ C(gen_C:Z3_1(x))

The following defined symbols remain to be analysed:
mul0

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

Proved the following rewrite lemma:
mul0(gen_C:Z3_1(n524_1), gen_C:Z3_1(b)) → gen_C:Z3_1(*(n524_1, b)), rt ∈ Ω(1 + b·n524₁² + n524₁)

Induction Base:
mul0(gen_C:Z3_1(0), gen_C:Z3_1(b)) →_R^Ω(1)
Z

Induction Step:
mul0(gen_C:Z3_1(+(n524_1, 1)), gen_C:Z3_1(b)) →_R^Ω(1)
add0(mul0(gen_C:Z3_1(n524_1), gen_C:Z3_1(b)), gen_C:Z3_1(b)) →_IH
add0(gen_C:Z3_1(*(c525_1, b)), gen_C:Z3_1(b)) →_L^{Ω(1 + b·n524₁)}
gen_C:Z3_1(+(*(n524_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(C(y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(y), y') → add0(y, C(y'))
add0(Z, y) → y
second(C(y)) → y
isZero(C(y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: C:Z → C:Z → C:Z
C :: C:Z → C:Z
add0 :: C:Z → C:Z → C:Z
Z :: C:Z
second :: C:Z → C:Z
isZero :: C:Z → False:True
False :: False:True
True :: False:True
goal :: C:Z → C:Z → C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat → C:Z

Lemmas:
add0(gen_C:Z3_1(n5_1), gen_C:Z3_1(b)) → gen_C:Z3_1(+(n5_1, b)), rt ∈ Ω(1 + n5₁)
mul0(gen_C:Z3_1(n524_1), gen_C:Z3_1(b)) → gen_C:Z3_1(*(n524_1, b)), rt ∈ Ω(1 + b·n524₁² + n524₁)

Generator Equations:
gen_C:Z3_1(0) ⇔ Z
gen_C:Z3_1(+(x, 1)) ⇔ C(gen_C:Z3_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_C:Z3_1(n524_1), gen_C:Z3_1(b)) → gen_C:Z3_1(*(n524_1, b)), rt ∈ Ω(1 + b·n524₁² + n524₁)

(18) BOUNDS(n^3, INF)

(19) Obligation:

Innermost TRS:
Rules:
mul0(C(y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(y), y') → add0(y, C(y'))
add0(Z, y) → y
second(C(y)) → y
isZero(C(y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: C:Z → C:Z → C:Z
C :: C:Z → C:Z
add0 :: C:Z → C:Z → C:Z
Z :: C:Z
second :: C:Z → C:Z
isZero :: C:Z → False:True
False :: False:True
True :: False:True
goal :: C:Z → C:Z → C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat → C:Z

Lemmas:
add0(gen_C:Z3_1(n5_1), gen_C:Z3_1(b)) → gen_C:Z3_1(+(n5_1, b)), rt ∈ Ω(1 + n5₁)
mul0(gen_C:Z3_1(n524_1), gen_C:Z3_1(b)) → gen_C:Z3_1(*(n524_1, b)), rt ∈ Ω(1 + b·n524₁² + n524₁)

Generator Equations:
gen_C:Z3_1(0) ⇔ Z
gen_C:Z3_1(+(x, 1)) ⇔ C(gen_C:Z3_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_C:Z3_1(n524_1), gen_C:Z3_1(b)) → gen_C:Z3_1(*(n524_1, b)), rt ∈ Ω(1 + b·n524₁² + n524₁)

(21) BOUNDS(n^3, INF)

(22) Obligation:

Innermost TRS:
Rules:
mul0(C(y), y') → add0(mul0(y, y'), y')
mul0(Z, y) → Z
add0(C(y), y') → add0(y, C(y'))
add0(Z, y) → y
second(C(y)) → y
isZero(C(y)) → False
isZero(Z) → True
goal(xs, ys) → mul0(xs, ys)

Types:
mul0 :: C:Z → C:Z → C:Z
C :: C:Z → C:Z
add0 :: C:Z → C:Z → C:Z
Z :: C:Z
second :: C:Z → C:Z
isZero :: C:Z → False:True
False :: False:True
True :: False:True
goal :: C:Z → C:Z → C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat → C:Z

Lemmas:
add0(gen_C:Z3_1(n5_1), gen_C:Z3_1(b)) → gen_C:Z3_1(+(n5_1, b)), rt ∈ Ω(1 + n5₁)

Generator Equations:
gen_C:Z3_1(0) ⇔ Z
gen_C:Z3_1(+(x, 1)) ⇔ C(gen_C:Z3_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_C:Z3_1(n5_1), gen_C:Z3_1(b)) → gen_C:Z3_1(+(n5_1, b)), rt ∈ Ω(1 + n5₁)

(24) BOUNDS(n^1, INF)