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

Renamed function symbols to avoid clashes with predefined symbol.

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

Sliced the following arguments:
C'/0

Runtime Complexity 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)

Rewrite Strategy: INNERMOST

Infered types.

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':Z'1 :: C':Z'
_hole_False':True'2 :: False':True'
_gen_C':Z'3 :: Nat → C':Z'

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

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

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':Z'1 :: C':Z'
_hole_False':True'2 :: False':True'
_gen_C':Z'3 :: Nat → C':Z'

Generator Equations:
_gen_C':Z'3(0) ⇔ Z'
_gen_C':Z'3(+(x, 1)) ⇔ C'(_gen_C':Z'3(x))

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

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

Could not prove a rewrite lemma for the defined symbol add0'.

The following conjecture could not be proven:

add0'(_gen_C':Z'3(_n5), _gen_C':Z'3(b)) →? _gen_C':Z'3(+(_n5, b))

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':Z'1 :: C':Z'
_hole_False':True'2 :: False':True'
_gen_C':Z'3 :: Nat → C':Z'

Generator Equations:
_gen_C':Z'3(0) ⇔ Z'
_gen_C':Z'3(+(x, 1)) ⇔ C'(_gen_C':Z'3(x))

The following defined symbols remain to be analysed:
mul0'

Proved the following rewrite lemma:
mul0'(_gen_C':Z'3(_n1055), _gen_C':Z'3(0)) → _gen_C':Z'3(0), rt ∈ Ω(1 + n1055)

Induction Base:
mul0'(_gen_C':Z'3(0), _gen_C':Z'3(0)) →RΩ(1)
Z'

Induction Step:
mul0'(_gen_C':Z'3(+(_$n1056, 1)), _gen_C':Z'3(0)) →RΩ(1)
add0'(mul0'(_gen_C':Z'3(_$n1056), _gen_C':Z'3(0)), _gen_C':Z'3(0)) →IH
add0'(_gen_C':Z'3(0), _gen_C':Z'3(0)) →RΩ(1)
_gen_C':Z'3(0)

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

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':Z'1 :: C':Z'
_hole_False':True'2 :: False':True'
_gen_C':Z'3 :: Nat → C':Z'

Lemmas:
mul0'(_gen_C':Z'3(_n1055), _gen_C':Z'3(0)) → _gen_C':Z'3(0), rt ∈ Ω(1 + n1055)

Generator Equations:
_gen_C':Z'3(0) ⇔ Z'
_gen_C':Z'3(+(x, 1)) ⇔ C'(_gen_C':Z'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
mul0'(_gen_C':Z'3(_n1055), _gen_C':Z'3(0)) → _gen_C':Z'3(0), rt ∈ Ω(1 + n1055)