The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))

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:

times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))

Types:
times :: s:0' → s:0' → s:0'
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
times, plus

They will be analysed ascendingly in the following order:
plus < times

(6) Obligation:

Innermost TRS:
Rules:
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))

Types:
times :: s:0' → s:0' → s:0'
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
plus, times

They will be analysed ascendingly in the following order:
plus < times

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

Proved the following rewrite lemma:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Induction Base:
plus(gen_s:0'2_0(a), gen_s:0'2_0(0)) →_R^Ω(1)
gen_s:0'2_0(a)

Induction Step:
plus(gen_s:0'2_0(a), gen_s:0'2_0(+(n4_0, 1))) →_R^Ω(1)
s(plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0))) →_IH
s(gen_s:0'2_0(+(a, c5_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:
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))

Types:
times :: s:0' → s:0' → s:0'
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
times

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

Proved the following rewrite lemma:
times(gen_s:0'2_0(a), gen_s:0'2_0(n471_0)) → gen_s:0'2_0(*(n471_0, a)), rt ∈ Ω(1 + a·n471₀ + n471₀)

Induction Base:
times(gen_s:0'2_0(a), gen_s:0'2_0(0)) →_R^Ω(1)
0'

Induction Step:
times(gen_s:0'2_0(a), gen_s:0'2_0(+(n471_0, 1))) →_R^Ω(1)
plus(times(gen_s:0'2_0(a), gen_s:0'2_0(n471_0)), gen_s:0'2_0(a)) →_IH
plus(gen_s:0'2_0(*(c472_0, a)), gen_s:0'2_0(a)) →_L^{Ω(1 + a)}
gen_s:0'2_0(+(a, *(n471_0, a)))

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:
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))

Types:
times :: s:0' → s:0' → s:0'
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
times(gen_s:0'2_0(a), gen_s:0'2_0(n471_0)) → gen_s:0'2_0(*(n471_0, a)), rt ∈ Ω(1 + a·n471₀ + n471₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_s:0'2_0(a), gen_s:0'2_0(n471_0)) → gen_s:0'2_0(*(n471_0, a)), rt ∈ Ω(1 + a·n471₀ + n471₀)

(14) BOUNDS(n^2, INF)

(15) Obligation:

Innermost TRS:
Rules:
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))

Types:
times :: s:0' → s:0' → s:0'
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
times(gen_s:0'2_0(a), gen_s:0'2_0(n471_0)) → gen_s:0'2_0(*(n471_0, a)), rt ∈ Ω(1 + a·n471₀ + n471₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_s:0'2_0(a), gen_s:0'2_0(n471_0)) → gen_s:0'2_0(*(n471_0, a)), rt ∈ Ω(1 + a·n471₀ + n471₀)

(17) BOUNDS(n^2, INF)

(18) Obligation:

Innermost TRS:
Rules:
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))

Types:
times :: s:0' → s:0' → s:0'
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(20) BOUNDS(n^1, INF)