The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))

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:

div(0', y) → 0'
div(x, y) → quot(x, y, y)
quot(0', s(y), z) → 0'
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0', s(z)) → s(div(x, s(z)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
div(0', y) → 0'
div(x, y) → quot(x, y, y)
quot(0', s(y), z) → 0'
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0', s(z)) → s(div(x, s(z)))

Types:
div :: 0':s → 0':s → 0':s
0' :: 0':s
quot :: 0':s → 0':s → 0':s → 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
div, quot

They will be analysed ascendingly in the following order:
div = quot

(6) Obligation:

Innermost TRS:
Rules:
div(0', y) → 0'
div(x, y) → quot(x, y, y)
quot(0', s(y), z) → 0'
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0', s(z)) → s(div(x, s(z)))

Types:
div :: 0':s → 0':s → 0':s
0' :: 0':s
quot :: 0':s → 0':s → 0':s → 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
quot, div

They will be analysed ascendingly in the following order:
div = quot

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

Proved the following rewrite lemma:
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

Induction Base:
quot(gen_0':s2_0(0), gen_0':s2_0(+(1, 0)), gen_0':s2_0(c)) →_R^Ω(1)
0'

Induction Step:
quot(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(1, +(n4_0, 1))), gen_0':s2_0(c)) →_R^Ω(1)
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) →_IH
gen_0':s2_0(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:
div(0', y) → 0'
div(x, y) → quot(x, y, y)
quot(0', s(y), z) → 0'
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0', s(z)) → s(div(x, s(z)))

Types:
div :: 0':s → 0':s → 0':s
0' :: 0':s
quot :: 0':s → 0':s → 0':s → 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
div

They will be analysed ascendingly in the following order:
div = quot

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol div.

(11) Obligation:

Innermost TRS:
Rules:
div(0', y) → 0'
div(x, y) → quot(x, y, y)
quot(0', s(y), z) → 0'
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0', s(z)) → s(div(x, s(z)))

Types:
div :: 0':s → 0':s → 0':s
0' :: 0':s
quot :: 0':s → 0':s → 0':s → 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
div(0', y) → 0'
div(x, y) → quot(x, y, y)
quot(0', s(y), z) → 0'
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0', s(z)) → s(div(x, s(z)))

Types:
div :: 0':s → 0':s → 0':s
0' :: 0':s
quot :: 0':s → 0':s → 0':s → 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)