The Runtime Complexity (full) 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:

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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
quot(s(x), s(y), z) →⁺ quot(x, y, z)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), y / s(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:

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

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

(7) OrderProof (LOWER BOUND(ID) transformation)

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

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

(8) Obligation:

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

Types:
quot :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':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:
plus, quot

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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

Induction Base:
plus(gen_0':s2_0(0), gen_0':s2_0(b)) →_R^Ω(1)
gen_0':s2_0(b)

Induction Step:
plus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(b)) →_R^Ω(1)
s(plus(gen_0':s2_0(n4_0), gen_0':s2_0(b))) →_IH
s(gen_0':s2_0(+(b, c5_0)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

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

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

Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), 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:
quot

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
quot(gen_0':s2_0(n421_0), gen_0':s2_0(+(1, n421_0)), gen_0':s2_0(1)) → gen_0':s2_0(0), rt ∈ Ω(1 + n421₀)

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

Induction Step:
quot(gen_0':s2_0(+(n421_0, 1)), gen_0':s2_0(+(1, +(n421_0, 1))), gen_0':s2_0(1)) →_R^Ω(1)
quot(gen_0':s2_0(n421_0), gen_0':s2_0(+(1, n421_0)), gen_0':s2_0(1)) →_IH
gen_0':s2_0(0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

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

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

Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
quot(gen_0':s2_0(n421_0), gen_0':s2_0(+(1, n421_0)), gen_0':s2_0(1)) → gen_0':s2_0(0), rt ∈ Ω(1 + n421₀)

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:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

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

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

Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
quot(gen_0':s2_0(n421_0), gen_0':s2_0(+(1, n421_0)), gen_0':s2_0(1)) → gen_0':s2_0(0), rt ∈ Ω(1 + n421₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

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

(19) BOUNDS(n^1, INF)

(20) Obligation:

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

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

Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), 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.

(21) LowerBoundsProof (EQUIVALENT transformation)

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

(22) BOUNDS(n^1, INF)