### (0) Obligation:

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

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

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
plus(s(x), y) →+ s(plus(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x)].
The result substitution is [ ].

### (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:

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

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

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

### (8) Obligation:

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

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

They will be analysed ascendingly in the following order:
plus < times
times < div
div = 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)) →RΩ(1)
gen_0':s2_0(b)

Induction Step:
plus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(b)) →RΩ(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 ircR ∈ Ω(n).

### (11) Obligation:

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

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

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

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

Proved the following rewrite lemma:
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)

Induction Base:
times(gen_0':s2_0(0), gen_0':s2_0(b)) →RΩ(1)
0'

Induction Step:
times(gen_0':s2_0(+(n577_0, 1)), gen_0':s2_0(b)) →RΩ(1)
plus(gen_0':s2_0(b), times(gen_0':s2_0(n577_0), gen_0':s2_0(b))) →IH
plus(gen_0':s2_0(b), gen_0':s2_0(*(c578_0, b))) →LΩ(1 + b)
gen_0':s2_0(+(b, *(n577_0, b)))

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (14) Obligation:

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

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 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)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)

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

### (15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n13700)

Induction Base:
quot(gen_0':s2_0(0), gen_0':s2_0(+(1, 0)), gen_0':s2_0(c)) →RΩ(1)
0'

Induction Step:
quot(gen_0':s2_0(+(n1370_0, 1)), gen_0':s2_0(+(1, +(n1370_0, 1))), gen_0':s2_0(c)) →RΩ(1)
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) →IH
gen_0':s2_0(0)

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

### (17) Obligation:

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

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 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)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n13700)

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

### (18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (19) Obligation:

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

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 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)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n13700)

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.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)

### (22) Obligation:

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

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 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)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) → gen_0':s2_0(0), rt ∈ Ω(1 + n13700)

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.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)

### (25) Obligation:

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

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 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)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)

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.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n577_0, b)), rt ∈ Ω(1 + b·n5770 + n5770)

### (28) Obligation:

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

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 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.

### (29) 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)