### (0) Obligation:

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

div(x, s(y)) → d(x, s(y), 0)
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0
ge(u, 0) → true
ge(0, s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
d, ge, plus

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

### (8) Obligation:

TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
ge, d, plus

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

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

Proved the following rewrite lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Induction Base:
ge(gen_s:0'3_0(0), gen_s:0'3_0(0)) →RΩ(1)
true

Induction Step:
ge(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) →RΩ(1)
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

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

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

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

Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n270_0)) → gen_s:0'3_0(+(n270_0, a)), rt ∈ Ω(1 + n2700)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) →RΩ(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n270_0, 1))) →RΩ(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n270_0))) →IH
s(gen_s:0'3_0(+(a, c271_0)))

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

### (14) Obligation:

TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n270_0)) → gen_s:0'3_0(+(n270_0, a)), rt ∈ Ω(1 + n2700)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
d

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

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

### (16) Obligation:

TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n270_0)) → gen_s:0'3_0(+(n270_0, a)), rt ∈ Ω(1 + n2700)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (19) Obligation:

TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n270_0)) → gen_s:0'3_0(+(n270_0, a)), rt ∈ Ω(1 + n2700)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (22) Obligation:

TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)