### (0) Obligation:

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

ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(0, y) → 0
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0)
div(x, y, z) → if(ge(y, s(0)), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

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(x), s(y)) →+ ge(x, y)
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 [ ].

### (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:

ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0') → x
minus(0', y) → 0'
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0')
div(x, y, z) → if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0') → x
minus(0', y) → 0'
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0')
div(x, y, z) → if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero → 0':s:div_by_zero → true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero → 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
id_inc :: 0':s:div_by_zero → 0':s:div_by_zero
quot :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
if :: true:false → true:false → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat → 0':s:div_by_zero

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

Heuristically decided to analyse the following defined symbols:
ge, minus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div

### (8) Obligation:

TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0') → x
minus(0', y) → 0'
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0')
div(x, y, z) → if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero → 0':s:div_by_zero → true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero → 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
id_inc :: 0':s:div_by_zero → 0':s:div_by_zero
quot :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
if :: true:false → true:false → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat → 0':s:div_by_zero

Generator Equations:
gen_0':s:div_by_zero3_0(0) ⇔ 0'
gen_0':s:div_by_zero3_0(+(x, 1)) ⇔ s(gen_0':s:div_by_zero3_0(x))

The following defined symbols remain to be analysed:
ge, minus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div

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

Proved the following rewrite lemma:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Induction Base:
ge(gen_0':s:div_by_zero3_0(0), gen_0':s:div_by_zero3_0(0)) →RΩ(1)
true

Induction Step:
ge(gen_0':s:div_by_zero3_0(+(n5_0, 1)), gen_0':s:div_by_zero3_0(+(n5_0, 1))) →RΩ(1)
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0') → x
minus(0', y) → 0'
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0')
div(x, y, z) → if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero → 0':s:div_by_zero → true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero → 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
id_inc :: 0':s:div_by_zero → 0':s:div_by_zero
quot :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
if :: true:false → true:false → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat → 0':s:div_by_zero

Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s:div_by_zero3_0(0) ⇔ 0'
gen_0':s:div_by_zero3_0(+(x, 1)) ⇔ s(gen_0':s:div_by_zero3_0(x))

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

They will be analysed ascendingly in the following order:
minus < div

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

Proved the following rewrite lemma:
minus(gen_0':s:div_by_zero3_0(n294_0), gen_0':s:div_by_zero3_0(n294_0)) → gen_0':s:div_by_zero3_0(0), rt ∈ Ω(1 + n2940)

Induction Base:
minus(gen_0':s:div_by_zero3_0(0), gen_0':s:div_by_zero3_0(0)) →RΩ(1)
gen_0':s:div_by_zero3_0(0)

Induction Step:
minus(gen_0':s:div_by_zero3_0(+(n294_0, 1)), gen_0':s:div_by_zero3_0(+(n294_0, 1))) →RΩ(1)
minus(gen_0':s:div_by_zero3_0(n294_0), gen_0':s:div_by_zero3_0(n294_0)) →IH
gen_0':s:div_by_zero3_0(0)

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

### (14) Obligation:

TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0') → x
minus(0', y) → 0'
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0')
div(x, y, z) → if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero → 0':s:div_by_zero → true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero → 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
id_inc :: 0':s:div_by_zero → 0':s:div_by_zero
quot :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
if :: true:false → true:false → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat → 0':s:div_by_zero

Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
minus(gen_0':s:div_by_zero3_0(n294_0), gen_0':s:div_by_zero3_0(n294_0)) → gen_0':s:div_by_zero3_0(0), rt ∈ Ω(1 + n2940)

Generator Equations:
gen_0':s:div_by_zero3_0(0) ⇔ 0'
gen_0':s:div_by_zero3_0(+(x, 1)) ⇔ s(gen_0':s:div_by_zero3_0(x))

The following defined symbols remain to be analysed:
div

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

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

### (16) Obligation:

TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0') → x
minus(0', y) → 0'
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0')
div(x, y, z) → if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero → 0':s:div_by_zero → true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero → 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
id_inc :: 0':s:div_by_zero → 0':s:div_by_zero
quot :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
if :: true:false → true:false → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat → 0':s:div_by_zero

Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
minus(gen_0':s:div_by_zero3_0(n294_0), gen_0':s:div_by_zero3_0(n294_0)) → gen_0':s:div_by_zero3_0(0), rt ∈ Ω(1 + n2940)

Generator Equations:
gen_0':s:div_by_zero3_0(0) ⇔ 0'
gen_0':s:div_by_zero3_0(+(x, 1)) ⇔ s(gen_0':s:div_by_zero3_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_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (19) Obligation:

TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0') → x
minus(0', y) → 0'
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0')
div(x, y, z) → if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero → 0':s:div_by_zero → true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero → 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
id_inc :: 0':s:div_by_zero → 0':s:div_by_zero
quot :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
if :: true:false → true:false → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat → 0':s:div_by_zero

Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
minus(gen_0':s:div_by_zero3_0(n294_0), gen_0':s:div_by_zero3_0(n294_0)) → gen_0':s:div_by_zero3_0(0), rt ∈ Ω(1 + n2940)

Generator Equations:
gen_0':s:div_by_zero3_0(0) ⇔ 0'
gen_0':s:div_by_zero3_0(+(x, 1)) ⇔ s(gen_0':s:div_by_zero3_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_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (22) Obligation:

TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0') → x
minus(0', y) → 0'
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0')
div(x, y, z) → if(ge(y, s(0')), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

Types:
ge :: 0':s:div_by_zero → 0':s:div_by_zero → true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero → 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
id_inc :: 0':s:div_by_zero → 0':s:div_by_zero
quot :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div :: 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
if :: true:false → true:false → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero → 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat → 0':s:div_by_zero

Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s:div_by_zero3_0(0) ⇔ 0'
gen_0':s:div_by_zero3_0(+(x, 1)) ⇔ s(gen_0':s:div_by_zero3_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_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) → true, rt ∈ Ω(1 + n50)