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

p(0) → 0
p(s(x)) → x
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → s(plus(p(s(x)), y))
plus(x, s(y)) → s(plus(x, p(s(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(zero(y), 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(zero(y), z))
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(y)) → false
eq(s(x), s(y)) → eq(x, y)
divides(y, x) → eq(x, times(div(x, y), y))
prime(s(s(x))) → pr(s(s(x)), s(x))
pr(x, s(0)) → true
pr(x, s(s(y))) → if(divides(s(s(y)), x), x, s(y))
if(true, x, y) → false
if(false, x, y) → pr(x, y)
zero(div(x, x)) → x
zero(divides(x, x)) → x
zero(times(x, x)) → x
zero(quot(x, x, x)) → x
zero(s(x)) → if(eq(x, s(0)), plus(zero(0), 0), s(plus(0, zero(0))))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Heuristically decided to analyse the following defined symbols:
plus', times', div', quot', zero', eq', divides', pr', if'

They will be analysed ascendingly in the following order:
plus' < times'
plus' < zero'
times' < div'
times' < divides'
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
eq' < zero'
zero' = divides'
zero' = pr'
zero' = if'
eq' < divides'
divides' = pr'
divides' = if'
pr' = if'

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
plus', times', div', quot', zero', eq', divides', pr', if'

They will be analysed ascendingly in the following order:
plus' < times'
plus' < zero'
times' < div'
times' < divides'
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
eq' < zero'
zero' = divides'
zero' = pr'
zero' = if'
eq' < divides'
divides' = pr'
divides' = if'
pr' = if'

Proved the following rewrite lemma:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)

Induction Base:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(0)) →RΩ(1)
_gen_0':s':true':false'2(a)

Induction Step:
plus'(_gen_0':s':true':false'2(_a243), _gen_0':s':true':false'2(+(_$n5, 1))) →RΩ(1)
s'(plus'(_gen_0':s':true':false'2(_a243), p'(s'(_gen_0':s':true':false'2(_$n5))))) →RΩ(1)
s'(plus'(_gen_0':s':true':false'2(_a243), _gen_0':s':true':false'2(_$n5))) →IH
s'(_gen_0':s':true':false'2(+(_$n5, _a243)))

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

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
times', div', quot', zero', eq', divides', pr', if'

They will be analysed ascendingly in the following order:
times' < div'
times' < divides'
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
eq' < zero'
zero' = divides'
zero' = pr'
zero' = if'
eq' < divides'
divides' = pr'
divides' = if'
pr' = if'

Proved the following rewrite lemma:
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)

Induction Base:
times'(_gen_0':s':true':false'2(0), _gen_0':s':true':false'2(b)) →RΩ(1)
0'

Induction Step:
times'(_gen_0':s':true':false'2(+(_$n2262, 1)), _gen_0':s':true':false'2(_b2584)) →RΩ(1)
plus'(_gen_0':s':true':false'2(_b2584), times'(_gen_0':s':true':false'2(_$n2262), _gen_0':s':true':false'2(_b2584))) →IH
plus'(_gen_0':s':true':false'2(_b2584), _gen_0':s':true':false'2(*(_$n2262, _b2584))) →LΩ(1 + $n2262·b2584)
_gen_0':s':true':false'2(+(*(_$n2262, _b2584), _b2584))

We have rt ∈ Ω(n3) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n3).

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
eq', div', quot', zero', divides', pr', if'

They will be analysed ascendingly in the following order:
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
eq' < zero'
zero' = divides'
zero' = pr'
zero' = if'
eq' < divides'
divides' = pr'
divides' = if'
pr' = if'

Proved the following rewrite lemma:
eq'(_gen_0':s':true':false'2(_n4822), _gen_0':s':true':false'2(_n4822)) → true', rt ∈ Ω(1 + n4822)

Induction Base:
eq'(_gen_0':s':true':false'2(0), _gen_0':s':true':false'2(0)) →RΩ(1)
true'

Induction Step:
eq'(_gen_0':s':true':false'2(+(_$n4823, 1)), _gen_0':s':true':false'2(+(_$n4823, 1))) →RΩ(1)
eq'(_gen_0':s':true':false'2(_$n4823), _gen_0':s':true':false'2(_$n4823)) →IH
true'

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

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)
eq'(_gen_0':s':true':false'2(_n4822), _gen_0':s':true':false'2(_n4822)) → true', rt ∈ Ω(1 + n4822)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
quot', div', zero', divides', pr', if'

They will be analysed ascendingly in the following order:
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
zero' = divides'
zero' = pr'
zero' = if'
divides' = pr'
divides' = if'
pr' = if'

Proved the following rewrite lemma:
quot'(_gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(1)) → _gen_0':s':true':false'2(1), rt ∈ Ω(1 + n6455)

Induction Base:
quot'(_gen_0':s':true':false'2(0), _gen_0':s':true':false'2(0), _gen_0':s':true':false'2(1)) →RΩ(1)
s'(div'(_gen_0':s':true':false'2(0), s'(_gen_0':s':true':false'2(0)))) →RΩ(1)
s'(0')

Induction Step:
quot'(_gen_0':s':true':false'2(+(_$n6456, 1)), _gen_0':s':true':false'2(+(_$n6456, 1)), _gen_0':s':true':false'2(1)) →RΩ(1)
quot'(_gen_0':s':true':false'2(_$n6456), _gen_0':s':true':false'2(_$n6456), _gen_0':s':true':false'2(1)) →IH
_gen_0':s':true':false'2(1)

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

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)
eq'(_gen_0':s':true':false'2(_n4822), _gen_0':s':true':false'2(_n4822)) → true', rt ∈ Ω(1 + n4822)
quot'(_gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(1)) → _gen_0':s':true':false'2(1), rt ∈ Ω(1 + n6455)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
div', zero', divides', pr', if'

They will be analysed ascendingly in the following order:
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
zero' = divides'
zero' = pr'
zero' = if'
divides' = pr'
divides' = if'
pr' = if'

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

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)
eq'(_gen_0':s':true':false'2(_n4822), _gen_0':s':true':false'2(_n4822)) → true', rt ∈ Ω(1 + n4822)
quot'(_gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(1)) → _gen_0':s':true':false'2(1), rt ∈ Ω(1 + n6455)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
zero', divides', pr', if'

They will be analysed ascendingly in the following order:
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
zero' = divides'
zero' = pr'
zero' = if'
divides' = pr'
divides' = if'
pr' = if'

Could not prove a rewrite lemma for the defined symbol zero'.

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)
eq'(_gen_0':s':true':false'2(_n4822), _gen_0':s':true':false'2(_n4822)) → true', rt ∈ Ω(1 + n4822)
quot'(_gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(1)) → _gen_0':s':true':false'2(1), rt ∈ Ω(1 + n6455)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
if', divides', pr'

They will be analysed ascendingly in the following order:
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
zero' = divides'
zero' = pr'
zero' = if'
divides' = pr'
divides' = if'
pr' = if'

Could not prove a rewrite lemma for the defined symbol if'.

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)
eq'(_gen_0':s':true':false'2(_n4822), _gen_0':s':true':false'2(_n4822)) → true', rt ∈ Ω(1 + n4822)
quot'(_gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(1)) → _gen_0':s':true':false'2(1), rt ∈ Ω(1 + n6455)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
pr', divides'

They will be analysed ascendingly in the following order:
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
zero' = divides'
zero' = pr'
zero' = if'
divides' = pr'
divides' = if'
pr' = if'

Could not prove a rewrite lemma for the defined symbol pr'.

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)
eq'(_gen_0':s':true':false'2(_n4822), _gen_0':s':true':false'2(_n4822)) → true', rt ∈ Ω(1 + n4822)
quot'(_gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(1)) → _gen_0':s':true':false'2(1), rt ∈ Ω(1 + n6455)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

The following defined symbols remain to be analysed:
divides'

They will be analysed ascendingly in the following order:
div' = quot'
div' = zero'
div' = divides'
div' = pr'
div' = if'
quot' = zero'
quot' = divides'
quot' = pr'
quot' = if'
zero' = divides'
zero' = pr'
zero' = if'
divides' = pr'
divides' = if'
pr' = if'

Could not prove a rewrite lemma for the defined symbol divides'.

Rules:
p'(0') → 0'
p'(s'(x)) → x
plus'(x, 0') → x
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
plus'(x, s'(y)) → s'(plus'(x, p'(s'(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'(zero'(y), 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'(zero'(y), z))
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(y)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
divides'(y, x) → eq'(x, times'(div'(x, y), y))
prime'(s'(s'(x))) → pr'(s'(s'(x)), s'(x))
pr'(x, s'(0')) → true'
pr'(x, s'(s'(y))) → if'(divides'(s'(s'(y)), x), x, s'(y))
if'(true', x, y) → false'
if'(false', x, y) → pr'(x, y)
zero'(div'(x, x)) → x
zero'(divides'(x, x)) → x
zero'(times'(x, x)) → x
zero'(quot'(x, x, x)) → x
zero'(s'(x)) → if'(eq'(x, s'(0')), plus'(zero'(0'), 0'), s'(plus'(0', zero'(0'))))

Types:
p' :: 0':s':true':false' → 0':s':true':false'
0' :: 0':s':true':false'
s' :: 0':s':true':false' → 0':s':true':false'
plus' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
times' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
div' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
quot' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
zero' :: 0':s':true':false' → 0':s':true':false'
eq' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
true' :: 0':s':true':false'
false' :: 0':s':true':false'
divides' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
prime' :: 0':s':true':false' → 0':s':true':false'
pr' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
if' :: 0':s':true':false' → 0':s':true':false' → 0':s':true':false' → 0':s':true':false'
_hole_0':s':true':false'1 :: 0':s':true':false'
_gen_0':s':true':false'2 :: Nat → 0':s':true':false'

Lemmas:
plus'(_gen_0':s':true':false'2(a), _gen_0':s':true':false'2(_n4)) → _gen_0':s':true':false'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)
eq'(_gen_0':s':true':false'2(_n4822), _gen_0':s':true':false'2(_n4822)) → true', rt ∈ Ω(1 + n4822)
quot'(_gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(_n6455), _gen_0':s':true':false'2(1)) → _gen_0':s':true':false'2(1), rt ∈ Ω(1 + n6455)

Generator Equations:
_gen_0':s':true':false'2(0) ⇔ 0'
_gen_0':s':true':false'2(+(x, 1)) ⇔ s'(_gen_0':s':true':false'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n3) was proven with the following lemma:
times'(_gen_0':s':true':false'2(_n2261), _gen_0':s':true':false'2(b)) → _gen_0':s':true':false'2(*(_n2261, b)), rt ∈ Ω(1 + b2584·n22612 + n2261)