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

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))

Rewrite Strategy: INNERMOST


Renamed function symbols to avoid clashes with predefined symbol.


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


minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))

Rewrite Strategy: INNERMOST


Infered types.


Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'


Heuristically decided to analyse the following defined symbols:
minus', quot'

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


Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

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

The following defined symbols remain to be analysed:
minus', quot'

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


Proved the following rewrite lemma:
minus'(_gen_0':s'2(_n4), _gen_0':s'2(_n4)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)

Induction Base:
minus'(_gen_0':s'2(0), _gen_0':s'2(0)) →RΩ(1)
_gen_0':s'2(0)

Induction Step:
minus'(_gen_0':s'2(+(_$n5, 1)), _gen_0':s'2(+(_$n5, 1))) →RΩ(1)
minus'(_gen_0':s'2(_$n5), _gen_0':s'2(_$n5)) →IH
_gen_0':s'2(0)

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


Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'2(_n4), _gen_0':s'2(_n4)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)

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

The following defined symbols remain to be analysed:
quot'


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


Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'2(_n4), _gen_0':s'2(_n4)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)

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

No more defined symbols left to analyse.


The lowerbound Ω(n) was proven with the following lemma:
minus'(_gen_0':s'2(_n4), _gen_0':s'2(_n4)) → _gen_0':s'2(0), rt ∈ Ω(1 + n4)