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

sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


sum'(0') → 0'
sum'(s'(x)) → +'(sum'(x), s'(x))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
sum'(0') → 0'
sum'(s'(x)) → +'(sum'(x), s'(x))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))

Types:
sum' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
+' :: 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:
sum', +'

They will be analysed ascendingly in the following order:
+' < sum'

Rules:
sum'(0') → 0'
sum'(s'(x)) → +'(sum'(x), s'(x))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))

Types:
sum' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
+' :: 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:
+', sum'

They will be analysed ascendingly in the following order:
+' < sum'

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

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

Induction Step:
+'(_gen_0':s'2(_a137), _gen_0':s'2(+(_$n5, 1))) →RΩ(1)
s'(+'(_gen_0':s'2(_a137), _gen_0':s'2(_$n5))) →IH
s'(_gen_0':s'2(+(_$n5, _a137)))

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

Rules:
sum'(0') → 0'
sum'(s'(x)) → +'(sum'(x), s'(x))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))

Types:
sum' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
+' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
+'(_gen_0':s'2(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), 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:
sum'

Proved the following rewrite lemma:
sum'(_gen_0':s'2(+(1, _n335))) → _*3, rt ∈ Ω(n335)

Induction Base:
sum'(_gen_0':s'2(+(1, 0)))

Induction Step:
sum'(_gen_0':s'2(+(1, +(_$n336, 1)))) →RΩ(1)
+'(sum'(_gen_0':s'2(+(1, _$n336))), s'(_gen_0':s'2(+(1, _$n336)))) →IH
+'(_*3, s'(_gen_0':s'2(+(1, _$n336))))

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

Rules:
sum'(0') → 0'
sum'(s'(x)) → +'(sum'(x), s'(x))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))

Types:
sum' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
+' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
+'(_gen_0':s'2(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)
sum'(_gen_0':s'2(+(1, _n335))) → _*3, rt ∈ Ω(n335)

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