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

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

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
+' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
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:
+'

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

Types:
+' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
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:
+'

Proved the following rewrite lemma:
+'(_gen_0':s'2(1), _gen_0':s'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)

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

Induction Step:
+'(_gen_0':s'2(1), _gen_0':s'2(+(1, +(_$n5, 1)))) →RΩ(1)
s'(+'(s'(_gen_0':s'2(0)), +'(_gen_0':s'2(+(1, _$n5)), 0'))) →RΩ(1)
s'(+'(s'(_gen_0':s'2(0)), s'(_gen_0':s'2(_$n5)))) →IH
s'(_*3)

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

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

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

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