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

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

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
double' :: 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:
double', +'

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

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

Types:
double' :: 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:
+', double'

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

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(_a161), _gen_0':s'2(+(_\$n5, 1))) →RΩ(1)
s'(+'(_gen_0':s'2(_a161), _gen_0':s'2(_\$n5))) →IH
s'(_gen_0':s'2(+(_\$n5, _a161)))

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

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

Types:
double' :: 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:
double'

Proved the following rewrite lemma:
double'(_gen_0':s'2(_n466)) → _gen_0':s'2(*(2, _n466)), rt ∈ Ω(1 + n466)

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

Induction Step:
double'(_gen_0':s'2(+(_\$n467, 1))) →RΩ(1)
s'(s'(double'(_gen_0':s'2(_\$n467)))) →IH
s'(s'(_gen_0':s'2(*(2, _\$n467))))

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

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

Types:
double' :: 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)
double'(_gen_0':s'2(_n466)) → _gen_0':s'2(*(2, _n466)), rt ∈ Ω(1 + n466)

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)