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

half(0) → 0
half(s(s(x))) → s(half(x))
log(s(0)) → 0
log(s(s(x))) → s(log(s(half(x))))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

half'(0') → 0'
half'(s'(s'(x))) → s'(half'(x))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(half'(x))))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
half'(0') → 0'
half'(s'(s'(x))) → s'(half'(x))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(half'(x))))

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
log' :: 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:
half', log'

They will be analysed ascendingly in the following order:
half' < log'

Rules:
half'(0') → 0'
half'(s'(s'(x))) → s'(half'(x))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(half'(x))))

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
log' :: 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:
half', log'

They will be analysed ascendingly in the following order:
half' < log'

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

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

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

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

Rules:
half'(0') → 0'
half'(s'(s'(x))) → s'(half'(x))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(half'(x))))

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
log' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
half'(_gen_0':s'2(*(2, _n4))) → _gen_0':s'2(_n4), 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:
log'

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

Rules:
half'(0') → 0'
half'(s'(s'(x))) → s'(half'(x))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(half'(x))))

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
log' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

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