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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
inc(0) → 0
inc(s(x)) → s(inc(x))
zero(0) → true
zero(s(x)) → false
p(0) → 0
p(s(x)) → x
bits(x) → bitIter(x, 0)
bitIter(x, y) → if(zero(x), x, inc(y))
if(true, x, y) → p(y)
if(false, x, y) → bitIter(half(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:

half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
zero'(0') → true'
zero'(s'(x)) → false'
p'(0') → 0'
p'(s'(x)) → x
bits'(x) → bitIter'(x, 0')
bitIter'(x, y) → if'(zero'(x), x, inc'(y))
if'(true', x, y) → p'(y)
if'(false', x, y) → bitIter'(half'(x), y)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
zero'(0') → true'
zero'(s'(x)) → false'
p'(0') → 0'
p'(s'(x)) → x
bits'(x) → bitIter'(x, 0')
bitIter'(x, y) → if'(zero'(x), x, inc'(y))
if'(true', x, y) → p'(y)
if'(false', x, y) → bitIter'(half'(x), y)

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
inc' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
p' :: 0':s' → 0':s'
bits' :: 0':s' → 0':s'
bitIter' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
half', inc', bitIter'

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

Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
zero'(0') → true'
zero'(s'(x)) → false'
p'(0') → 0'
p'(s'(x)) → x
bits'(x) → bitIter'(x, 0')
bitIter'(x, y) → if'(zero'(x), x, inc'(y))
if'(true', x, y) → p'(y)
if'(false', x, y) → bitIter'(half'(x), y)

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
inc' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
p' :: 0':s' → 0':s'
bits' :: 0':s' → 0':s'
bitIter' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
half', inc', bitIter'

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

Proved the following rewrite lemma:
half'(_gen_0':s'3(*(2, _n5))) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)

Induction Base:
half'(_gen_0':s'3(*(2, 0))) →RΩ(1)
0'

Induction Step:
half'(_gen_0':s'3(*(2, +(_\$n6, 1)))) →RΩ(1)
s'(half'(_gen_0':s'3(*(2, _\$n6)))) →IH
s'(_gen_0':s'3(_\$n6))

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

Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
zero'(0') → true'
zero'(s'(x)) → false'
p'(0') → 0'
p'(s'(x)) → x
bits'(x) → bitIter'(x, 0')
bitIter'(x, y) → if'(zero'(x), x, inc'(y))
if'(true', x, y) → p'(y)
if'(false', x, y) → bitIter'(half'(x), y)

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
inc' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
p' :: 0':s' → 0':s'
bits' :: 0':s' → 0':s'
bitIter' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
half'(_gen_0':s'3(*(2, _n5))) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
inc', bitIter'

They will be analysed ascendingly in the following order:
inc' < bitIter'

Proved the following rewrite lemma:
inc'(_gen_0':s'3(_n509)) → _gen_0':s'3(_n509), rt ∈ Ω(1 + n509)

Induction Base:
inc'(_gen_0':s'3(0)) →RΩ(1)
0'

Induction Step:
inc'(_gen_0':s'3(+(_\$n510, 1))) →RΩ(1)
s'(inc'(_gen_0':s'3(_\$n510))) →IH
s'(_gen_0':s'3(_\$n510))

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

Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
zero'(0') → true'
zero'(s'(x)) → false'
p'(0') → 0'
p'(s'(x)) → x
bits'(x) → bitIter'(x, 0')
bitIter'(x, y) → if'(zero'(x), x, inc'(y))
if'(true', x, y) → p'(y)
if'(false', x, y) → bitIter'(half'(x), y)

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
inc' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
p' :: 0':s' → 0':s'
bits' :: 0':s' → 0':s'
bitIter' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
half'(_gen_0':s'3(*(2, _n5))) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)
inc'(_gen_0':s'3(_n509)) → _gen_0':s'3(_n509), rt ∈ Ω(1 + n509)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
bitIter'

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

Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
zero'(0') → true'
zero'(s'(x)) → false'
p'(0') → 0'
p'(s'(x)) → x
bits'(x) → bitIter'(x, 0')
bitIter'(x, y) → if'(zero'(x), x, inc'(y))
if'(true', x, y) → p'(y)
if'(false', x, y) → bitIter'(half'(x), y)

Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
inc' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
p' :: 0':s' → 0':s'
bits' :: 0':s' → 0':s'
bitIter' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
half'(_gen_0':s'3(*(2, _n5))) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)
inc'(_gen_0':s'3(_n509)) → _gen_0':s'3(_n509), rt ∈ Ω(1 + n509)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
half'(_gen_0':s'3(*(2, _n5))) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)