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

cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(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:

cond'(true', x) → cond'(odd'(x), p'(x))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x

Rewrite Strategy: INNERMOST

Infered types.

Rules:
cond'(true', x) → cond'(odd'(x), p'(x))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x

Types:
cond' :: true':false' → 0':s' → cond'
true' :: true':false'
odd' :: 0':s' → true':false'
p' :: 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
cond', odd'

They will be analysed ascendingly in the following order:
odd' < cond'

Rules:
cond'(true', x) → cond'(odd'(x), p'(x))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x

Types:
cond' :: true':false' → 0':s' → cond'
true' :: true':false'
odd' :: 0':s' → true':false'
p' :: 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
odd', cond'

They will be analysed ascendingly in the following order:
odd' < cond'

Proved the following rewrite lemma:
odd'(_gen_0':s'4(*(2, _n6))) → false', rt ∈ Ω(1 + n6)

Induction Base:
odd'(_gen_0':s'4(*(2, 0))) →RΩ(1)
false'

Induction Step:
odd'(_gen_0':s'4(*(2, +(_\$n7, 1)))) →RΩ(1)
odd'(_gen_0':s'4(*(2, _\$n7))) →IH
false'

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

Rules:
cond'(true', x) → cond'(odd'(x), p'(x))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x

Types:
cond' :: true':false' → 0':s' → cond'
true' :: true':false'
odd' :: 0':s' → true':false'
p' :: 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
odd'(_gen_0':s'4(*(2, _n6))) → false', rt ∈ Ω(1 + n6)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
cond'

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

Rules:
cond'(true', x) → cond'(odd'(x), p'(x))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x

Types:
cond' :: true':false' → 0':s' → cond'
true' :: true':false'
odd' :: 0':s' → true':false'
p' :: 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
odd'(_gen_0':s'4(*(2, _n6))) → false', rt ∈ Ω(1 + n6)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
odd'(_gen_0':s'4(*(2, _n6))) → false', rt ∈ Ω(1 + n6)