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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(0, y) → 0
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0
if_minus(false, s(x), y) → s(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(0', y) → 0'
minus'(s'(x), y) → if_minus'(le'(s'(x), y), s'(x), y)
if_minus'(true', s'(x), y) → 0'
if_minus'(false', s'(x), y) → s'(minus'(x, y))
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(quot'(x, s'(s'(0'))))))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(0', y) → 0'
minus'(s'(x), y) → if_minus'(le'(s'(x), y), s'(x), y)
if_minus'(true', s'(x), y) → 0'
if_minus'(false', s'(x), y) → s'(minus'(x, y))
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(quot'(x, s'(s'(0'))))))

Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
minus' :: 0':s' → 0':s' → 0':s'
if_minus' :: true':false' → 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
log' :: 0':s' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
le', minus', quot', log'

They will be analysed ascendingly in the following order:
le' < minus'
minus' < quot'
quot' < log'

Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(0', y) → 0'
minus'(s'(x), y) → if_minus'(le'(s'(x), y), s'(x), y)
if_minus'(true', s'(x), y) → 0'
if_minus'(false', s'(x), y) → s'(minus'(x, y))
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(quot'(x, s'(s'(0'))))))

Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
minus' :: 0':s' → 0':s' → 0':s'
if_minus' :: true':false' → 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
log' :: 0':s' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_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:
le', minus', quot', log'

They will be analysed ascendingly in the following order:
le' < minus'
minus' < quot'
quot' < log'

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

Induction Base:
le'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
true'

Induction Step:
le'(_gen_0':s'3(+(_\$n6, 1)), _gen_0':s'3(+(_\$n6, 1))) →RΩ(1)
le'(_gen_0':s'3(_\$n6), _gen_0':s'3(_\$n6)) →IH
true'

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

Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(0', y) → 0'
minus'(s'(x), y) → if_minus'(le'(s'(x), y), s'(x), y)
if_minus'(true', s'(x), y) → 0'
if_minus'(false', s'(x), y) → s'(minus'(x, y))
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(quot'(x, s'(s'(0'))))))

Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
minus' :: 0':s' → 0':s' → 0':s'
if_minus' :: true':false' → 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
log' :: 0':s' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → true', 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:
minus', quot', log'

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

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

Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(0', y) → 0'
minus'(s'(x), y) → if_minus'(le'(s'(x), y), s'(x), y)
if_minus'(true', s'(x), y) → 0'
if_minus'(false', s'(x), y) → s'(minus'(x, y))
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(quot'(x, s'(s'(0'))))))

Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
minus' :: 0':s' → 0':s' → 0':s'
if_minus' :: true':false' → 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
log' :: 0':s' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → true', 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:
quot', log'

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

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

Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(0', y) → 0'
minus'(s'(x), y) → if_minus'(le'(s'(x), y), s'(x), y)
if_minus'(true', s'(x), y) → 0'
if_minus'(false', s'(x), y) → s'(minus'(x, y))
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(quot'(x, s'(s'(0'))))))

Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
minus' :: 0':s' → 0':s' → 0':s'
if_minus' :: true':false' → 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
log' :: 0':s' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → true', 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:
log'

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

Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
minus'(0', y) → 0'
minus'(s'(x), y) → if_minus'(le'(s'(x), y), s'(x), y)
if_minus'(true', s'(x), y) → 0'
if_minus'(false', s'(x), y) → s'(minus'(x, y))
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(s'(0')) → 0'
log'(s'(s'(x))) → s'(log'(s'(quot'(x, s'(s'(0'))))))

Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
minus' :: 0':s' → 0':s' → 0':s'
if_minus' :: true':false' → 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
log' :: 0':s' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → true', rt ∈ Ω(1 + n5)

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:
le'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → true', rt ∈ Ω(1 + n5)