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

a(b(x)) → b(a(x))
a(c(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:

a'(b'(x)) → b'(a'(x))
a'(c'(x)) → x

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a'(b'(x)) → b'(a'(x))
a'(c'(x)) → x

Types:
a' :: b':c' → b':c'
b' :: b':c' → b':c'
c' :: b':c' → b':c'
_hole_b':c'1 :: b':c'
_gen_b':c'2 :: Nat → b':c'

Heuristically decided to analyse the following defined symbols:
a'

Rules:
a'(b'(x)) → b'(a'(x))
a'(c'(x)) → x

Types:
a' :: b':c' → b':c'
b' :: b':c' → b':c'
c' :: b':c' → b':c'
_hole_b':c'1 :: b':c'
_gen_b':c'2 :: Nat → b':c'

Generator Equations:
_gen_b':c'2(0) ⇔ _hole_b':c'1
_gen_b':c'2(+(x, 1)) ⇔ b'(_gen_b':c'2(x))

The following defined symbols remain to be analysed:
a'

Proved the following rewrite lemma:
a'(_gen_b':c'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)

Induction Base:
a'(_gen_b':c'2(+(1, 0)))

Induction Step:
a'(_gen_b':c'2(+(1, +(_\$n5, 1)))) →RΩ(1)
b'(a'(_gen_b':c'2(+(1, _\$n5)))) →IH
b'(_*3)

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

Rules:
a'(b'(x)) → b'(a'(x))
a'(c'(x)) → x

Types:
a' :: b':c' → b':c'
b' :: b':c' → b':c'
c' :: b':c' → b':c'
_hole_b':c'1 :: b':c'
_gen_b':c'2 :: Nat → b':c'

Lemmas:
a'(_gen_b':c'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)

Generator Equations:
_gen_b':c'2(0) ⇔ _hole_b':c'1
_gen_b':c'2(+(x, 1)) ⇔ b'(_gen_b':c'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
a'(_gen_b':c'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)