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

a(b(x)) → b(b(a(a(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'(b'(a'(a'(x))))

Rewrite Strategy: INNERMOST


Infered types.


Rules:
a'(b'(x)) → b'(b'(a'(a'(x))))

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


Heuristically decided to analyse the following defined symbols:
a'


Rules:
a'(b'(x)) → b'(b'(a'(a'(x))))

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

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

The following defined symbols remain to be analysed:
a'


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

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

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

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


Rules:
a'(b'(x)) → b'(b'(a'(a'(x))))

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

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

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

No more defined symbols left to analyse.


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