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

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

g'(f'(x), y) → f'(h'(x, y))
h'(x, y) → g'(x, f'(y))

Rewrite Strategy: INNERMOST

Sliced the following arguments:
g'/1
h'/1

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

g'(f'(x)) → f'(h'(x))
h'(x) → g'(x)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
g'(f'(x)) → f'(h'(x))
h'(x) → g'(x)

Types:
g' :: f' → f'
f' :: f' → f'
h' :: f' → f'
_hole_f'1 :: f'
_gen_f'2 :: Nat → f'

Heuristically decided to analyse the following defined symbols:
g', h'

They will be analysed ascendingly in the following order:
g' = h'

Rules:
g'(f'(x)) → f'(h'(x))
h'(x) → g'(x)

Types:
g' :: f' → f'
f' :: f' → f'
h' :: f' → f'
_hole_f'1 :: f'
_gen_f'2 :: Nat → f'

Generator Equations:
_gen_f'2(0) ⇔ _hole_f'1
_gen_f'2(+(x, 1)) ⇔ f'(_gen_f'2(x))

The following defined symbols remain to be analysed:
h', g'

They will be analysed ascendingly in the following order:
g' = h'

Proved the following rewrite lemma:
h'(_gen_f'2(_n4)) → _*3, rt ∈ Ω(n4)

Induction Base:
h'(_gen_f'2(0))

Induction Step:
h'(_gen_f'2(+(_\$n5, 1))) →RΩ(1)
g'(_gen_f'2(+(_\$n5, 1))) →RΩ(1)
f'(h'(_gen_f'2(_\$n5))) →IH
f'(_*3)

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

Rules:
g'(f'(x)) → f'(h'(x))
h'(x) → g'(x)

Types:
g' :: f' → f'
f' :: f' → f'
h' :: f' → f'
_hole_f'1 :: f'
_gen_f'2 :: Nat → f'

Lemmas:
h'(_gen_f'2(_n4)) → _*3, rt ∈ Ω(n4)

Generator Equations:
_gen_f'2(0) ⇔ _hole_f'1
_gen_f'2(+(x, 1)) ⇔ f'(_gen_f'2(x))

The following defined symbols remain to be analysed:
g'

They will be analysed ascendingly in the following order:
g' = h'

Proved the following rewrite lemma:
g'(_gen_f'2(+(1, _n212))) → _*3, rt ∈ Ω(n212)

Induction Base:
g'(_gen_f'2(+(1, 0)))

Induction Step:
g'(_gen_f'2(+(1, +(_\$n213, 1)))) →RΩ(1)
f'(h'(_gen_f'2(+(1, _\$n213)))) →RΩ(1)
f'(g'(_gen_f'2(+(1, _\$n213)))) →IH
f'(_*3)

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

Rules:
g'(f'(x)) → f'(h'(x))
h'(x) → g'(x)

Types:
g' :: f' → f'
f' :: f' → f'
h' :: f' → f'
_hole_f'1 :: f'
_gen_f'2 :: Nat → f'

Lemmas:
h'(_gen_f'2(_n4)) → _*3, rt ∈ Ω(n4)
g'(_gen_f'2(+(1, _n212))) → _*3, rt ∈ Ω(n212)

Generator Equations:
_gen_f'2(0) ⇔ _hole_f'1
_gen_f'2(+(x, 1)) ⇔ f'(_gen_f'2(x))

The following defined symbols remain to be analysed:
h'

They will be analysed ascendingly in the following order:
g' = h'

Proved the following rewrite lemma:
h'(_gen_f'2(_n604)) → _*3, rt ∈ Ω(n604)

Induction Base:
h'(_gen_f'2(0))

Induction Step:
h'(_gen_f'2(+(_\$n605, 1))) →RΩ(1)
g'(_gen_f'2(+(_\$n605, 1))) →RΩ(1)
f'(h'(_gen_f'2(_\$n605))) →IH
f'(_*3)

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

Rules:
g'(f'(x)) → f'(h'(x))
h'(x) → g'(x)

Types:
g' :: f' → f'
f' :: f' → f'
h' :: f' → f'
_hole_f'1 :: f'
_gen_f'2 :: Nat → f'

Lemmas:
h'(_gen_f'2(_n604)) → _*3, rt ∈ Ω(n604)
g'(_gen_f'2(+(1, _n212))) → _*3, rt ∈ Ω(n212)

Generator Equations:
_gen_f'2(0) ⇔ _hole_f'1
_gen_f'2(+(x, 1)) ⇔ f'(_gen_f'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
h'(_gen_f'2(_n604)) → _*3, rt ∈ Ω(n604)