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

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

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


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

Rewrite Strategy: INNERMOST

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

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


h'(f'(x)) → f'(g'(x))
g'(x) → h'(x)

Rewrite Strategy: INNERMOST

Infered types.

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

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

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

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

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

Types:
h' :: f' → f'
f' :: f' → f'
g' :: 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:
g', h'

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

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

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

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

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

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

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

Lemmas:
g'(_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:
h'

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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