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

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


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

Rewrite Strategy: INNERMOST

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

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


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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
f' :: h' → h'
g' :: h' → h'
h' :: h' → h'
_hole_h'1 :: h'
_gen_h'2 :: Nat → h'

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

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

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

Types:
f' :: h' → h'
g' :: h' → h'
h' :: h' → h'
_hole_h'1 :: h'
_gen_h'2 :: Nat → h'

Generator Equations:
_gen_h'2(0) ⇔ _hole_h'1
_gen_h'2(+(x, 1)) ⇔ h'(_gen_h'2(x))

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

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

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

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

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

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

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

Types:
f' :: h' → h'
g' :: h' → h'
h' :: h' → h'
_hole_h'1 :: h'
_gen_h'2 :: Nat → h'

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

Generator Equations:
_gen_h'2(0) ⇔ _hole_h'1
_gen_h'2(+(x, 1)) ⇔ h'(_gen_h'2(x))

The following defined symbols remain to be analysed:
f'

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

Proved the following rewrite lemma:
f'(_gen_h'2(_n415)) → _*3, rt ∈ Ω(n415)

Induction Base:
f'(_gen_h'2(0))

Induction Step:
f'(_gen_h'2(+(_$n416, 1))) →RΩ(1)
g'(_gen_h'2(+(_$n416, 1))) →RΩ(1)
h'(f'(_gen_h'2(_$n416))) →IH
h'(_*3)

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

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

Types:
f' :: h' → h'
g' :: h' → h'
h' :: h' → h'
_hole_h'1 :: h'
_gen_h'2 :: Nat → h'

Lemmas:
g'(_gen_h'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
f'(_gen_h'2(_n415)) → _*3, rt ∈ Ω(n415)

Generator Equations:
_gen_h'2(0) ⇔ _hole_h'1
_gen_h'2(+(x, 1)) ⇔ h'(_gen_h'2(x))

The following defined symbols remain to be analysed:
g'

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

Proved the following rewrite lemma:
g'(_gen_h'2(+(1, _n833))) → _*3, rt ∈ Ω(n833)

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

Induction Step:
g'(_gen_h'2(+(1, +(_$n834, 1)))) →RΩ(1)
h'(g'(_gen_h'2(+(1, _$n834)))) →IH
h'(_*3)

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

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

Types:
f' :: h' → h'
g' :: h' → h'
h' :: h' → h'
_hole_h'1 :: h'
_gen_h'2 :: Nat → h'

Lemmas:
g'(_gen_h'2(+(1, _n833))) → _*3, rt ∈ Ω(n833)
f'(_gen_h'2(_n415)) → _*3, rt ∈ Ω(n415)

Generator Equations:
_gen_h'2(0) ⇔ _hole_h'1
_gen_h'2(+(x, 1)) ⇔ h'(_gen_h'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
g'(_gen_h'2(+(1, _n833))) → _*3, rt ∈ Ω(n833)