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

f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


f'(g'(X)) → g'(f'(f'(X)))
f'(h'(X)) → h'(g'(X))

Rewrite Strategy: INNERMOST

Sliced the following arguments:
h'/0

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


f'(g'(X)) → g'(f'(f'(X)))
f'(h') → h'

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
f' :: g':h' → g':h'
g' :: g':h' → g':h'
h' :: g':h'
_hole_g':h'1 :: g':h'
_gen_g':h'2 :: Nat → g':h'

Heuristically decided to analyse the following defined symbols:
f'

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

Types:
f' :: g':h' → g':h'
g' :: g':h' → g':h'
h' :: g':h'
_hole_g':h'1 :: g':h'
_gen_g':h'2 :: Nat → g':h'

Generator Equations:
_gen_g':h'2(0) ⇔ h'
_gen_g':h'2(+(x, 1)) ⇔ g'(_gen_g':h'2(x))

The following defined symbols remain to be analysed:
f'

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

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

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

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

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

Types:
f' :: g':h' → g':h'
g' :: g':h' → g':h'
h' :: g':h'
_hole_g':h'1 :: g':h'
_gen_g':h'2 :: Nat → g':h'

Lemmas:
f'(_gen_g':h'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)

Generator Equations:
_gen_g':h'2(0) ⇔ h'
_gen_g':h'2(+(x, 1)) ⇔ g'(_gen_g':h'2(x))

No more defined symbols left to analyse.

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