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

a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


a__f'(g'(X), Y) → a__f'(mark'(X), f'(g'(X), Y))
mark'(f'(X1, X2)) → a__f'(mark'(X1), X2)
mark'(g'(X)) → g'(mark'(X))
a__f'(X1, X2) → f'(X1, X2)

Rewrite Strategy: INNERMOST

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

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


a__f'(g'(X)) → a__f'(mark'(X))
mark'(f'(X1)) → a__f'(mark'(X1))
mark'(g'(X)) → g'(mark'(X))
a__f'(X1) → f'(X1)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__f'(g'(X)) → a__f'(mark'(X))
mark'(f'(X1)) → a__f'(mark'(X1))
mark'(g'(X)) → g'(mark'(X))
a__f'(X1) → f'(X1)

Types:
a__f' :: g':f' → g':f'
g' :: g':f' → g':f'
mark' :: g':f' → g':f'
f' :: g':f' → g':f'
_hole_g':f'1 :: g':f'
_gen_g':f'2 :: Nat → g':f'

Heuristically decided to analyse the following defined symbols:
a__f', mark'

They will be analysed ascendingly in the following order:
a__f' = mark'

Rules:
a__f'(g'(X)) → a__f'(mark'(X))
mark'(f'(X1)) → a__f'(mark'(X1))
mark'(g'(X)) → g'(mark'(X))
a__f'(X1) → f'(X1)

Types:
a__f' :: g':f' → g':f'
g' :: g':f' → g':f'
mark' :: g':f' → g':f'
f' :: g':f' → g':f'
_hole_g':f'1 :: g':f'
_gen_g':f'2 :: Nat → g':f'

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

The following defined symbols remain to be analysed:
mark', a__f'

They will be analysed ascendingly in the following order:
a__f' = mark'

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

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

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

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

Rules:
a__f'(g'(X)) → a__f'(mark'(X))
mark'(f'(X1)) → a__f'(mark'(X1))
mark'(g'(X)) → g'(mark'(X))
a__f'(X1) → f'(X1)

Types:
a__f' :: g':f' → g':f'
g' :: g':f' → g':f'
mark' :: g':f' → g':f'
f' :: g':f' → g':f'
_hole_g':f'1 :: g':f'
_gen_g':f'2 :: Nat → g':f'

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

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

The following defined symbols remain to be analysed:
a__f'

They will be analysed ascendingly in the following order:
a__f' = mark'

Could not prove a rewrite lemma for the defined symbol a__f'.

Rules:
a__f'(g'(X)) → a__f'(mark'(X))
mark'(f'(X1)) → a__f'(mark'(X1))
mark'(g'(X)) → g'(mark'(X))
a__f'(X1) → f'(X1)

Types:
a__f' :: g':f' → g':f'
g' :: g':f' → g':f'
mark' :: g':f' → g':f'
f' :: g':f' → g':f'
_hole_g':f'1 :: g':f'
_gen_g':f'2 :: Nat → g':f'

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

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

No more defined symbols left to analyse.

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