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

a__f(f(X)) → a__c(f(g(f(X))))
a__c(X) → d(X)
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(h(X)) → a__h(mark(X))
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
a__c(X) → c(X)
a__h(X) → h(X)

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'(f'(X)) → a__c'(f'(g'(f'(X))))
a__c'(X) → d'(X)
a__h'(X) → a__c'(d'(X))
mark'(f'(X)) → a__f'(mark'(X))
mark'(c'(X)) → a__c'(X)
mark'(h'(X)) → a__h'(mark'(X))
mark'(g'(X)) → g'(X)
mark'(d'(X)) → d'(X)
a__f'(X) → f'(X)
a__c'(X) → c'(X)
a__h'(X) → h'(X)

Rewrite Strategy: INNERMOST

Sliced the following arguments:
a__c'/0
g'/0
d'/0
c'/0

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


a__f'(f'(X)) → a__c'
a__c' → d'
a__h'(X) → a__c'
mark'(f'(X)) → a__f'(mark'(X))
mark'(c') → a__c'
mark'(h'(X)) → a__h'(mark'(X))
mark'(g') → g'
mark'(d') → d'
a__f'(X) → f'(X)
a__c' → c'
a__h'(X) → h'(X)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__f'(f'(X)) → a__c'
a__c' → d'
a__h'(X) → a__c'
mark'(f'(X)) → a__f'(mark'(X))
mark'(c') → a__c'
mark'(h'(X)) → a__h'(mark'(X))
mark'(g') → g'
mark'(d') → d'
a__f'(X) → f'(X)
a__c' → c'
a__h'(X) → h'(X)

Types:
a__f' :: f':d':c':h':g' → f':d':c':h':g'
f' :: f':d':c':h':g' → f':d':c':h':g'
a__c' :: f':d':c':h':g'
d' :: f':d':c':h':g'
a__h' :: f':d':c':h':g' → f':d':c':h':g'
mark' :: f':d':c':h':g' → f':d':c':h':g'
c' :: f':d':c':h':g'
h' :: f':d':c':h':g' → f':d':c':h':g'
g' :: f':d':c':h':g'
_hole_f':d':c':h':g'1 :: f':d':c':h':g'
_gen_f':d':c':h':g'2 :: Nat → f':d':c':h':g'

Heuristically decided to analyse the following defined symbols:
mark'

Rules:
a__f'(f'(X)) → a__c'
a__c' → d'
a__h'(X) → a__c'
mark'(f'(X)) → a__f'(mark'(X))
mark'(c') → a__c'
mark'(h'(X)) → a__h'(mark'(X))
mark'(g') → g'
mark'(d') → d'
a__f'(X) → f'(X)
a__c' → c'
a__h'(X) → h'(X)

Types:
a__f' :: f':d':c':h':g' → f':d':c':h':g'
f' :: f':d':c':h':g' → f':d':c':h':g'
a__c' :: f':d':c':h':g'
d' :: f':d':c':h':g'
a__h' :: f':d':c':h':g' → f':d':c':h':g'
mark' :: f':d':c':h':g' → f':d':c':h':g'
c' :: f':d':c':h':g'
h' :: f':d':c':h':g' → f':d':c':h':g'
g' :: f':d':c':h':g'
_hole_f':d':c':h':g'1 :: f':d':c':h':g'
_gen_f':d':c':h':g'2 :: Nat → f':d':c':h':g'

Generator Equations:
_gen_f':d':c':h':g'2(0) ⇔ d'
_gen_f':d':c':h':g'2(+(x, 1)) ⇔ f'(_gen_f':d':c':h':g'2(x))

The following defined symbols remain to be analysed:
mark'

Proved the following rewrite lemma:
mark'(_gen_f':d':c':h':g'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)

Induction Base:
mark'(_gen_f':d':c':h':g'2(+(1, 0)))

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

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

Rules:
a__f'(f'(X)) → a__c'
a__c' → d'
a__h'(X) → a__c'
mark'(f'(X)) → a__f'(mark'(X))
mark'(c') → a__c'
mark'(h'(X)) → a__h'(mark'(X))
mark'(g') → g'
mark'(d') → d'
a__f'(X) → f'(X)
a__c' → c'
a__h'(X) → h'(X)

Types:
a__f' :: f':d':c':h':g' → f':d':c':h':g'
f' :: f':d':c':h':g' → f':d':c':h':g'
a__c' :: f':d':c':h':g'
d' :: f':d':c':h':g'
a__h' :: f':d':c':h':g' → f':d':c':h':g'
mark' :: f':d':c':h':g' → f':d':c':h':g'
c' :: f':d':c':h':g'
h' :: f':d':c':h':g' → f':d':c':h':g'
g' :: f':d':c':h':g'
_hole_f':d':c':h':g'1 :: f':d':c':h':g'
_gen_f':d':c':h':g'2 :: Nat → f':d':c':h':g'

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

Generator Equations:
_gen_f':d':c':h':g'2(0) ⇔ d'
_gen_f':d':c':h':g'2(+(x, 1)) ⇔ f'(_gen_f':d':c':h':g'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_f':d':c':h':g'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)