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

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(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'(0') → cons'(0', f'(s'(0')))
a__f'(s'(0')) → a__f'(a__p'(s'(0')))
a__p'(s'(X)) → mark'(X)
mark'(f'(X)) → a__f'(mark'(X))
mark'(p'(X)) → a__p'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
a__f'(X) → f'(X)
a__p'(X) → p'(X)

Rewrite Strategy: INNERMOST

Sliced the following arguments:
cons'/1

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


a__f'(0') → cons'(0')
a__f'(s'(0')) → a__f'(a__p'(s'(0')))
a__p'(s'(X)) → mark'(X)
mark'(f'(X)) → a__f'(mark'(X))
mark'(p'(X)) → a__p'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(s'(X)) → s'(mark'(X))
a__f'(X) → f'(X)
a__p'(X) → p'(X)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__f'(0') → cons'(0')
a__f'(s'(0')) → a__f'(a__p'(s'(0')))
a__p'(s'(X)) → mark'(X)
mark'(f'(X)) → a__f'(mark'(X))
mark'(p'(X)) → a__p'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(s'(X)) → s'(mark'(X))
a__f'(X) → f'(X)
a__p'(X) → p'(X)

Types:
a__f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
0' :: 0':cons':s':f':p'
cons' :: 0':cons':s':f':p' → 0':cons':s':f':p'
s' :: 0':cons':s':f':p' → 0':cons':s':f':p'
a__p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
mark' :: 0':cons':s':f':p' → 0':cons':s':f':p'
f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
_hole_0':cons':s':f':p'1 :: 0':cons':s':f':p'
_gen_0':cons':s':f':p'2 :: Nat → 0':cons':s':f':p'

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

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

Rules:
a__f'(0') → cons'(0')
a__f'(s'(0')) → a__f'(a__p'(s'(0')))
a__p'(s'(X)) → mark'(X)
mark'(f'(X)) → a__f'(mark'(X))
mark'(p'(X)) → a__p'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(s'(X)) → s'(mark'(X))
a__f'(X) → f'(X)
a__p'(X) → p'(X)

Types:
a__f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
0' :: 0':cons':s':f':p'
cons' :: 0':cons':s':f':p' → 0':cons':s':f':p'
s' :: 0':cons':s':f':p' → 0':cons':s':f':p'
a__p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
mark' :: 0':cons':s':f':p' → 0':cons':s':f':p'
f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
_hole_0':cons':s':f':p'1 :: 0':cons':s':f':p'
_gen_0':cons':s':f':p'2 :: Nat → 0':cons':s':f':p'

Generator Equations:
_gen_0':cons':s':f':p'2(0) ⇔ 0'
_gen_0':cons':s':f':p'2(+(x, 1)) ⇔ cons'(_gen_0':cons':s':f':p'2(x))

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

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

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

Rules:
a__f'(0') → cons'(0')
a__f'(s'(0')) → a__f'(a__p'(s'(0')))
a__p'(s'(X)) → mark'(X)
mark'(f'(X)) → a__f'(mark'(X))
mark'(p'(X)) → a__p'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(s'(X)) → s'(mark'(X))
a__f'(X) → f'(X)
a__p'(X) → p'(X)

Types:
a__f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
0' :: 0':cons':s':f':p'
cons' :: 0':cons':s':f':p' → 0':cons':s':f':p'
s' :: 0':cons':s':f':p' → 0':cons':s':f':p'
a__p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
mark' :: 0':cons':s':f':p' → 0':cons':s':f':p'
f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
_hole_0':cons':s':f':p'1 :: 0':cons':s':f':p'
_gen_0':cons':s':f':p'2 :: Nat → 0':cons':s':f':p'

Generator Equations:
_gen_0':cons':s':f':p'2(0) ⇔ 0'
_gen_0':cons':s':f':p'2(+(x, 1)) ⇔ cons'(_gen_0':cons':s':f':p'2(x))

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

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

Proved the following rewrite lemma:
mark'(_gen_0':cons':s':f':p'2(_n14)) → _gen_0':cons':s':f':p'2(_n14), rt ∈ Ω(1 + n14)

Induction Base:
mark'(_gen_0':cons':s':f':p'2(0)) →RΩ(1)
0'

Induction Step:
mark'(_gen_0':cons':s':f':p'2(+(_$n15, 1))) →RΩ(1)
cons'(mark'(_gen_0':cons':s':f':p'2(_$n15))) →IH
cons'(_gen_0':cons':s':f':p'2(_$n15))

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

Rules:
a__f'(0') → cons'(0')
a__f'(s'(0')) → a__f'(a__p'(s'(0')))
a__p'(s'(X)) → mark'(X)
mark'(f'(X)) → a__f'(mark'(X))
mark'(p'(X)) → a__p'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(s'(X)) → s'(mark'(X))
a__f'(X) → f'(X)
a__p'(X) → p'(X)

Types:
a__f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
0' :: 0':cons':s':f':p'
cons' :: 0':cons':s':f':p' → 0':cons':s':f':p'
s' :: 0':cons':s':f':p' → 0':cons':s':f':p'
a__p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
mark' :: 0':cons':s':f':p' → 0':cons':s':f':p'
f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
_hole_0':cons':s':f':p'1 :: 0':cons':s':f':p'
_gen_0':cons':s':f':p'2 :: Nat → 0':cons':s':f':p'

Lemmas:
mark'(_gen_0':cons':s':f':p'2(_n14)) → _gen_0':cons':s':f':p'2(_n14), rt ∈ Ω(1 + n14)

Generator Equations:
_gen_0':cons':s':f':p'2(0) ⇔ 0'
_gen_0':cons':s':f':p'2(+(x, 1)) ⇔ cons'(_gen_0':cons':s':f':p'2(x))

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

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

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

Rules:
a__f'(0') → cons'(0')
a__f'(s'(0')) → a__f'(a__p'(s'(0')))
a__p'(s'(X)) → mark'(X)
mark'(f'(X)) → a__f'(mark'(X))
mark'(p'(X)) → a__p'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(s'(X)) → s'(mark'(X))
a__f'(X) → f'(X)
a__p'(X) → p'(X)

Types:
a__f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
0' :: 0':cons':s':f':p'
cons' :: 0':cons':s':f':p' → 0':cons':s':f':p'
s' :: 0':cons':s':f':p' → 0':cons':s':f':p'
a__p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
mark' :: 0':cons':s':f':p' → 0':cons':s':f':p'
f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
_hole_0':cons':s':f':p'1 :: 0':cons':s':f':p'
_gen_0':cons':s':f':p'2 :: Nat → 0':cons':s':f':p'

Lemmas:
mark'(_gen_0':cons':s':f':p'2(_n14)) → _gen_0':cons':s':f':p'2(_n14), rt ∈ Ω(1 + n14)

Generator Equations:
_gen_0':cons':s':f':p'2(0) ⇔ 0'
_gen_0':cons':s':f':p'2(+(x, 1)) ⇔ cons'(_gen_0':cons':s':f':p'2(x))

The following defined symbols remain to be analysed:
a__p'

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

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

Rules:
a__f'(0') → cons'(0')
a__f'(s'(0')) → a__f'(a__p'(s'(0')))
a__p'(s'(X)) → mark'(X)
mark'(f'(X)) → a__f'(mark'(X))
mark'(p'(X)) → a__p'(mark'(X))
mark'(0') → 0'
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(s'(X)) → s'(mark'(X))
a__f'(X) → f'(X)
a__p'(X) → p'(X)

Types:
a__f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
0' :: 0':cons':s':f':p'
cons' :: 0':cons':s':f':p' → 0':cons':s':f':p'
s' :: 0':cons':s':f':p' → 0':cons':s':f':p'
a__p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
mark' :: 0':cons':s':f':p' → 0':cons':s':f':p'
f' :: 0':cons':s':f':p' → 0':cons':s':f':p'
p' :: 0':cons':s':f':p' → 0':cons':s':f':p'
_hole_0':cons':s':f':p'1 :: 0':cons':s':f':p'
_gen_0':cons':s':f':p'2 :: Nat → 0':cons':s':f':p'

Lemmas:
mark'(_gen_0':cons':s':f':p'2(_n14)) → _gen_0':cons':s':f':p'2(_n14), rt ∈ Ω(1 + n14)

Generator Equations:
_gen_0':cons':s':f':p'2(0) ⇔ 0'
_gen_0':cons':s':f':p'2(+(x, 1)) ⇔ cons'(_gen_0':cons':s':f':p'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_0':cons':s':f':p'2(_n14)) → _gen_0':cons':s':f':p'2(_n14), rt ∈ Ω(1 + n14)