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

inc(Cons(x, xs)) → Cons(Cons(Nil, Nil), inc(xs))
nestinc(Nil) → number17(Nil)
nestinc(Cons(x, xs)) → nestinc(inc(Cons(x, xs)))
inc(Nil) → Cons(Nil, Nil)
number17(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestinc(x)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


inc'(Cons'(x, xs)) → Cons'(Cons'(Nil', Nil'), inc'(xs))
nestinc'(Nil') → number17'(Nil')
nestinc'(Cons'(x, xs)) → nestinc'(inc'(Cons'(x, xs)))
inc'(Nil') → Cons'(Nil', Nil')
number17'(x) → Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Cons'(Nil', Nil')))))))))))))))))
goal'(x) → nestinc'(x)

Rewrite Strategy: INNERMOST

Sliced the following arguments:
Cons'/0
number17'/0

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


inc'(Cons'(xs)) → Cons'(inc'(xs))
nestinc'(Nil') → number17'
nestinc'(Cons'(xs)) → nestinc'(inc'(Cons'(xs)))
inc'(Nil') → Cons'(Nil')
number17' → Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Nil')))))))))))))))))
goal'(x) → nestinc'(x)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
inc'(Cons'(xs)) → Cons'(inc'(xs))
nestinc'(Nil') → number17'
nestinc'(Cons'(xs)) → nestinc'(inc'(Cons'(xs)))
inc'(Nil') → Cons'(Nil')
number17' → Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Nil')))))))))))))))))
goal'(x) → nestinc'(x)

Types:
inc' :: Cons':Nil' → Cons':Nil'
Cons' :: Cons':Nil' → Cons':Nil'
nestinc' :: Cons':Nil' → Cons':Nil'
Nil' :: Cons':Nil'
number17' :: Cons':Nil'
goal' :: Cons':Nil' → Cons':Nil'
_hole_Cons':Nil'1 :: Cons':Nil'
_gen_Cons':Nil'2 :: Nat → Cons':Nil'

Heuristically decided to analyse the following defined symbols:
inc', nestinc'

They will be analysed ascendingly in the following order:
inc' < nestinc'

Rules:
inc'(Cons'(xs)) → Cons'(inc'(xs))
nestinc'(Nil') → number17'
nestinc'(Cons'(xs)) → nestinc'(inc'(Cons'(xs)))
inc'(Nil') → Cons'(Nil')
number17' → Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Nil')))))))))))))))))
goal'(x) → nestinc'(x)

Types:
inc' :: Cons':Nil' → Cons':Nil'
Cons' :: Cons':Nil' → Cons':Nil'
nestinc' :: Cons':Nil' → Cons':Nil'
Nil' :: Cons':Nil'
number17' :: Cons':Nil'
goal' :: Cons':Nil' → Cons':Nil'
_hole_Cons':Nil'1 :: Cons':Nil'
_gen_Cons':Nil'2 :: Nat → Cons':Nil'

Generator Equations:
_gen_Cons':Nil'2(0) ⇔ Nil'
_gen_Cons':Nil'2(+(x, 1)) ⇔ Cons'(_gen_Cons':Nil'2(x))

The following defined symbols remain to be analysed:
inc', nestinc'

They will be analysed ascendingly in the following order:
inc' < nestinc'

Proved the following rewrite lemma:
inc'(_gen_Cons':Nil'2(_n4)) → _gen_Cons':Nil'2(+(1, _n4)), rt ∈ Ω(1 + n4)

Induction Base:
inc'(_gen_Cons':Nil'2(0)) →RΩ(1)
Cons'(Nil')

Induction Step:
inc'(_gen_Cons':Nil'2(+(_$n5, 1))) →RΩ(1)
Cons'(inc'(_gen_Cons':Nil'2(_$n5))) →IH
Cons'(_gen_Cons':Nil'2(+(1, _$n5)))

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

Rules:
inc'(Cons'(xs)) → Cons'(inc'(xs))
nestinc'(Nil') → number17'
nestinc'(Cons'(xs)) → nestinc'(inc'(Cons'(xs)))
inc'(Nil') → Cons'(Nil')
number17' → Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Nil')))))))))))))))))
goal'(x) → nestinc'(x)

Types:
inc' :: Cons':Nil' → Cons':Nil'
Cons' :: Cons':Nil' → Cons':Nil'
nestinc' :: Cons':Nil' → Cons':Nil'
Nil' :: Cons':Nil'
number17' :: Cons':Nil'
goal' :: Cons':Nil' → Cons':Nil'
_hole_Cons':Nil'1 :: Cons':Nil'
_gen_Cons':Nil'2 :: Nat → Cons':Nil'

Lemmas:
inc'(_gen_Cons':Nil'2(_n4)) → _gen_Cons':Nil'2(+(1, _n4)), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_Cons':Nil'2(0) ⇔ Nil'
_gen_Cons':Nil'2(+(x, 1)) ⇔ Cons'(_gen_Cons':Nil'2(x))

The following defined symbols remain to be analysed:
nestinc'

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

Rules:
inc'(Cons'(xs)) → Cons'(inc'(xs))
nestinc'(Nil') → number17'
nestinc'(Cons'(xs)) → nestinc'(inc'(Cons'(xs)))
inc'(Nil') → Cons'(Nil')
number17' → Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Cons'(Nil')))))))))))))))))
goal'(x) → nestinc'(x)

Types:
inc' :: Cons':Nil' → Cons':Nil'
Cons' :: Cons':Nil' → Cons':Nil'
nestinc' :: Cons':Nil' → Cons':Nil'
Nil' :: Cons':Nil'
number17' :: Cons':Nil'
goal' :: Cons':Nil' → Cons':Nil'
_hole_Cons':Nil'1 :: Cons':Nil'
_gen_Cons':Nil'2 :: Nat → Cons':Nil'

Lemmas:
inc'(_gen_Cons':Nil'2(_n4)) → _gen_Cons':Nil'2(+(1, _n4)), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_Cons':Nil'2(0) ⇔ Nil'
_gen_Cons':Nil'2(+(x, 1)) ⇔ Cons'(_gen_Cons':Nil'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
inc'(_gen_Cons':Nil'2(_n4)) → _gen_Cons':Nil'2(+(1, _n4)), rt ∈ Ω(1 + n4)