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

app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
sum(plus(cons(0, x), cons(y, l))) → pred(sum(cons(s(x), cons(y, l))))
pred(cons(s(x), nil)) → cons(x, nil)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
sum'(plus'(cons'(0', x), cons'(y, l))) → pred'(sum'(cons'(s'(x), cons'(y, l))))
pred'(cons'(s'(x), nil')) → cons'(x, nil')

Rewrite Strategy: INNERMOST

Infered types.

Rules:
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
sum'(plus'(cons'(0', x), cons'(y, l))) → pred'(sum'(cons'(s'(x), cons'(y, l))))
pred'(cons'(s'(x), nil')) → cons'(x, nil')

Types:
app' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
nil' :: nil':cons':0':s'
cons' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
sum' :: nil':cons':0':s' → nil':cons':0':s'
plus' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
0' :: nil':cons':0':s'
s' :: nil':cons':0':s' → nil':cons':0':s'
pred' :: nil':cons':0':s' → nil':cons':0':s'
_hole_nil':cons':0':s'1 :: nil':cons':0':s'
_gen_nil':cons':0':s'2 :: Nat → nil':cons':0':s'

Heuristically decided to analyse the following defined symbols:
app', sum', plus'

They will be analysed ascendingly in the following order:
app' < sum'
plus' < sum'

Rules:
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
sum'(plus'(cons'(0', x), cons'(y, l))) → pred'(sum'(cons'(s'(x), cons'(y, l))))
pred'(cons'(s'(x), nil')) → cons'(x, nil')

Types:
app' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
nil' :: nil':cons':0':s'
cons' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
sum' :: nil':cons':0':s' → nil':cons':0':s'
plus' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
0' :: nil':cons':0':s'
s' :: nil':cons':0':s' → nil':cons':0':s'
pred' :: nil':cons':0':s' → nil':cons':0':s'
_hole_nil':cons':0':s'1 :: nil':cons':0':s'
_gen_nil':cons':0':s'2 :: Nat → nil':cons':0':s'

Generator Equations:
_gen_nil':cons':0':s'2(0) ⇔ nil'
_gen_nil':cons':0':s'2(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':0':s'2(x))

The following defined symbols remain to be analysed:
app', sum', plus'

They will be analysed ascendingly in the following order:
app' < sum'
plus' < sum'

Proved the following rewrite lemma:
app'(_gen_nil':cons':0':s'2(_n4), _gen_nil':cons':0':s'2(b)) → _gen_nil':cons':0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)

Induction Base:
app'(_gen_nil':cons':0':s'2(0), _gen_nil':cons':0':s'2(b)) →RΩ(1)
_gen_nil':cons':0':s'2(b)

Induction Step:
app'(_gen_nil':cons':0':s'2(+(_\$n5, 1)), _gen_nil':cons':0':s'2(_b161)) →RΩ(1)
cons'(nil', app'(_gen_nil':cons':0':s'2(_\$n5), _gen_nil':cons':0':s'2(_b161))) →IH
cons'(nil', _gen_nil':cons':0':s'2(+(_\$n5, _b161)))

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

Rules:
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
sum'(plus'(cons'(0', x), cons'(y, l))) → pred'(sum'(cons'(s'(x), cons'(y, l))))
pred'(cons'(s'(x), nil')) → cons'(x, nil')

Types:
app' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
nil' :: nil':cons':0':s'
cons' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
sum' :: nil':cons':0':s' → nil':cons':0':s'
plus' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
0' :: nil':cons':0':s'
s' :: nil':cons':0':s' → nil':cons':0':s'
pred' :: nil':cons':0':s' → nil':cons':0':s'
_hole_nil':cons':0':s'1 :: nil':cons':0':s'
_gen_nil':cons':0':s'2 :: Nat → nil':cons':0':s'

Lemmas:
app'(_gen_nil':cons':0':s'2(_n4), _gen_nil':cons':0':s'2(b)) → _gen_nil':cons':0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_nil':cons':0':s'2(0) ⇔ nil'
_gen_nil':cons':0':s'2(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':0':s'2(x))

The following defined symbols remain to be analysed:
plus', sum'

They will be analysed ascendingly in the following order:
plus' < sum'

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

Rules:
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
sum'(plus'(cons'(0', x), cons'(y, l))) → pred'(sum'(cons'(s'(x), cons'(y, l))))
pred'(cons'(s'(x), nil')) → cons'(x, nil')

Types:
app' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
nil' :: nil':cons':0':s'
cons' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
sum' :: nil':cons':0':s' → nil':cons':0':s'
plus' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
0' :: nil':cons':0':s'
s' :: nil':cons':0':s' → nil':cons':0':s'
pred' :: nil':cons':0':s' → nil':cons':0':s'
_hole_nil':cons':0':s'1 :: nil':cons':0':s'
_gen_nil':cons':0':s'2 :: Nat → nil':cons':0':s'

Lemmas:
app'(_gen_nil':cons':0':s'2(_n4), _gen_nil':cons':0':s'2(b)) → _gen_nil':cons':0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_nil':cons':0':s'2(0) ⇔ nil'
_gen_nil':cons':0':s'2(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':0':s'2(x))

The following defined symbols remain to be analysed:
sum'

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

The following conjecture could not be proven:

sum'(_gen_nil':cons':0':s'2(+(2, _n884))) →? _*3

Rules:
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
sum'(plus'(cons'(0', x), cons'(y, l))) → pred'(sum'(cons'(s'(x), cons'(y, l))))
pred'(cons'(s'(x), nil')) → cons'(x, nil')

Types:
app' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
nil' :: nil':cons':0':s'
cons' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
sum' :: nil':cons':0':s' → nil':cons':0':s'
plus' :: nil':cons':0':s' → nil':cons':0':s' → nil':cons':0':s'
0' :: nil':cons':0':s'
s' :: nil':cons':0':s' → nil':cons':0':s'
pred' :: nil':cons':0':s' → nil':cons':0':s'
_hole_nil':cons':0':s'1 :: nil':cons':0':s'
_gen_nil':cons':0':s'2 :: Nat → nil':cons':0':s'

Lemmas:
app'(_gen_nil':cons':0':s'2(_n4), _gen_nil':cons':0':s'2(b)) → _gen_nil':cons':0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_nil':cons':0':s'2(0) ⇔ nil'
_gen_nil':cons':0':s'2(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':0':s'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
app'(_gen_nil':cons':0':s'2(_n4), _gen_nil':cons':0':s'2(b)) → _gen_nil':cons':0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)