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

lessElements(l, t) → lessE(l, t, 0)
lessE(l, t, n) → if(le(length(l), n), le(length(toList(t)), n), l, t, n)
if(true, b, l, t, n) → l
if(false, true, l, t, n) → t
if(false, false, l, t, n) → lessE(l, t, s(n))
length(nil) → 0
length(cons(n, l)) → s(length(l))
toList(leaf) → nil
toList(node(t1, n, t2)) → append(toList(t1), cons(n, toList(t2)))
append(nil, l2) → l2
append(cons(n, l1), l2) → cons(n, append(l1, l2))
le(s(n), 0) → false
le(0, m) → true
le(s(n), s(m)) → le(n, m)
ac

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(n, l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, n, t2)) → append'(toList'(t1), cons'(n, toList'(t2)))
append'(nil', l2) → l2
append'(cons'(n, l1), l2) → cons'(n, append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Rewrite Strategy: INNERMOST

Sliced the following arguments:
cons'/0
node'/1

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

lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, t2)) → append'(toList'(t1), cons'(toList'(t2)))
append'(nil', l2) → l2
append'(cons'(l1), l2) → cons'(append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Rewrite Strategy: INNERMOST

Infered types.

Rules:
lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, t2)) → append'(toList'(t1), cons'(toList'(t2)))
append'(nil', l2) → l2
append'(cons'(l1), l2) → cons'(append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Types:
lessElements' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
lessE' :: nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
0' :: 0':s'
if' :: true':false' → true':false' → nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
le' :: 0':s' → 0':s' → true':false'
length' :: nil':cons':leaf':node' → 0':s'
toList' :: nil':cons':leaf':node' → nil':cons':leaf':node'
true' :: true':false'
false' :: true':false'
s' :: 0':s' → 0':s'
nil' :: nil':cons':leaf':node'
cons' :: nil':cons':leaf':node' → nil':cons':leaf':node'
leaf' :: nil':cons':leaf':node'
node' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
append' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
a' :: c':d'
c' :: c':d'
d' :: c':d'
_hole_nil':cons':leaf':node'1 :: nil':cons':leaf':node'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_c':d'4 :: c':d'
_gen_nil':cons':leaf':node'5 :: Nat → nil':cons':leaf':node'
_gen_0':s'6 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
lessE', le', length', toList', append'

They will be analysed ascendingly in the following order:
le' < lessE'
length' < lessE'
toList' < lessE'
append' < toList'

Rules:
lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, t2)) → append'(toList'(t1), cons'(toList'(t2)))
append'(nil', l2) → l2
append'(cons'(l1), l2) → cons'(append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Types:
lessElements' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
lessE' :: nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
0' :: 0':s'
if' :: true':false' → true':false' → nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
le' :: 0':s' → 0':s' → true':false'
length' :: nil':cons':leaf':node' → 0':s'
toList' :: nil':cons':leaf':node' → nil':cons':leaf':node'
true' :: true':false'
false' :: true':false'
s' :: 0':s' → 0':s'
nil' :: nil':cons':leaf':node'
cons' :: nil':cons':leaf':node' → nil':cons':leaf':node'
leaf' :: nil':cons':leaf':node'
node' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
append' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
a' :: c':d'
c' :: c':d'
d' :: c':d'
_hole_nil':cons':leaf':node'1 :: nil':cons':leaf':node'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_c':d'4 :: c':d'
_gen_nil':cons':leaf':node'5 :: Nat → nil':cons':leaf':node'
_gen_0':s'6 :: Nat → 0':s'

Generator Equations:
_gen_nil':cons':leaf':node'5(0) ⇔ nil'
_gen_nil':cons':leaf':node'5(+(x, 1)) ⇔ cons'(_gen_nil':cons':leaf':node'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
le', lessE', length', toList', append'

They will be analysed ascendingly in the following order:
le' < lessE'
length' < lessE'
toList' < lessE'
append' < toList'

Proved the following rewrite lemma:
le'(_gen_0':s'6(+(1, _n8)), _gen_0':s'6(_n8)) → false', rt ∈ Ω(1 + n8)

Induction Base:
le'(_gen_0':s'6(+(1, 0)), _gen_0':s'6(0)) →RΩ(1)
false'

Induction Step:
le'(_gen_0':s'6(+(1, +(_\$n9, 1))), _gen_0':s'6(+(_\$n9, 1))) →RΩ(1)
le'(_gen_0':s'6(+(1, _\$n9)), _gen_0':s'6(_\$n9)) →IH
false'

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

Rules:
lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, t2)) → append'(toList'(t1), cons'(toList'(t2)))
append'(nil', l2) → l2
append'(cons'(l1), l2) → cons'(append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Types:
lessElements' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
lessE' :: nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
0' :: 0':s'
if' :: true':false' → true':false' → nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
le' :: 0':s' → 0':s' → true':false'
length' :: nil':cons':leaf':node' → 0':s'
toList' :: nil':cons':leaf':node' → nil':cons':leaf':node'
true' :: true':false'
false' :: true':false'
s' :: 0':s' → 0':s'
nil' :: nil':cons':leaf':node'
cons' :: nil':cons':leaf':node' → nil':cons':leaf':node'
leaf' :: nil':cons':leaf':node'
node' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
append' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
a' :: c':d'
c' :: c':d'
d' :: c':d'
_hole_nil':cons':leaf':node'1 :: nil':cons':leaf':node'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_c':d'4 :: c':d'
_gen_nil':cons':leaf':node'5 :: Nat → nil':cons':leaf':node'
_gen_0':s'6 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'6(+(1, _n8)), _gen_0':s'6(_n8)) → false', rt ∈ Ω(1 + n8)

Generator Equations:
_gen_nil':cons':leaf':node'5(0) ⇔ nil'
_gen_nil':cons':leaf':node'5(+(x, 1)) ⇔ cons'(_gen_nil':cons':leaf':node'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
length', lessE', toList', append'

They will be analysed ascendingly in the following order:
length' < lessE'
toList' < lessE'
append' < toList'

Proved the following rewrite lemma:
length'(_gen_nil':cons':leaf':node'5(_n906)) → _gen_0':s'6(_n906), rt ∈ Ω(1 + n906)

Induction Base:
length'(_gen_nil':cons':leaf':node'5(0)) →RΩ(1)
0'

Induction Step:
length'(_gen_nil':cons':leaf':node'5(+(_\$n907, 1))) →RΩ(1)
s'(length'(_gen_nil':cons':leaf':node'5(_\$n907))) →IH
s'(_gen_0':s'6(_\$n907))

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

Rules:
lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, t2)) → append'(toList'(t1), cons'(toList'(t2)))
append'(nil', l2) → l2
append'(cons'(l1), l2) → cons'(append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Types:
lessElements' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
lessE' :: nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
0' :: 0':s'
if' :: true':false' → true':false' → nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
le' :: 0':s' → 0':s' → true':false'
length' :: nil':cons':leaf':node' → 0':s'
toList' :: nil':cons':leaf':node' → nil':cons':leaf':node'
true' :: true':false'
false' :: true':false'
s' :: 0':s' → 0':s'
nil' :: nil':cons':leaf':node'
cons' :: nil':cons':leaf':node' → nil':cons':leaf':node'
leaf' :: nil':cons':leaf':node'
node' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
append' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
a' :: c':d'
c' :: c':d'
d' :: c':d'
_hole_nil':cons':leaf':node'1 :: nil':cons':leaf':node'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_c':d'4 :: c':d'
_gen_nil':cons':leaf':node'5 :: Nat → nil':cons':leaf':node'
_gen_0':s'6 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'6(+(1, _n8)), _gen_0':s'6(_n8)) → false', rt ∈ Ω(1 + n8)
length'(_gen_nil':cons':leaf':node'5(_n906)) → _gen_0':s'6(_n906), rt ∈ Ω(1 + n906)

Generator Equations:
_gen_nil':cons':leaf':node'5(0) ⇔ nil'
_gen_nil':cons':leaf':node'5(+(x, 1)) ⇔ cons'(_gen_nil':cons':leaf':node'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
append', lessE', toList'

They will be analysed ascendingly in the following order:
toList' < lessE'
append' < toList'

Proved the following rewrite lemma:
append'(_gen_nil':cons':leaf':node'5(_n1502), _gen_nil':cons':leaf':node'5(b)) → _gen_nil':cons':leaf':node'5(+(_n1502, b)), rt ∈ Ω(1 + n1502)

Induction Base:
append'(_gen_nil':cons':leaf':node'5(0), _gen_nil':cons':leaf':node'5(b)) →RΩ(1)
_gen_nil':cons':leaf':node'5(b)

Induction Step:
append'(_gen_nil':cons':leaf':node'5(+(_\$n1503, 1)), _gen_nil':cons':leaf':node'5(_b1671)) →RΩ(1)
cons'(append'(_gen_nil':cons':leaf':node'5(_\$n1503), _gen_nil':cons':leaf':node'5(_b1671))) →IH
cons'(_gen_nil':cons':leaf':node'5(+(_\$n1503, _b1671)))

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

Rules:
lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, t2)) → append'(toList'(t1), cons'(toList'(t2)))
append'(nil', l2) → l2
append'(cons'(l1), l2) → cons'(append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Types:
lessElements' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
lessE' :: nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
0' :: 0':s'
if' :: true':false' → true':false' → nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
le' :: 0':s' → 0':s' → true':false'
length' :: nil':cons':leaf':node' → 0':s'
toList' :: nil':cons':leaf':node' → nil':cons':leaf':node'
true' :: true':false'
false' :: true':false'
s' :: 0':s' → 0':s'
nil' :: nil':cons':leaf':node'
cons' :: nil':cons':leaf':node' → nil':cons':leaf':node'
leaf' :: nil':cons':leaf':node'
node' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
append' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
a' :: c':d'
c' :: c':d'
d' :: c':d'
_hole_nil':cons':leaf':node'1 :: nil':cons':leaf':node'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_c':d'4 :: c':d'
_gen_nil':cons':leaf':node'5 :: Nat → nil':cons':leaf':node'
_gen_0':s'6 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'6(+(1, _n8)), _gen_0':s'6(_n8)) → false', rt ∈ Ω(1 + n8)
length'(_gen_nil':cons':leaf':node'5(_n906)) → _gen_0':s'6(_n906), rt ∈ Ω(1 + n906)
append'(_gen_nil':cons':leaf':node'5(_n1502), _gen_nil':cons':leaf':node'5(b)) → _gen_nil':cons':leaf':node'5(+(_n1502, b)), rt ∈ Ω(1 + n1502)

Generator Equations:
_gen_nil':cons':leaf':node'5(0) ⇔ nil'
_gen_nil':cons':leaf':node'5(+(x, 1)) ⇔ cons'(_gen_nil':cons':leaf':node'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
toList', lessE'

They will be analysed ascendingly in the following order:
toList' < lessE'

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

Rules:
lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, t2)) → append'(toList'(t1), cons'(toList'(t2)))
append'(nil', l2) → l2
append'(cons'(l1), l2) → cons'(append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Types:
lessElements' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
lessE' :: nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
0' :: 0':s'
if' :: true':false' → true':false' → nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
le' :: 0':s' → 0':s' → true':false'
length' :: nil':cons':leaf':node' → 0':s'
toList' :: nil':cons':leaf':node' → nil':cons':leaf':node'
true' :: true':false'
false' :: true':false'
s' :: 0':s' → 0':s'
nil' :: nil':cons':leaf':node'
cons' :: nil':cons':leaf':node' → nil':cons':leaf':node'
leaf' :: nil':cons':leaf':node'
node' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
append' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
a' :: c':d'
c' :: c':d'
d' :: c':d'
_hole_nil':cons':leaf':node'1 :: nil':cons':leaf':node'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_c':d'4 :: c':d'
_gen_nil':cons':leaf':node'5 :: Nat → nil':cons':leaf':node'
_gen_0':s'6 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'6(+(1, _n8)), _gen_0':s'6(_n8)) → false', rt ∈ Ω(1 + n8)
length'(_gen_nil':cons':leaf':node'5(_n906)) → _gen_0':s'6(_n906), rt ∈ Ω(1 + n906)
append'(_gen_nil':cons':leaf':node'5(_n1502), _gen_nil':cons':leaf':node'5(b)) → _gen_nil':cons':leaf':node'5(+(_n1502, b)), rt ∈ Ω(1 + n1502)

Generator Equations:
_gen_nil':cons':leaf':node'5(0) ⇔ nil'
_gen_nil':cons':leaf':node'5(+(x, 1)) ⇔ cons'(_gen_nil':cons':leaf':node'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
lessE'

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

Rules:
lessElements'(l, t) → lessE'(l, t, 0')
lessE'(l, t, n) → if'(le'(length'(l), n), le'(length'(toList'(t)), n), l, t, n)
if'(true', b, l, t, n) → l
if'(false', true', l, t, n) → t
if'(false', false', l, t, n) → lessE'(l, t, s'(n))
length'(nil') → 0'
length'(cons'(l)) → s'(length'(l))
toList'(leaf') → nil'
toList'(node'(t1, t2)) → append'(toList'(t1), cons'(toList'(t2)))
append'(nil', l2) → l2
append'(cons'(l1), l2) → cons'(append'(l1, l2))
le'(s'(n), 0') → false'
le'(0', m) → true'
le'(s'(n), s'(m)) → le'(n, m)
a'c'
a'd'

Types:
lessElements' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
lessE' :: nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
0' :: 0':s'
if' :: true':false' → true':false' → nil':cons':leaf':node' → nil':cons':leaf':node' → 0':s' → nil':cons':leaf':node'
le' :: 0':s' → 0':s' → true':false'
length' :: nil':cons':leaf':node' → 0':s'
toList' :: nil':cons':leaf':node' → nil':cons':leaf':node'
true' :: true':false'
false' :: true':false'
s' :: 0':s' → 0':s'
nil' :: nil':cons':leaf':node'
cons' :: nil':cons':leaf':node' → nil':cons':leaf':node'
leaf' :: nil':cons':leaf':node'
node' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
append' :: nil':cons':leaf':node' → nil':cons':leaf':node' → nil':cons':leaf':node'
a' :: c':d'
c' :: c':d'
d' :: c':d'
_hole_nil':cons':leaf':node'1 :: nil':cons':leaf':node'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_c':d'4 :: c':d'
_gen_nil':cons':leaf':node'5 :: Nat → nil':cons':leaf':node'
_gen_0':s'6 :: Nat → 0':s'

Lemmas:
le'(_gen_0':s'6(+(1, _n8)), _gen_0':s'6(_n8)) → false', rt ∈ Ω(1 + n8)
length'(_gen_nil':cons':leaf':node'5(_n906)) → _gen_0':s'6(_n906), rt ∈ Ω(1 + n906)
append'(_gen_nil':cons':leaf':node'5(_n1502), _gen_nil':cons':leaf':node'5(b)) → _gen_nil':cons':leaf':node'5(+(_n1502, b)), rt ∈ Ω(1 + n1502)

Generator Equations:
_gen_nil':cons':leaf':node'5(0) ⇔ nil'
_gen_nil':cons':leaf':node'5(+(x, 1)) ⇔ cons'(_gen_nil':cons':leaf':node'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
le'(_gen_0':s'6(+(1, _n8)), _gen_0':s'6(_n8)) → false', rt ∈ Ω(1 + n8)