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

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(fib(X)) → fib(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(fib1(X1, X2)) → fib1(active(X1), X2)
active(fib1(X1, X2)) → fib1(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fib(mark(X)) → mark(fib(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fib(X)) → fib(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) → fib1(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Heuristically decided to analyse the following defined symbols:
active', sel', fib1', s', cons', add', fib', proper', top'

They will be analysed ascendingly in the following order:
sel' < active'
fib1' < active'
s' < active'
cons' < active'
add' < active'
fib' < active'
active' < top'
sel' < proper'
fib1' < proper'
s' < proper'
cons' < proper'
add' < proper'
fib' < proper'
proper' < top'

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
sel', active', fib1', s', cons', add', fib', proper', top'

They will be analysed ascendingly in the following order:
sel' < active'
fib1' < active'
s' < active'
cons' < active'
add' < active'
fib' < active'
active' < top'
sel' < proper'
fib1' < proper'
s' < proper'
cons' < proper'
add' < proper'
fib' < proper'
proper' < top'

Proved the following rewrite lemma:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)

Induction Base:
sel'(_gen_0':mark':ok'3(+(1, 0)), _gen_0':mark':ok'3(b))

Induction Step:
sel'(_gen_0':mark':ok'3(+(1, +(_$n6, 1))), _gen_0':mark':ok'3(_b826)) →RΩ(1)
mark'(sel'(_gen_0':mark':ok'3(+(1, _$n6)), _gen_0':mark':ok'3(_b826))) →IH
mark'(_*4)

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
fib1', active', s', cons', add', fib', proper', top'

They will be analysed ascendingly in the following order:
fib1' < active'
s' < active'
cons' < active'
add' < active'
fib' < active'
active' < top'
fib1' < proper'
s' < proper'
cons' < proper'
add' < proper'
fib' < proper'
proper' < top'

Proved the following rewrite lemma:
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)

Induction Base:
fib1'(_gen_0':mark':ok'3(+(1, 0)), _gen_0':mark':ok'3(b))

Induction Step:
fib1'(_gen_0':mark':ok'3(+(1, +(_$n2670, 1))), _gen_0':mark':ok'3(_b3814)) →RΩ(1)
mark'(fib1'(_gen_0':mark':ok'3(+(1, _$n2670)), _gen_0':mark':ok'3(_b3814))) →IH
mark'(_*4)

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
s', active', cons', add', fib', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
cons' < active'
add' < active'
fib' < active'
active' < top'
s' < proper'
cons' < proper'
add' < proper'
fib' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_0':mark':ok'3(+(1, _n5701))) → _*4, rt ∈ Ω(n5701)

Induction Base:
s'(_gen_0':mark':ok'3(+(1, 0)))

Induction Step:
s'(_gen_0':mark':ok'3(+(1, +(_$n5702, 1)))) →RΩ(1)
mark'(s'(_gen_0':mark':ok'3(+(1, _$n5702)))) →IH
mark'(_*4)

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)
s'(_gen_0':mark':ok'3(+(1, _n5701))) → _*4, rt ∈ Ω(n5701)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
cons', active', add', fib', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
add' < active'
fib' < active'
active' < top'
cons' < proper'
add' < proper'
fib' < proper'
proper' < top'

Proved the following rewrite lemma:
cons'(_gen_0':mark':ok'3(+(1, _n7600)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n7600)

Induction Base:
cons'(_gen_0':mark':ok'3(+(1, 0)), _gen_0':mark':ok'3(b))

Induction Step:
cons'(_gen_0':mark':ok'3(+(1, +(_$n7601, 1))), _gen_0':mark':ok'3(_b9069)) →RΩ(1)
mark'(cons'(_gen_0':mark':ok'3(+(1, _$n7601)), _gen_0':mark':ok'3(_b9069))) →IH
mark'(_*4)

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)
s'(_gen_0':mark':ok'3(+(1, _n5701))) → _*4, rt ∈ Ω(n5701)
cons'(_gen_0':mark':ok'3(+(1, _n7600)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n7600)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
add', active', fib', proper', top'

They will be analysed ascendingly in the following order:
add' < active'
fib' < active'
active' < top'
add' < proper'
fib' < proper'
proper' < top'

Proved the following rewrite lemma:
add'(_gen_0':mark':ok'3(+(1, _n11022)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n11022)

Induction Base:
add'(_gen_0':mark':ok'3(+(1, 0)), _gen_0':mark':ok'3(b))

Induction Step:
add'(_gen_0':mark':ok'3(+(1, +(_$n11023, 1))), _gen_0':mark':ok'3(_b13031)) →RΩ(1)
mark'(add'(_gen_0':mark':ok'3(+(1, _$n11023)), _gen_0':mark':ok'3(_b13031))) →IH
mark'(_*4)

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)
s'(_gen_0':mark':ok'3(+(1, _n5701))) → _*4, rt ∈ Ω(n5701)
cons'(_gen_0':mark':ok'3(+(1, _n7600)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n7600)
add'(_gen_0':mark':ok'3(+(1, _n11022)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n11022)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
fib', active', proper', top'

They will be analysed ascendingly in the following order:
fib' < active'
active' < top'
fib' < proper'
proper' < top'

Proved the following rewrite lemma:
fib'(_gen_0':mark':ok'3(+(1, _n15031))) → _*4, rt ∈ Ω(n15031)

Induction Base:
fib'(_gen_0':mark':ok'3(+(1, 0)))

Induction Step:
fib'(_gen_0':mark':ok'3(+(1, +(_$n15032, 1)))) →RΩ(1)
mark'(fib'(_gen_0':mark':ok'3(+(1, _$n15032)))) →IH
mark'(_*4)

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)
s'(_gen_0':mark':ok'3(+(1, _n5701))) → _*4, rt ∈ Ω(n5701)
cons'(_gen_0':mark':ok'3(+(1, _n7600)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n7600)
add'(_gen_0':mark':ok'3(+(1, _n11022)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n11022)
fib'(_gen_0':mark':ok'3(+(1, _n15031))) → _*4, rt ∈ Ω(n15031)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
active', proper', top'

They will be analysed ascendingly in the following order:
active' < top'
proper' < top'

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)
s'(_gen_0':mark':ok'3(+(1, _n5701))) → _*4, rt ∈ Ω(n5701)
cons'(_gen_0':mark':ok'3(+(1, _n7600)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n7600)
add'(_gen_0':mark':ok'3(+(1, _n11022)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n11022)
fib'(_gen_0':mark':ok'3(+(1, _n15031))) → _*4, rt ∈ Ω(n15031)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
proper', top'

They will be analysed ascendingly in the following order:
proper' < top'

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)
s'(_gen_0':mark':ok'3(+(1, _n5701))) → _*4, rt ∈ Ω(n5701)
cons'(_gen_0':mark':ok'3(+(1, _n7600)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n7600)
add'(_gen_0':mark':ok'3(+(1, _n11022)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n11022)
fib'(_gen_0':mark':ok'3(+(1, _n15031))) → _*4, rt ∈ Ω(n15031)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

The following defined symbols remain to be analysed:
top'

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

Rules:
active'(fib'(N)) → mark'(sel'(N, fib1'(s'(0'), s'(0'))))
active'(fib1'(X, Y)) → mark'(cons'(X, fib1'(Y, add'(X, Y))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(fib'(X)) → fib'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(fib1'(X1, X2)) → fib1'(active'(X1), X2)
active'(fib1'(X1, X2)) → fib1'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
fib'(mark'(X)) → mark'(fib'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
fib1'(mark'(X1), X2) → mark'(fib1'(X1, X2))
fib1'(X1, mark'(X2)) → mark'(fib1'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
proper'(fib'(X)) → fib'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(fib1'(X1, X2)) → fib1'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
fib'(ok'(X)) → ok'(fib'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
fib1'(ok'(X1), ok'(X2)) → ok'(fib1'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':ok' → 0':mark':ok'
fib' :: 0':mark':ok' → 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
sel' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
fib1' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
add' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'

Lemmas:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
fib1'(_gen_0':mark':ok'3(+(1, _n2669)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n2669)
s'(_gen_0':mark':ok'3(+(1, _n5701))) → _*4, rt ∈ Ω(n5701)
cons'(_gen_0':mark':ok'3(+(1, _n7600)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n7600)
add'(_gen_0':mark':ok'3(+(1, _n11022)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n11022)
fib'(_gen_0':mark':ok'3(+(1, _n15031))) → _*4, rt ∈ Ω(n15031)

Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
sel'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)