### (0) Obligation:

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(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)
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))
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))
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))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
fib(mark(X)) →+ mark(fib(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative 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(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)
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))
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))
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))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

### (7) OrderProof (LOWER BOUND(ID) transformation)

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
fib < active
active < top
sel < proper
fib1 < proper
s < proper
cons < proper
fib < proper
proper < top

### (8) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(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
fib < active
active < top
sel < proper
fib1 < proper
s < proper
cons < proper
fib < proper
proper < top

### (9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

Induction Base:
sel(gen_0':mark:ok3_0(+(1, 0)), gen_0':mark:ok3_0(b))

Induction Step:
sel(gen_0':mark:ok3_0(+(1, +(n5_0, 1))), gen_0':mark:ok3_0(b)) →RΩ(1)
mark(sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (11) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(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
fib < active
active < top
fib1 < proper
s < proper
cons < proper
fib < proper
proper < top

### (12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)

Induction Base:
fib1(gen_0':mark:ok3_0(+(1, 0)), gen_0':mark:ok3_0(b))

Induction Step:
fib1(gen_0':mark:ok3_0(+(1, +(n1209_0, 1))), gen_0':mark:ok3_0(b)) →RΩ(1)
mark(fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (14) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(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
fib < active
active < top
s < proper
cons < proper
fib < proper
proper < top

### (15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)

Induction Base:
s(gen_0':mark:ok3_0(+(1, 0)))

Induction Step:
s(gen_0':mark:ok3_0(+(1, +(n2717_0, 1)))) →RΩ(1)
mark(s(gen_0':mark:ok3_0(+(1, n2717_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (17) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(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
fib < active
active < top
cons < proper
fib < proper
proper < top

### (18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)

Induction Base:
cons(gen_0':mark:ok3_0(+(1, 0)), gen_0':mark:ok3_0(b))

Induction Step:
cons(gen_0':mark:ok3_0(+(1, +(n3401_0, 1))), gen_0':mark:ok3_0(b)) →RΩ(1)
mark(cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (20) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

The following defined symbols remain to be analysed:

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

### (21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
add(gen_0':mark:ok3_0(+(1, n5220_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n52200)

Induction Base:

Induction Step:
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (23) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)
add(gen_0':mark:ok3_0(+(1, n5220_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n52200)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(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

### (24) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
fib(gen_0':mark:ok3_0(+(1, n7542_0))) → *4_0, rt ∈ Ω(n75420)

Induction Base:
fib(gen_0':mark:ok3_0(+(1, 0)))

Induction Step:
fib(gen_0':mark:ok3_0(+(1, +(n7542_0, 1)))) →RΩ(1)
mark(fib(gen_0':mark:ok3_0(+(1, n7542_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (26) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)
add(gen_0':mark:ok3_0(+(1, n5220_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n52200)
fib(gen_0':mark:ok3_0(+(1, n7542_0))) → *4_0, rt ∈ Ω(n75420)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(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

### (27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (28) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)
add(gen_0':mark:ok3_0(+(1, n5220_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n52200)
fib(gen_0':mark:ok3_0(+(1, n7542_0))) → *4_0, rt ∈ Ω(n75420)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

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

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

### (29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (30) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)
add(gen_0':mark:ok3_0(+(1, n5220_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n52200)
fib(gen_0':mark:ok3_0(+(1, n7542_0))) → *4_0, rt ∈ Ω(n75420)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

The following defined symbols remain to be analysed:
top

### (31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (32) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)
add(gen_0':mark:ok3_0(+(1, n5220_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n52200)
fib(gen_0':mark:ok3_0(+(1, n7542_0))) → *4_0, rt ∈ Ω(n75420)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

### (35) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)
add(gen_0':mark:ok3_0(+(1, n5220_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n52200)
fib(gen_0':mark:ok3_0(+(1, n7542_0))) → *4_0, rt ∈ Ω(n75420)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

No more defined symbols left to analyse.

### (36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

### (38) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)
add(gen_0':mark:ok3_0(+(1, n5220_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n52200)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

No more defined symbols left to analyse.

### (39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

### (41) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)
cons(gen_0':mark:ok3_0(+(1, n3401_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n34010)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

No more defined symbols left to analyse.

### (42) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

### (44) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)
s(gen_0':mark:ok3_0(+(1, n2717_0))) → *4_0, rt ∈ Ω(n27170)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

No more defined symbols left to analyse.

### (45) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

### (47) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
fib1(gen_0':mark:ok3_0(+(1, n1209_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n12090)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

No more defined symbols left to analyse.

### (48) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

### (50) Obligation:

TRS:
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(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)
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))
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))
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))
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:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

Lemmas:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_0':mark:ok3_0(0) ⇔ 0'
gen_0':mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:ok3_0(x))

No more defined symbols left to analyse.

### (51) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
sel(gen_0':mark:ok3_0(+(1, n5_0)), gen_0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)