### (0) Obligation:

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

active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
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
first(mark(X1), X2) →+ mark(first(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
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(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
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(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

Heuristically decided to analyse the following defined symbols:
active, cons, first, from, s, proper, top

They will be analysed ascendingly in the following order:
cons < active
first < active
from < active
s < active
active < top
cons < proper
first < proper
from < proper
s < proper
proper < top

### (8) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

The following defined symbols remain to be analysed:
cons, active, first, from, s, proper, top

They will be analysed ascendingly in the following order:
cons < active
first < active
from < active
s < active
active < top
cons < proper
first < proper
from < proper
s < proper
proper < top

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

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

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

Induction Step:
cons(gen_0':nil:mark:ok3_0(+(1, +(n5_0, 1))), gen_0':nil:mark:ok3_0(b)) →RΩ(1)
mark(cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil: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(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

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

The following defined symbols remain to be analysed:
first, active, from, s, proper, top

They will be analysed ascendingly in the following order:
first < active
from < active
s < active
active < top
first < proper
from < proper
s < proper
proper < top

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

Proved the following rewrite lemma:
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)

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

Induction Step:
first(gen_0':nil:mark:ok3_0(+(1, +(n842_0, 1))), gen_0':nil:mark:ok3_0(b)) →RΩ(1)
mark(first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil: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(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)

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

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

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

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

Proved the following rewrite lemma:
from(gen_0':nil:mark:ok3_0(+(1, n2182_0))) → *4_0, rt ∈ Ω(n21820)

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

Induction Step:
from(gen_0':nil:mark:ok3_0(+(1, +(n2182_0, 1)))) →RΩ(1)
mark(from(gen_0':nil:mark:ok3_0(+(1, n2182_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(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)
from(gen_0':nil:mark:ok3_0(+(1, n2182_0))) → *4_0, rt ∈ Ω(n21820)

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

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

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

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

Proved the following rewrite lemma:
s(gen_0':nil:mark:ok3_0(+(1, n2824_0))) → *4_0, rt ∈ Ω(n28240)

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

Induction Step:
s(gen_0':nil:mark:ok3_0(+(1, +(n2824_0, 1)))) →RΩ(1)
mark(s(gen_0':nil:mark:ok3_0(+(1, n2824_0)))) →IH
mark(*4_0)

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

### (20) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)
from(gen_0':nil:mark:ok3_0(+(1, n2182_0))) → *4_0, rt ∈ Ω(n21820)
s(gen_0':nil:mark:ok3_0(+(1, n2824_0))) → *4_0, rt ∈ Ω(n28240)

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

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

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

### (22) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)
from(gen_0':nil:mark:ok3_0(+(1, n2182_0))) → *4_0, rt ∈ Ω(n21820)
s(gen_0':nil:mark:ok3_0(+(1, n2824_0))) → *4_0, rt ∈ Ω(n28240)

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

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

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

### (24) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)
from(gen_0':nil:mark:ok3_0(+(1, n2182_0))) → *4_0, rt ∈ Ω(n21820)
s(gen_0':nil:mark:ok3_0(+(1, n2824_0))) → *4_0, rt ∈ Ω(n28240)

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

The following defined symbols remain to be analysed:
top

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

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

### (26) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)
from(gen_0':nil:mark:ok3_0(+(1, n2182_0))) → *4_0, rt ∈ Ω(n21820)
s(gen_0':nil:mark:ok3_0(+(1, n2824_0))) → *4_0, rt ∈ Ω(n28240)

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

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

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

### (29) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)
from(gen_0':nil:mark:ok3_0(+(1, n2182_0))) → *4_0, rt ∈ Ω(n21820)
s(gen_0':nil:mark:ok3_0(+(1, n2824_0))) → *4_0, rt ∈ Ω(n28240)

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

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

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

### (32) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)
from(gen_0':nil:mark:ok3_0(+(1, n2182_0))) → *4_0, rt ∈ Ω(n21820)

Generator Equations:
gen_0':nil:mark:ok3_0(0) ⇔ 0'
gen_0':nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':nil: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:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

### (35) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

Lemmas:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
first(gen_0':nil:mark:ok3_0(+(1, n842_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n8420)

Generator Equations:
gen_0':nil:mark:ok3_0(0) ⇔ 0'
gen_0':nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':nil: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:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)

### (38) Obligation:

TRS:
Rules:
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
first :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

Generator Equations:
gen_0':nil:mark:ok3_0(0) ⇔ 0'
gen_0':nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_0':nil: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:
cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)