The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0)
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(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
from(mark(X)) →⁺ mark(from(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 [ ].

(2) BOUNDS(n^1, INF)

(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(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

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

Heuristically decided to analyse the following defined symbols:
active, cons, from, s, length1, length, proper, top

They will be analysed ascendingly in the following order:
cons < active
from < active
s < active
length1 < active
length < active
active < top
cons < proper
from < proper
s < proper
length1 < proper
length < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

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

The following defined symbols remain to be analysed:
cons, active, from, s, length1, length, proper, top

They will be analysed ascendingly in the following order:
cons < active
from < active
s < active
length1 < active
length < active
active < top
cons < proper
from < proper
s < proper
length1 < proper
length < proper
proper < top

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

Proved the following rewrite lemma:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

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

Induction Step:
cons(gen_mark:nil:0':ok3_0(+(1, +(n5_0, 1))), gen_mark:nil:0':ok3_0(b)) →_R^Ω(1)
mark(cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b))) →_IH
mark(*4_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

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

The following defined symbols remain to be analysed:
from, active, s, length1, length, proper, top

They will be analysed ascendingly in the following order:
from < active
s < active
length1 < active
length < active
active < top
from < proper
s < proper
length1 < proper
length < proper
proper < top

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

Proved the following rewrite lemma:
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)

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

Induction Step:
from(gen_mark:nil:0':ok3_0(+(1, +(n830_0, 1)))) →_R^Ω(1)
mark(from(gen_mark:nil:0':ok3_0(+(1, n830_0)))) →_IH
mark(*4_0)

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)

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

The following defined symbols remain to be analysed:
s, active, length1, length, proper, top

They will be analysed ascendingly in the following order:
s < active
length1 < active
length < active
active < top
s < proper
length1 < proper
length < proper
proper < top

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

Proved the following rewrite lemma:
s(gen_mark:nil:0':ok3_0(+(1, n1319_0))) → *4_0, rt ∈ Ω(n1319₀)

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

Induction Step:
s(gen_mark:nil:0':ok3_0(+(1, +(n1319_0, 1)))) →_R^Ω(1)
mark(s(gen_mark:nil:0':ok3_0(+(1, n1319_0)))) →_IH
mark(*4_0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)
s(gen_mark:nil:0':ok3_0(+(1, n1319_0))) → *4_0, rt ∈ Ω(n1319₀)

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

The following defined symbols remain to be analysed:
length1, active, length, proper, top

They will be analysed ascendingly in the following order:
length1 < active
length < active
active < top
length1 < proper
length < proper
proper < top

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol length1.

(19) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)
s(gen_mark:nil:0':ok3_0(+(1, n1319_0))) → *4_0, rt ∈ Ω(n1319₀)

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

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

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

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(21) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)
s(gen_mark:nil:0':ok3_0(+(1, n1319_0))) → *4_0, rt ∈ Ω(n1319₀)

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

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(23) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)
s(gen_mark:nil:0':ok3_0(+(1, n1319_0))) → *4_0, rt ∈ Ω(n1319₀)

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

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

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

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(25) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)
s(gen_mark:nil:0':ok3_0(+(1, n1319_0))) → *4_0, rt ∈ Ω(n1319₀)

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

The following defined symbols remain to be analysed:
top

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(27) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)
s(gen_mark:nil:0':ok3_0(+(1, n1319_0))) → *4_0, rt ∈ Ω(n1319₀)

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

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(29) BOUNDS(n^1, INF)

(30) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)
s(gen_mark:nil:0':ok3_0(+(1, n1319_0))) → *4_0, rt ∈ Ω(n1319₀)

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

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(32) BOUNDS(n^1, INF)

(33) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:nil:0':ok3_0(+(1, n830_0))) → *4_0, rt ∈ Ω(n830₀)

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

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(35) BOUNDS(n^1, INF)

(36) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0')
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0') → ok(0')
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:0':ok → mark:nil:0':ok
from :: mark:nil:0':ok → mark:nil:0':ok
mark :: mark:nil:0':ok → mark:nil:0':ok
cons :: mark:nil:0':ok → mark:nil:0':ok → mark:nil:0':ok
s :: mark:nil:0':ok → mark:nil:0':ok
length :: mark:nil:0':ok → mark:nil:0':ok
nil :: mark:nil:0':ok
0' :: mark:nil:0':ok
length1 :: mark:nil:0':ok → mark:nil:0':ok
proper :: mark:nil:0':ok → mark:nil:0':ok
ok :: mark:nil:0':ok → mark:nil:0':ok
top :: mark:nil:0':ok → top
hole_mark:nil:0':ok1_0 :: mark:nil:0':ok
hole_top2_0 :: top
gen_mark:nil:0':ok3_0 :: Nat → mark:nil:0':ok

Lemmas:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

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

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:nil:0':ok3_0(+(1, n5_0)), gen_mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(38) BOUNDS(n^1, INF)