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(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(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
cons(mark(X1), X2) →⁺ mark(cons(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 [ ].

(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(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(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(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

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

Heuristically decided to analyse the following defined symbols:
active, cons, tail, proper, top

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

(8) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

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

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

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

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

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

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

Induction Step:
cons(gen_zeros:0':mark:ok3_0(+(1, +(n5_0, 1))), gen_zeros:0':mark:ok3_0(b)) →_R^Ω(1)
mark(cons(gen_zeros:0':mark:ok3_0(+(1, n5_0)), gen_zeros:0':mark: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(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

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

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

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

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

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

Proved the following rewrite lemma:
tail(gen_zeros:0':mark:ok3_0(+(1, n710_0))) → *4_0, rt ∈ Ω(n710₀)

Induction Base:
tail(gen_zeros:0':mark:ok3_0(+(1, 0)))

Induction Step:
tail(gen_zeros:0':mark:ok3_0(+(1, +(n710_0, 1)))) →_R^Ω(1)
mark(tail(gen_zeros:0':mark:ok3_0(+(1, n710_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(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

Lemmas:
cons(gen_zeros:0':mark:ok3_0(+(1, n5_0)), gen_zeros:0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
tail(gen_zeros:0':mark:ok3_0(+(1, n710_0))) → *4_0, rt ∈ Ω(n710₀)

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

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

Lemmas:
cons(gen_zeros:0':mark:ok3_0(+(1, n5_0)), gen_zeros:0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
tail(gen_zeros:0':mark:ok3_0(+(1, n710_0))) → *4_0, rt ∈ Ω(n710₀)

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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

Lemmas:
cons(gen_zeros:0':mark:ok3_0(+(1, n5_0)), gen_zeros:0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
tail(gen_zeros:0':mark:ok3_0(+(1, n710_0))) → *4_0, rt ∈ Ω(n710₀)

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

The following defined symbols remain to be analysed:
top

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

Lemmas:
cons(gen_zeros:0':mark:ok3_0(+(1, n5_0)), gen_zeros:0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
tail(gen_zeros:0':mark:ok3_0(+(1, n710_0))) → *4_0, rt ∈ Ω(n710₀)

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

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

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

(22) BOUNDS(n^1, INF)

(23) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

Lemmas:
cons(gen_zeros:0':mark:ok3_0(+(1, n5_0)), gen_zeros:0':mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
tail(gen_zeros:0':mark:ok3_0(+(1, n710_0))) → *4_0, rt ∈ Ω(n710₀)

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

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

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

(25) BOUNDS(n^1, INF)

(26) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:ok → zeros:0':mark:ok
zeros :: zeros:0':mark:ok
mark :: zeros:0':mark:ok → zeros:0':mark:ok
cons :: zeros:0':mark:ok → zeros:0':mark:ok → zeros:0':mark:ok
0' :: zeros:0':mark:ok
tail :: zeros:0':mark:ok → zeros:0':mark:ok
proper :: zeros:0':mark:ok → zeros:0':mark:ok
ok :: zeros:0':mark:ok → zeros:0':mark:ok
top :: zeros:0':mark:ok → top
hole_zeros:0':mark:ok1_0 :: zeros:0':mark:ok
hole_top2_0 :: top
gen_zeros:0':mark:ok3_0 :: Nat → zeros:0':mark:ok

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

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

(28) BOUNDS(n^1, INF)