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(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
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(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
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(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

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

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

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

(8) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds:ok3_0(x))

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

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

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

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

Induction Base:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)), gen_nats:0':mark:pairs:odds:ok3_0(b))

Induction Step:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n5_0, 1))), gen_nats:0':mark:pairs:odds:ok3_0(b)) →_R^Ω(1)
mark(cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds: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(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

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

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds:ok3_0(x))

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

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

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

Proved the following rewrite lemma:
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)

Induction Base:
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)))

Induction Step:
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n926_0, 1)))) →_R^Ω(1)
mark(incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_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(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds:ok3_0(x))

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

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

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

Proved the following rewrite lemma:
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)

Induction Base:
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)))

Induction Step:
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n1439_0, 1)))) →_R^Ω(1)
mark(s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_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(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds:ok3_0(x))

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

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

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

Proved the following rewrite lemma:
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0))) → *4_0, rt ∈ Ω(n2053₀)

Induction Base:
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)))

Induction Step:
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n2053_0, 1)))) →_R^Ω(1)
mark(head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0)))) →_IH
mark(*4_0)

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

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0))) → *4_0, rt ∈ Ω(n2053₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds: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

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

Proved the following rewrite lemma:
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2768_0))) → *4_0, rt ∈ Ω(n2768₀)

Induction Base:
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)))

Induction Step:
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n2768_0, 1)))) →_R^Ω(1)
mark(tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2768_0)))) →_IH
mark(*4_0)

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

(22) Complex Obligation (BEST)

(23) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0))) → *4_0, rt ∈ Ω(n2053₀)
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2768_0))) → *4_0, rt ∈ Ω(n2768₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds: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

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

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

(25) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0))) → *4_0, rt ∈ Ω(n2053₀)
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2768_0))) → *4_0, rt ∈ Ω(n2768₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds:ok3_0(x))

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

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

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

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

(27) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0))) → *4_0, rt ∈ Ω(n2053₀)
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2768_0))) → *4_0, rt ∈ Ω(n2768₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds:ok3_0(x))

The following defined symbols remain to be analysed:
top

(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(29) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0))) → *4_0, rt ∈ Ω(n2053₀)
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2768_0))) → *4_0, rt ∈ Ω(n2768₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds: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_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0))) → *4_0, rt ∈ Ω(n2053₀)
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2768_0))) → *4_0, rt ∈ Ω(n2768₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds: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_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(34) BOUNDS(n^1, INF)

(35) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2053_0))) → *4_0, rt ∈ Ω(n2053₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds: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_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(37) BOUNDS(n^1, INF)

(38) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1439_0))) → *4_0, rt ∈ Ω(n1439₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds: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_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(40) BOUNDS(n^1, INF)

(41) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n926_0))) → *4_0, rt ∈ Ω(n926₀)

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds:ok3_0(x))

No more defined symbols left to analyse.

(42) LowerBoundsProof (EQUIVALENT transformation)

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

(43) BOUNDS(n^1, INF)

(44) Obligation:

TRS:
Rules:
active(nats) → mark(cons(0', incr(nats)))
active(pairs) → mark(cons(0', incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok → nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok → top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat → nats:0':mark:pairs:odds:ok

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

Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) ⇔ nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) ⇔ mark(gen_nats:0':mark:pairs:odds:ok3_0(x))

No more defined symbols left to analyse.

(45) LowerBoundsProof (EQUIVALENT transformation)

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

(46) BOUNDS(n^1, INF)