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(adx(zeros))
active(zeros) → mark(cons(0, zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(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
adx(mark(X)) →⁺ mark(adx(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(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(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(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

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

Heuristically decided to analyse the following defined symbols:
active, adx, cons, s, incr, hd, tl, proper, top

They will be analysed ascendingly in the following order:
adx < active
cons < active
s < active
incr < active
hd < active
tl < active
active < top
adx < proper
cons < proper
s < proper
incr < proper
hd < proper
tl < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

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

The following defined symbols remain to be analysed:
adx, active, cons, s, incr, hd, tl, proper, top

They will be analysed ascendingly in the following order:
adx < active
cons < active
s < active
incr < active
hd < active
tl < active
active < top
adx < proper
cons < proper
s < proper
incr < proper
hd < proper
tl < proper
proper < top

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

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

Induction Base:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, 0)))

Induction Step:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
mark(adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0)))) →_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(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

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

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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

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

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

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

The following defined symbols remain to be analysed:
incr, active, hd, tl, proper, top

They will be analysed ascendingly in the following order:
incr < active
hd < active
tl < active
active < top
incr < proper
hd < proper
tl < proper
proper < top

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)

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

Induction Step:
incr(gen_nats:zeros:mark:0':ok3_0(+(1, +(n384_0, 1)))) →_R^Ω(1)
mark(incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0)))) →_IH
mark(*4_0)

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

(17) Complex Obligation (BEST)

(18) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)

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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0))) → *4_0, rt ∈ Ω(n845₀)

Induction Base:
hd(gen_nats:zeros:mark:0':ok3_0(+(1, 0)))

Induction Step:
hd(gen_nats:zeros:mark:0':ok3_0(+(1, +(n845_0, 1)))) →_R^Ω(1)
mark(hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0)))) →_IH
mark(*4_0)

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

(20) Complex Obligation (BEST)

(21) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0))) → *4_0, rt ∈ Ω(n845₀)

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

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

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

(22) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1407_0))) → *4_0, rt ∈ Ω(n1407₀)

Induction Base:
tl(gen_nats:zeros:mark:0':ok3_0(+(1, 0)))

Induction Step:
tl(gen_nats:zeros:mark:0':ok3_0(+(1, +(n1407_0, 1)))) →_R^Ω(1)
mark(tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1407_0)))) →_IH
mark(*4_0)

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

(23) Complex Obligation (BEST)

(24) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0))) → *4_0, rt ∈ Ω(n845₀)
tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1407_0))) → *4_0, rt ∈ Ω(n1407₀)

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

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

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

(26) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0))) → *4_0, rt ∈ Ω(n845₀)
tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1407_0))) → *4_0, rt ∈ Ω(n1407₀)

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

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

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

(28) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0))) → *4_0, rt ∈ Ω(n845₀)
tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1407_0))) → *4_0, rt ∈ Ω(n1407₀)

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

The following defined symbols remain to be analysed:
top

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

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

(30) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0))) → *4_0, rt ∈ Ω(n845₀)
tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1407_0))) → *4_0, rt ∈ Ω(n1407₀)

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

(32) BOUNDS(n^1, INF)

(33) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0))) → *4_0, rt ∈ Ω(n845₀)
tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1407_0))) → *4_0, rt ∈ Ω(n1407₀)

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

(35) BOUNDS(n^1, INF)

(36) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n845_0))) → *4_0, rt ∈ Ω(n845₀)

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

(38) BOUNDS(n^1, INF)

(39) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n384_0))) → *4_0, rt ∈ Ω(n384₀)

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

No more defined symbols left to analyse.

(40) LowerBoundsProof (EQUIVALENT transformation)

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

(41) BOUNDS(n^1, INF)

(42) Obligation:

TRS:
Rules:
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0', zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok → nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok → top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat → nats:zeros:mark:0':ok

Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

No more defined symbols left to analyse.

(43) LowerBoundsProof (EQUIVALENT transformation)

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

(44) BOUNDS(n^1, INF)