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(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(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
f(mark(X)) →⁺ mark(f(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(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(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(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

Heuristically decided to analyse the following defined symbols:
active, cons, f, s, p, proper, top

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

(8) Obligation:

TRS:
Rules:
active(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

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

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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
active(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

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

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

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
active(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

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

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

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

TRS:
Rules:
active(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

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

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

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

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

(16) Obligation:

TRS:
Rules:
active(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

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

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

(18) Obligation:

TRS:
Rules:
active(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

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

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

(20) Obligation:

TRS:
Rules:
active(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

The following defined symbols remain to be analysed:
top

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

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

(22) Obligation:

TRS:
Rules:
active(f(0')) → mark(cons(0', f(s(0'))))
active(f(s(0'))) → mark(f(p(s(0'))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0') → ok(0')
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:ok → 0':mark:ok
f :: 0':mark:ok → 0':mark:ok
0' :: 0':mark:ok
mark :: 0':mark:ok → 0':mark:ok
cons :: 0':mark:ok → 0':mark:ok → 0':mark:ok
s :: 0':mark:ok → 0':mark:ok
p :: 0':mark:ok → 0':mark:ok
proper :: 0':mark:ok → 0':mark:ok
ok :: 0':mark:ok → 0':mark:ok
top :: 0':mark:ok → top
hole_0':mark:ok1_0 :: 0':mark:ok
hole_top2_0 :: top
gen_0':mark:ok3_0 :: Nat → 0':mark:ok

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

No more defined symbols left to analyse.