The Runtime Complexity (innermost) 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:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__f, a__p, mark

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

(6) Obligation:

Innermost TRS:
Rules:
a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p

Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_0(x), 0')

The following defined symbols remain to be analysed:
a__p, a__f, mark

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

Innermost TRS:
Rules:
a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p

Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_0(x), 0')

The following defined symbols remain to be analysed:
mark, a__f

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

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

Proved the following rewrite lemma:
mark(gen_0':s:f:cons:p2_0(n10_0)) → gen_0':s:f:cons:p2_0(n10_0), rt ∈ Ω(1 + n10₀)

Induction Base:
mark(gen_0':s:f:cons:p2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_0':s:f:cons:p2_0(+(n10_0, 1))) →_R^Ω(1)
cons(mark(gen_0':s:f:cons:p2_0(n10_0)), 0') →_IH
cons(gen_0':s:f:cons:p2_0(c11_0), 0')

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p

Lemmas:
mark(gen_0':s:f:cons:p2_0(n10_0)) → gen_0':s:f:cons:p2_0(n10_0), rt ∈ Ω(1 + n10₀)

Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_0(x), 0')

The following defined symbols remain to be analysed:
a__f, a__p

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

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

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

(13) Obligation:

Innermost TRS:
Rules:
a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p

Lemmas:
mark(gen_0':s:f:cons:p2_0(n10_0)) → gen_0':s:f:cons:p2_0(n10_0), rt ∈ Ω(1 + n10₀)

Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_0(x), 0')

The following defined symbols remain to be analysed:
a__p

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

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

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

(15) Obligation:

Innermost TRS:
Rules:
a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p

Lemmas:
mark(gen_0':s:f:cons:p2_0(n10_0)) → gen_0':s:f:cons:p2_0(n10_0), rt ∈ Ω(1 + n10₀)

Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_0(x), 0')

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_0':s:f:cons:p2_0(n10_0)) → gen_0':s:f:cons:p2_0(n10_0), rt ∈ Ω(1 + n10₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p

Lemmas:
mark(gen_0':s:f:cons:p2_0(n10_0)) → gen_0':s:f:cons:p2_0(n10_0), rt ∈ Ω(1 + n10₀)

Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_0(x), 0')

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_0':s:f:cons:p2_0(n10_0)) → gen_0':s:f:cons:p2_0(n10_0), rt ∈ Ω(1 + n10₀)

(20) BOUNDS(n^1, INF)