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:

a__zeros → cons(0, zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeros → zeros
a__tail(X) → tail(X)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a__tail(cons(X, tail(cons(X134_3, X235_3)))) →⁺ a__tail(cons(mark(X134_3), X235_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X235_3 / tail(cons(X134_3, X235_3))].
The result substitution is [X / mark(X134_3)].

(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:

a__zeros → cons(0', zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
a__zeros → zeros
a__tail(X) → tail(X)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
a__zeros → zeros
a__tail(X) → tail(X)

Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail

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

Heuristically decided to analyse the following defined symbols:
a__tail, mark

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

(8) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
a__zeros → zeros
a__tail(X) → tail(X)

Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail

Generator Equations:
gen_0':zeros:cons:tail2_0(0) ⇔ 0'
gen_0':zeros:cons:tail2_0(+(x, 1)) ⇔ cons(gen_0':zeros:cons:tail2_0(x), 0')

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

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

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

Proved the following rewrite lemma:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
mark(gen_0':zeros:cons:tail2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_0':zeros:cons:tail2_0(+(n4_0, 1))) →_R^Ω(1)
cons(mark(gen_0':zeros:cons:tail2_0(n4_0)), 0') →_IH
cons(gen_0':zeros:cons:tail2_0(c5_0), 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:
a__zeros → cons(0', zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
a__zeros → zeros
a__tail(X) → tail(X)

Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail

Lemmas:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':zeros:cons:tail2_0(0) ⇔ 0'
gen_0':zeros:cons:tail2_0(+(x, 1)) ⇔ cons(gen_0':zeros:cons:tail2_0(x), 0')

The following defined symbols remain to be analysed:
a__tail

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

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

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

(13) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
a__zeros → zeros
a__tail(X) → tail(X)

Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail

Lemmas:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':zeros:cons:tail2_0(0) ⇔ 0'
gen_0':zeros:cons:tail2_0(+(x, 1)) ⇔ cons(gen_0':zeros:cons:tail2_0(x), 0')

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
a__zeros → cons(0', zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
a__zeros → zeros
a__tail(X) → tail(X)

Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail

Lemmas:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':zeros:cons:tail2_0(0) ⇔ 0'
gen_0':zeros:cons:tail2_0(+(x, 1)) ⇔ cons(gen_0':zeros:cons:tail2_0(x), 0')

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)