The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

a(C(x1, x2), y) → C(a(x1, y), a(x2, C(x1, x2)))
a(Z, y) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1

The (relative) TRS S consists of the following rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a(C(x1, x2), y) →⁺ C(a(x1, y), a(x2, C(x1, x2)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x1 / C(x1, x2)].
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:

a(C(x1, x2), y) → C(a(x1, y), a(x2, C(x1, x2)))
a(Z, y) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1

The (relative) TRS S consists of the following rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Rewrite Strategy: INNERMOST

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
a/1

(6) Obligation:

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

a(C(x1, x2)) → C(a(x1), a(x2))
a(Z) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1

The (relative) TRS S consists of the following rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Innermost TRS:
Rules:
a(C(x1, x2)) → C(a(x1), a(x2))
a(Z) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
a :: C:Z → C:Z
C :: C:Z → C:Z → C:Z
Z :: C:Z
eqZList :: C:Z → C:Z → False:True
and :: False:True → False:True → False:True
False :: False:True
True :: False:True
second :: C:Z → C:Z
first :: C:Z → C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat → C:Z

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a, eqZList

(10) Obligation:

Innermost TRS:
Rules:
a(C(x1, x2)) → C(a(x1), a(x2))
a(Z) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
a :: C:Z → C:Z
C :: C:Z → C:Z → C:Z
Z :: C:Z
eqZList :: C:Z → C:Z → False:True
and :: False:True → False:True → False:True
False :: False:True
True :: False:True
second :: C:Z → C:Z
first :: C:Z → C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat → C:Z

Generator Equations:
gen_C:Z3_0(0) ⇔ Z
gen_C:Z3_0(+(x, 1)) ⇔ C(Z, gen_C:Z3_0(x))

The following defined symbols remain to be analysed:
a, eqZList

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a(gen_C:Z3_0(n5_0)) → gen_C:Z3_0(n5_0), rt ∈ Ω(1 + n5₀)

Induction Base:
a(gen_C:Z3_0(0)) →_R^Ω(1)
Z

Induction Step:
a(gen_C:Z3_0(+(n5_0, 1))) →_R^Ω(1)
C(a(Z), a(gen_C:Z3_0(n5_0))) →_R^Ω(1)
C(Z, a(gen_C:Z3_0(n5_0))) →_IH
C(Z, gen_C:Z3_0(c6_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
a(C(x1, x2)) → C(a(x1), a(x2))
a(Z) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
a :: C:Z → C:Z
C :: C:Z → C:Z → C:Z
Z :: C:Z
eqZList :: C:Z → C:Z → False:True
and :: False:True → False:True → False:True
False :: False:True
True :: False:True
second :: C:Z → C:Z
first :: C:Z → C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat → C:Z

Lemmas:
a(gen_C:Z3_0(n5_0)) → gen_C:Z3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_C:Z3_0(0) ⇔ Z
gen_C:Z3_0(+(x, 1)) ⇔ C(Z, gen_C:Z3_0(x))

The following defined symbols remain to be analysed:
eqZList

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
eqZList(gen_C:Z3_0(+(1, n148_0)), gen_C:Z3_0(n148_0)) → False, rt ∈ Ω(1 + n148₀)

Induction Base:
eqZList(gen_C:Z3_0(+(1, 0)), gen_C:Z3_0(0)) →_R^Ω(1)
False

Induction Step:
eqZList(gen_C:Z3_0(+(1, +(n148_0, 1))), gen_C:Z3_0(+(n148_0, 1))) →_R^Ω(1)
and(eqZList(Z, Z), eqZList(gen_C:Z3_0(+(1, n148_0)), gen_C:Z3_0(n148_0))) →_R^Ω(1)
and(True, eqZList(gen_C:Z3_0(+(1, n148_0)), gen_C:Z3_0(n148_0))) →_IH
and(True, False) →_R^Ω(0)
False

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

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
a(C(x1, x2)) → C(a(x1), a(x2))
a(Z) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
a :: C:Z → C:Z
C :: C:Z → C:Z → C:Z
Z :: C:Z
eqZList :: C:Z → C:Z → False:True
and :: False:True → False:True → False:True
False :: False:True
True :: False:True
second :: C:Z → C:Z
first :: C:Z → C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat → C:Z

Lemmas:
a(gen_C:Z3_0(n5_0)) → gen_C:Z3_0(n5_0), rt ∈ Ω(1 + n5₀)
eqZList(gen_C:Z3_0(+(1, n148_0)), gen_C:Z3_0(n148_0)) → False, rt ∈ Ω(1 + n148₀)

Generator Equations:
gen_C:Z3_0(0) ⇔ Z
gen_C:Z3_0(+(x, 1)) ⇔ C(Z, gen_C:Z3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a(gen_C:Z3_0(n5_0)) → gen_C:Z3_0(n5_0), rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

Innermost TRS:
Rules:
a(C(x1, x2)) → C(a(x1), a(x2))
a(Z) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
a :: C:Z → C:Z
C :: C:Z → C:Z → C:Z
Z :: C:Z
eqZList :: C:Z → C:Z → False:True
and :: False:True → False:True → False:True
False :: False:True
True :: False:True
second :: C:Z → C:Z
first :: C:Z → C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat → C:Z

Lemmas:
a(gen_C:Z3_0(n5_0)) → gen_C:Z3_0(n5_0), rt ∈ Ω(1 + n5₀)
eqZList(gen_C:Z3_0(+(1, n148_0)), gen_C:Z3_0(n148_0)) → False, rt ∈ Ω(1 + n148₀)

Generator Equations:
gen_C:Z3_0(0) ⇔ Z
gen_C:Z3_0(+(x, 1)) ⇔ C(Z, gen_C:Z3_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a(gen_C:Z3_0(n5_0)) → gen_C:Z3_0(n5_0), rt ∈ Ω(1 + n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
a(C(x1, x2)) → C(a(x1), a(x2))
a(Z) → Z
eqZList(C(x1, x2), C(y1, y2)) → and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) → False
eqZList(Z, C(y1, y2)) → False
eqZList(Z, Z) → True
second(C(x1, x2)) → x2
first(C(x1, x2)) → x1
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
a :: C:Z → C:Z
C :: C:Z → C:Z → C:Z
Z :: C:Z
eqZList :: C:Z → C:Z → False:True
and :: False:True → False:True → False:True
False :: False:True
True :: False:True
second :: C:Z → C:Z
first :: C:Z → C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat → C:Z

Lemmas:
a(gen_C:Z3_0(n5_0)) → gen_C:Z3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_C:Z3_0(0) ⇔ Z
gen_C:Z3_0(+(x, 1)) ⇔ C(Z, gen_C:Z3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a(gen_C:Z3_0(n5_0)) → gen_C:Z3_0(n5_0), rt ∈ Ω(1 + n5₀)

(24) BOUNDS(n^1, INF)