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:

revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(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:

revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V(n), rest) → revconsapp(rest, V(n))
deeprevapp(N, rest) → rest
revconsapp(V(n), r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V(n)) → V(n)
deeprev(N) → N
second(V(n)) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V(n)) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V(n)) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V(n)) → False
isEmptyT(N) → True
first(V(n)) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
V/0

(4) Obligation:

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

revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V, rest) → revconsapp(rest, V)
deeprevapp(N, rest) → rest
revconsapp(V, r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V) → V
deeprev(N) → N
second(V) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V) → False
isEmptyT(N) → True
first(V) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V, rest) → revconsapp(rest, V)
deeprevapp(N, rest) → rest
revconsapp(V, r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V) → V
deeprev(N) → N
second(V) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V) → False
isEmptyT(N) → True
first(V) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_False:True2_0 :: False:True
gen_C:V:N3_0 :: Nat → C:V:N

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

Heuristically decided to analyse the following defined symbols:
revconsapp, deeprevapp

They will be analysed ascendingly in the following order:
revconsapp < deeprevapp

(8) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V, rest) → revconsapp(rest, V)
deeprevapp(N, rest) → rest
revconsapp(V, r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V) → V
deeprev(N) → N
second(V) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V) → False
isEmptyT(N) → True
first(V) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_False:True2_0 :: False:True
gen_C:V:N3_0 :: Nat → C:V:N

Generator Equations:
gen_C:V:N3_0(0) ⇔ V
gen_C:V:N3_0(+(x, 1)) ⇔ C(V, gen_C:V:N3_0(x))

The following defined symbols remain to be analysed:
revconsapp, deeprevapp

They will be analysed ascendingly in the following order:
revconsapp < deeprevapp

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

Proved the following rewrite lemma:
revconsapp(gen_C:V:N3_0(n5_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
revconsapp(gen_C:V:N3_0(0), gen_C:V:N3_0(b)) →_R^Ω(1)
gen_C:V:N3_0(b)

Induction Step:
revconsapp(gen_C:V:N3_0(+(n5_0, 1)), gen_C:V:N3_0(b)) →_R^Ω(1)
revconsapp(gen_C:V:N3_0(n5_0), C(V, gen_C:V:N3_0(b))) →_IH
gen_C:V:N3_0(+(+(b, 1), c6_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:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V, rest) → revconsapp(rest, V)
deeprevapp(N, rest) → rest
revconsapp(V, r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V) → V
deeprev(N) → N
second(V) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V) → False
isEmptyT(N) → True
first(V) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_False:True2_0 :: False:True
gen_C:V:N3_0 :: Nat → C:V:N

Lemmas:
revconsapp(gen_C:V:N3_0(n5_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_C:V:N3_0(0) ⇔ V
gen_C:V:N3_0(+(x, 1)) ⇔ C(V, gen_C:V:N3_0(x))

The following defined symbols remain to be analysed:
deeprevapp

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
deeprevapp(gen_C:V:N3_0(n939_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n939_0, b)), rt ∈ Ω(1 + b + n939₀)

Induction Base:
deeprevapp(gen_C:V:N3_0(0), gen_C:V:N3_0(b)) →_R^Ω(1)
revconsapp(gen_C:V:N3_0(b), V) →_L^{Ω(1 + b)}
gen_C:V:N3_0(+(b, 0))

Induction Step:
deeprevapp(gen_C:V:N3_0(+(n939_0, 1)), gen_C:V:N3_0(b)) →_R^Ω(1)
deeprevapp(gen_C:V:N3_0(n939_0), C(V, gen_C:V:N3_0(b))) →_IH
gen_C:V:N3_0(+(+(b, 1), c940_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V, rest) → revconsapp(rest, V)
deeprevapp(N, rest) → rest
revconsapp(V, r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V) → V
deeprev(N) → N
second(V) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V) → False
isEmptyT(N) → True
first(V) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_False:True2_0 :: False:True
gen_C:V:N3_0 :: Nat → C:V:N

Lemmas:
revconsapp(gen_C:V:N3_0(n5_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
deeprevapp(gen_C:V:N3_0(n939_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n939_0, b)), rt ∈ Ω(1 + b + n939₀)

Generator Equations:
gen_C:V:N3_0(0) ⇔ V
gen_C:V:N3_0(+(x, 1)) ⇔ C(V, gen_C:V:N3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
revconsapp(gen_C:V:N3_0(n5_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V, rest) → revconsapp(rest, V)
deeprevapp(N, rest) → rest
revconsapp(V, r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V) → V
deeprev(N) → N
second(V) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V) → False
isEmptyT(N) → True
first(V) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_False:True2_0 :: False:True
gen_C:V:N3_0 :: Nat → C:V:N

Lemmas:
revconsapp(gen_C:V:N3_0(n5_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
deeprevapp(gen_C:V:N3_0(n939_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n939_0, b)), rt ∈ Ω(1 + b + n939₀)

Generator Equations:
gen_C:V:N3_0(0) ⇔ V
gen_C:V:N3_0(+(x, 1)) ⇔ C(V, gen_C:V:N3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
revconsapp(gen_C:V:N3_0(n5_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
revconsapp(C(x1, x2), r) → revconsapp(x2, C(x1, r))
deeprevapp(C(x1, x2), rest) → deeprevapp(x2, C(x1, rest))
deeprevapp(V, rest) → revconsapp(rest, V)
deeprevapp(N, rest) → rest
revconsapp(V, r) → r
revconsapp(N, r) → r
deeprev(C(x1, x2)) → deeprevapp(C(x1, x2), N)
deeprev(V) → V
deeprev(N) → N
second(V) → N
second(C(x1, x2)) → x2
isVal(C(x1, x2)) → False
isVal(V) → True
isVal(N) → False
isNotEmptyT(C(x1, x2)) → True
isNotEmptyT(V) → False
isNotEmptyT(N) → False
isEmptyT(C(x1, x2)) → False
isEmptyT(V) → False
isEmptyT(N) → True
first(V) → N
first(C(x1, x2)) → x1
goal(x) → deeprev(x)

Types:
revconsapp :: C:V:N → C:V:N → C:V:N
C :: C:V:N → C:V:N → C:V:N
deeprevapp :: C:V:N → C:V:N → C:V:N
V :: C:V:N
N :: C:V:N
deeprev :: C:V:N → C:V:N
second :: C:V:N → C:V:N
isVal :: C:V:N → False:True
False :: False:True
True :: False:True
isNotEmptyT :: C:V:N → False:True
isEmptyT :: C:V:N → False:True
first :: C:V:N → C:V:N
goal :: C:V:N → C:V:N
hole_C:V:N1_0 :: C:V:N
hole_False:True2_0 :: False:True
gen_C:V:N3_0 :: Nat → C:V:N

Lemmas:
revconsapp(gen_C:V:N3_0(n5_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_C:V:N3_0(0) ⇔ V
gen_C:V:N3_0(+(x, 1)) ⇔ C(V, gen_C:V:N3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
revconsapp(gen_C:V:N3_0(n5_0), gen_C:V:N3_0(b)) → gen_C:V:N3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(22) BOUNDS(n^1, INF)