The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

naiverev(Cons(x, xs)) → app(naiverev(xs), Cons(x, Nil))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
naiverev(Cons(x, xs)) →⁺ app(naiverev(xs), Cons(x, Nil))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [xs / Cons(x, xs)].
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:

naiverev(Cons(x, xs)) → app(naiverev(xs), Cons(x, Nil))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

S is empty.
Rewrite Strategy: INNERMOST

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

Sliced the following arguments:
Cons/0

(6) Obligation:

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

naiverev(Cons(xs)) → app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) → Cons(app(xs, ys))
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) → app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) → Cons(app(xs, ys))
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

Types:
naiverev :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
app :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
naiverev, app

They will be analysed ascendingly in the following order:
app < naiverev

(10) Obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) → app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) → Cons(app(xs, ys))
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

Types:
naiverev :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
app :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
app, naiverev

They will be analysed ascendingly in the following order:
app < naiverev

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

Proved the following rewrite lemma:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
app(gen_Cons:Nil3_0(0), gen_Cons:Nil3_0(b)) →_R^Ω(1)
gen_Cons:Nil3_0(b)

Induction Step:
app(gen_Cons:Nil3_0(+(n5_0, 1)), gen_Cons:Nil3_0(b)) →_R^Ω(1)
Cons(app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b))) →_IH
Cons(gen_Cons:Nil3_0(+(b, 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:
naiverev(Cons(xs)) → app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) → Cons(app(xs, ys))
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

Types:
naiverev :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
app :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
naiverev

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

Proved the following rewrite lemma:
naiverev(gen_Cons:Nil3_0(n499_0)) → gen_Cons:Nil3_0(n499_0), rt ∈ Ω(1 + n499₀ + n499₀²)

Induction Base:
naiverev(gen_Cons:Nil3_0(0)) →_R^Ω(1)
Nil

Induction Step:
naiverev(gen_Cons:Nil3_0(+(n499_0, 1))) →_R^Ω(1)
app(naiverev(gen_Cons:Nil3_0(n499_0)), Cons(Nil)) →_IH
app(gen_Cons:Nil3_0(c500_0), Cons(Nil)) →_L^{Ω(1 + n499₀)}
gen_Cons:Nil3_0(+(n499_0, +(0, 1)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) → app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) → Cons(app(xs, ys))
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

Types:
naiverev :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
app :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
naiverev(gen_Cons:Nil3_0(n499_0)) → gen_Cons:Nil3_0(n499_0), rt ∈ Ω(1 + n499₀ + n499₀²)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
naiverev(gen_Cons:Nil3_0(n499_0)) → gen_Cons:Nil3_0(n499_0), rt ∈ Ω(1 + n499₀ + n499₀²)

(18) BOUNDS(n^2, INF)

(19) Obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) → app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) → Cons(app(xs, ys))
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

Types:
naiverev :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
app :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
naiverev(gen_Cons:Nil3_0(n499_0)) → gen_Cons:Nil3_0(n499_0), rt ∈ Ω(1 + n499₀ + n499₀²)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
naiverev(gen_Cons:Nil3_0(n499_0)) → gen_Cons:Nil3_0(n499_0), rt ∈ Ω(1 + n499₀ + n499₀²)

(21) BOUNDS(n^2, INF)

(22) Obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) → app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) → Cons(app(xs, ys))
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
naiverev(Nil) → Nil
app(Nil, ys) → ys
goal(xs) → naiverev(xs)

Types:
naiverev :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
app :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(24) BOUNDS(n^1, INF)