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:

odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(x)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
odd(Cons(x, Cons(x3_0, xs4_0))) →⁺ odd(xs4_0)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [xs4_0 / Cons(x, Cons(x3_0, xs4_0))].
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:

odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(x)

Types:
odd :: Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
even :: Cons:Nil → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
evenodd :: Cons:Nil → False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_a3_0 :: a
gen_Cons:Nil4_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
odd, even

They will be analysed ascendingly in the following order:
odd = even

(8) Obligation:

Innermost TRS:
Rules:
odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(x)

Types:
odd :: Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
even :: Cons:Nil → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
evenodd :: Cons:Nil → False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_a3_0 :: a
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil4_0(x))

The following defined symbols remain to be analysed:
even, odd

They will be analysed ascendingly in the following order:
odd = even

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

Proved the following rewrite lemma:
even(gen_Cons:Nil4_0(*(2, n6_0))) → True, rt ∈ Ω(1 + n6₀)

Induction Base:
even(gen_Cons:Nil4_0(*(2, 0))) →_R^Ω(1)
True

Induction Step:
even(gen_Cons:Nil4_0(*(2, +(n6_0, 1)))) →_R^Ω(1)
odd(gen_Cons:Nil4_0(+(1, *(2, n6_0)))) →_R^Ω(1)
even(gen_Cons:Nil4_0(*(2, n6_0))) →_IH
True

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:
odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(x)

Types:
odd :: Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
even :: Cons:Nil → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
evenodd :: Cons:Nil → False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_a3_0 :: a
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
even(gen_Cons:Nil4_0(*(2, n6_0))) → True, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil4_0(x))

The following defined symbols remain to be analysed:
odd

They will be analysed ascendingly in the following order:
odd = even

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

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

(13) Obligation:

Innermost TRS:
Rules:
odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(x)

Types:
odd :: Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
even :: Cons:Nil → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
evenodd :: Cons:Nil → False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_a3_0 :: a
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
even(gen_Cons:Nil4_0(*(2, n6_0))) → True, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil4_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
even(gen_Cons:Nil4_0(*(2, n6_0))) → True, rt ∈ Ω(1 + n6₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(x)

Types:
odd :: Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
even :: Cons:Nil → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
evenodd :: Cons:Nil → False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_a3_0 :: a
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
even(gen_Cons:Nil4_0(*(2, n6_0))) → True, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil4_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
even(gen_Cons:Nil4_0(*(2, n6_0))) → True, rt ∈ Ω(1 + n6₀)

(18) BOUNDS(n^1, INF)