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:

lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))

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
lte(Cons(x', xs'), Cons(x, xs)) →⁺ lte(xs', xs)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [xs' / Cons(x', xs'), 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:

lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
lte :: Cons:Nil → Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil → False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: Cons:Nil → Cons:Nil → False:True
and :: False:True → False:True → 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:
lte, even

(8) Obligation:

Innermost TRS:
Rules:
lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
lte :: Cons:Nil → Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil → False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: Cons:Nil → Cons:Nil → False:True
and :: False:True → False:True → 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:
lte, even

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

Proved the following rewrite lemma:
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) → False, rt ∈ Ω(1 + n6₀)

Induction Base:
lte(gen_Cons:Nil4_0(+(1, 0)), gen_Cons:Nil4_0(0)) →_R^Ω(1)
False

Induction Step:
lte(gen_Cons:Nil4_0(+(1, +(n6_0, 1))), gen_Cons:Nil4_0(+(n6_0, 1))) →_R^Ω(1)
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) →_IH
False

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:
lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
lte :: Cons:Nil → Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil → False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: Cons:Nil → Cons:Nil → False:True
and :: False:True → False:True → 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:
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) → False, 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:
even

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

Proved the following rewrite lemma:
even(gen_Cons:Nil4_0(+(1, *(2, n295_0)))) → False, rt ∈ Ω(1 + n295₀)

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

Induction Step:
even(gen_Cons:Nil4_0(+(1, *(2, +(n295_0, 1))))) →_R^Ω(1)
even(gen_Cons:Nil4_0(+(1, *(2, n295_0)))) →_IH
False

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:
lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
lte :: Cons:Nil → Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil → False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: Cons:Nil → Cons:Nil → False:True
and :: False:True → False:True → 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:
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) → False, rt ∈ Ω(1 + n6₀)
even(gen_Cons:Nil4_0(+(1, *(2, n295_0)))) → False, rt ∈ Ω(1 + n295₀)

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.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) → False, rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
lte :: Cons:Nil → Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil → False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: Cons:Nil → Cons:Nil → False:True
and :: False:True → False:True → 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:
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) → False, rt ∈ Ω(1 + n6₀)
even(gen_Cons:Nil4_0(+(1, *(2, n295_0)))) → False, rt ∈ Ω(1 + n295₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) → False, rt ∈ Ω(1 + n6₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
lte :: Cons:Nil → Cons:Nil → False:True
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil → False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: Cons:Nil → Cons:Nil → False:True
and :: False:True → False:True → 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:
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) → False, 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.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lte(gen_Cons:Nil4_0(+(1, n6_0)), gen_Cons:Nil4_0(n6_0)) → False, rt ∈ Ω(1 + n6₀)

(22) BOUNDS(n^1, INF)