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:

member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)

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

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0, S(y)) → False
!EQ(S(x), 0) → False
!EQ(0, 0) → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

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:

member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)

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

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Types:
member :: S:0' → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
member[Ite][True][Ite] :: False:True → S:0' → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: S:0' → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat → S:0'
gen_Cons:Nil5_0 :: Nat → Cons:Nil

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
member, !EQ

They will be analysed ascendingly in the following order:
!EQ < member

(6) Obligation:

Innermost TRS:
Rules:
member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Types:
member :: S:0' → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
member[Ite][True][Ite] :: False:True → S:0' → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: S:0' → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat → S:0'
gen_Cons:Nil5_0 :: Nat → Cons:Nil

Generator Equations:
gen_S:0'4_0(0) ⇔ 0'
gen_S:0'4_0(+(x, 1)) ⇔ S(gen_S:0'4_0(x))
gen_Cons:Nil5_0(0) ⇔ Nil
gen_Cons:Nil5_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil5_0(x))

The following defined symbols remain to be analysed:
!EQ, member

They will be analysed ascendingly in the following order:
!EQ < member

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Induction Base:
!EQ(gen_S:0'4_0(0), gen_S:0'4_0(+(1, 0))) →_R^Ω(0)
False

Induction Step:
!EQ(gen_S:0'4_0(+(n7_0, 1)), gen_S:0'4_0(+(1, +(n7_0, 1)))) →_R^Ω(0)
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) →_IH
False

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n⁰).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Types:
member :: S:0' → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
member[Ite][True][Ite] :: False:True → S:0' → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: S:0' → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat → S:0'
gen_Cons:Nil5_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_S:0'4_0(0) ⇔ 0'
gen_S:0'4_0(+(x, 1)) ⇔ S(gen_S:0'4_0(x))
gen_Cons:Nil5_0(0) ⇔ Nil
gen_Cons:Nil5_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil5_0(x))

The following defined symbols remain to be analysed:
member

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(n310_0)) → False, rt ∈ Ω(1 + n310₀)

Induction Base:
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(0)) →_R^Ω(1)
False

Induction Step:
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(+(n310_0, 1))) →_R^Ω(1)
member[Ite][True][Ite](!EQ(gen_S:0'4_0(1), 0'), gen_S:0'4_0(1), Cons(0', gen_Cons:Nil5_0(n310_0))) →_R^Ω(0)
member[Ite][True][Ite](False, gen_S:0'4_0(1), Cons(0', gen_Cons:Nil5_0(n310_0))) →_R^Ω(0)
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(n310_0)) →_IH
False

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Types:
member :: S:0' → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
member[Ite][True][Ite] :: False:True → S:0' → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: S:0' → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat → S:0'
gen_Cons:Nil5_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) → False, rt ∈ Ω(0)
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(n310_0)) → False, rt ∈ Ω(1 + n310₀)

Generator Equations:
gen_S:0'4_0(0) ⇔ 0'
gen_S:0'4_0(+(x, 1)) ⇔ S(gen_S:0'4_0(x))
gen_Cons:Nil5_0(0) ⇔ Nil
gen_Cons:Nil5_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil5_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(n310_0)) → False, rt ∈ Ω(1 + n310₀)

(14) BOUNDS(n^1, INF)

(15) Obligation:

Innermost TRS:
Rules:
member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Types:
member :: S:0' → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
member[Ite][True][Ite] :: False:True → S:0' → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: S:0' → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat → S:0'
gen_Cons:Nil5_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) → False, rt ∈ Ω(0)
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(n310_0)) → False, rt ∈ Ω(1 + n310₀)

Generator Equations:
gen_S:0'4_0(0) ⇔ 0'
gen_S:0'4_0(+(x, 1)) ⇔ S(gen_S:0'4_0(x))
gen_Cons:Nil5_0(0) ⇔ Nil
gen_Cons:Nil5_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil5_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(n310_0)) → False, rt ∈ Ω(1 + n310₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Types:
member :: S:0' → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
member[Ite][True][Ite] :: False:True → S:0' → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil → False:True
True :: False:True
goal :: S:0' → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat → S:0'
gen_Cons:Nil5_0 :: Nat → Cons:Nil

Lemmas:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_S:0'4_0(0) ⇔ 0'
gen_S:0'4_0(+(x, 1)) ⇔ S(gen_S:0'4_0(x))
gen_Cons:Nil5_0(0) ⇔ Nil
gen_Cons:Nil5_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil5_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(1) was proven with the following lemma:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) → False, rt ∈ Ω(0)

(20) BOUNDS(1, INF)