The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 typed CpxTrs
↳5 OrderProof (LOWER BOUND(ID), 0 ms)
↳6 typed CpxTrs
↳7 RewriteLemmaProof (LOWER BOUND(ID), 255 ms)
↳8 BEST
    ↳9 typed CpxTrs
        ↳10 LowerBoundsProof (⇔, 0 ms)
        ↳11 BOUNDS(n^1, INF)
    ↳12 typed CpxTrs
        ↳13 LowerBoundsProof (⇔, 0 ms)
        ↳14 BOUNDS(n^1, INF)

(0) Obligation:

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

list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(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:

list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(x)

Types:
list :: Cons:Nil → True:isEmpty[Match]:False
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: Cons:Nil → True:isEmpty[Match]:False
notEmpty :: Cons:Nil → True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil → True:isEmpty[Match]:False
hole_True:isEmpty[Match]:False1_0 :: True:isEmpty[Match]:False
hole_Cons:Nil2_0 :: Cons:Nil
hole_a3_0 :: a
gen_Cons:Nil4_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
list

(6) Obligation:

Innermost TRS:
Rules:
list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(x)

Types:
list :: Cons:Nil → True:isEmpty[Match]:False
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: Cons:Nil → True:isEmpty[Match]:False
notEmpty :: Cons:Nil → True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil → True:isEmpty[Match]:False
hole_True:isEmpty[Match]:False1_0 :: True:isEmpty[Match]:False
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:
list

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

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

Induction Base:
list(gen_Cons:Nil4_0(0)) →RΩ(1)
True

Induction Step:
list(gen_Cons:Nil4_0(+(n6_0, 1))) →RΩ(1)
list(gen_Cons:Nil4_0(n6_0)) →IH
True

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(x)

Types:
list :: Cons:Nil → True:isEmpty[Match]:False
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: Cons:Nil → True:isEmpty[Match]:False
notEmpty :: Cons:Nil → True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil → True:isEmpty[Match]:False
hole_True:isEmpty[Match]:False1_0 :: True:isEmpty[Match]:False
hole_Cons:Nil2_0 :: Cons:Nil
hole_a3_0 :: a
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
list(gen_Cons:Nil4_0(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.

(10) LowerBoundsProof (EQUIVALENT transformation)

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

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
Rules:
list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(x)

Types:
list :: Cons:Nil → True:isEmpty[Match]:False
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: Cons:Nil → True:isEmpty[Match]:False
notEmpty :: Cons:Nil → True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil → True:isEmpty[Match]:False
hole_True:isEmpty[Match]:False1_0 :: True:isEmpty[Match]:False
hole_Cons:Nil2_0 :: Cons:Nil
hole_a3_0 :: a
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
list(gen_Cons:Nil4_0(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.

(13) LowerBoundsProof (EQUIVALENT transformation)

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

(14) BOUNDS(n^1, INF)