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

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 93 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 SlicingProof (LOWER BOUND(ID), 0 ms)
    ↳6 CpxRelTRS
    ↳7 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳10 typed CpxTrs
    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 153 ms)
    ↳12 BEST
        ↳13 typed CpxTrs
            ↳14 LowerBoundsProof (⇔, 0 ms)
            ↳15 BOUNDS(n^1, INF)
        ↳16 typed CpxTrs
            ↳17 LowerBoundsProof (⇔, 0 ms)
            ↳18 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) DecreasingLoopProof (EQUIVALENT transformation)

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

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

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

Sliced the following arguments:
Cons/0
isEmpty[Match]/0

(6) Obligation:

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

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

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

Types:
list :: Cons:Nil → True:isEmpty[Match]:False
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: 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
gen_Cons:Nil3_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
list

(10) Obligation:

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

Types:
list :: Cons:Nil → True:isEmpty[Match]:False
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: 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
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:
list

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

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

Induction Base:
list(gen_Cons:Nil3_0(0)) →RΩ(1)
True

Induction Step:
list(gen_Cons:Nil3_0(+(n5_0, 1))) →RΩ(1)
list(gen_Cons:Nil3_0(n5_0)) →IH
True

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
list :: Cons:Nil → True:isEmpty[Match]:False
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: 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
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
list(gen_Cons:Nil3_0(n5_0)) → True, 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.

(14) LowerBoundsProof (EQUIVALENT transformation)

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

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
list :: Cons:Nil → True:isEmpty[Match]:False
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: 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
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
list(gen_Cons:Nil3_0(n5_0)) → True, 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.

(17) LowerBoundsProof (EQUIVALENT transformation)

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

(18) BOUNDS(n^1, INF)