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 SlicingProof (LOWER BOUND(ID), 0 ms)
↳4 CpxRelTRS
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 typed CpxTrs
↳7 OrderProof (LOWER BOUND(ID), 0 ms)
↳8 typed CpxTrs
↳9 RewriteLemmaProof (LOWER BOUND(ID), 1188 ms)
↳10 BEST
    ↳11 typed CpxTrs
        ↳12 LowerBoundsProof (⇔, 0 ms)
        ↳13 BOUNDS(n^1, INF)
    ↳14 typed CpxTrs
        ↳15 LowerBoundsProof (⇔, 0 ms)
        ↳16 BOUNDS(n^1, INF)

(0) Obligation:

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

decrease(Cons(x, xs)) → decrease(xs)
decrease(Nil) → number42(Nil)
number42(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(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:

decrease(Cons(x, xs)) → decrease(xs)
decrease(Nil) → number42(Nil)
number42(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(x)

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0
number42/0

(4) Obligation:

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

decrease(Cons(xs)) → decrease(xs)
decrease(Nil) → number42
number42 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
decrease(Cons(xs)) → decrease(xs)
decrease(Nil) → number42
number42 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(x)

Types:
decrease :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number42 :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
decrease

(8) Obligation:

Innermost TRS:
Rules:
decrease(Cons(xs)) → decrease(xs)
decrease(Nil) → number42
number42 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(x)

Types:
decrease :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number42 :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

The following defined symbols remain to be analysed:
decrease

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

Proved the following rewrite lemma:
decrease(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(42), rt ∈ Ω(1 + n40)

Induction Base:
decrease(gen_Cons:Nil2_0(0)) →RΩ(1)
number42 →RΩ(1)
Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil))))))))))))))))))))))))))))))))))))))))))

Induction Step:
decrease(gen_Cons:Nil2_0(+(n4_0, 1))) →RΩ(1)
decrease(gen_Cons:Nil2_0(n4_0)) →IH
gen_Cons:Nil2_0(42)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
decrease(Cons(xs)) → decrease(xs)
decrease(Nil) → number42
number42 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(x)

Types:
decrease :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number42 :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
decrease(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(42), rt ∈ Ω(1 + n40)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
decrease(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(42), rt ∈ Ω(1 + n40)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
decrease(Cons(xs)) → decrease(xs)
decrease(Nil) → number42
number42 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(x)

Types:
decrease :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number42 :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
decrease(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(42), rt ∈ Ω(1 + n40)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
decrease(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(42), rt ∈ Ω(1 + n40)

(16) BOUNDS(n^1, INF)