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), 401 ms)
↳10 BEST
    ↳11 typed CpxTrs
        ↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 432 ms)
        ↳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:

inc(Cons(x, xs)) → Cons(Cons(Nil, Nil), inc(xs))
nestinc(Nil) → number17(Nil)
nestinc(Cons(x, xs)) → nestinc(inc(Cons(x, xs)))
inc(Nil) → Cons(Nil, Nil)
number17(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, Nil)))))))))))))))))
goal(x) → nestinc(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:

inc(Cons(x, xs)) → Cons(Cons(Nil, Nil), inc(xs))
nestinc(Nil) → number17(Nil)
nestinc(Cons(x, xs)) → nestinc(inc(Cons(x, xs)))
inc(Nil) → Cons(Nil, Nil)
number17(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, Nil)))))))))))))))))
goal(x) → nestinc(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Sliced the following arguments:
Cons/0
number17/0

(4) Obligation:

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

inc(Cons(xs)) → Cons(inc(xs))
nestinc(Nil) → number17
nestinc(Cons(xs)) → nestinc(inc(Cons(xs)))
inc(Nil) → Cons(Nil)
number17 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil)))))))))))))))))
goal(x) → nestinc(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
inc(Cons(xs)) → Cons(inc(xs))
nestinc(Nil) → number17
nestinc(Cons(xs)) → nestinc(inc(Cons(xs)))
inc(Nil) → Cons(Nil)
number17 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil)))))))))))))))))
goal(x) → nestinc(x)

Types:
inc :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
nestinc :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number17 :: 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:
inc, nestinc

They will be analysed ascendingly in the following order:
inc < nestinc

(8) Obligation:

Innermost TRS:
Rules:
inc(Cons(xs)) → Cons(inc(xs))
nestinc(Nil) → number17
nestinc(Cons(xs)) → nestinc(inc(Cons(xs)))
inc(Nil) → Cons(Nil)
number17 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil)))))))))))))))))
goal(x) → nestinc(x)

Types:
inc :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
nestinc :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number17 :: 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:
inc, nestinc

They will be analysed ascendingly in the following order:
inc < nestinc

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

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

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

Induction Step:
inc(gen_Cons:Nil2_0(+(n4_0, 1))) →RΩ(1)
Cons(inc(gen_Cons:Nil2_0(n4_0))) →IH
Cons(gen_Cons:Nil2_0(+(1, c5_0)))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
inc(Cons(xs)) → Cons(inc(xs))
nestinc(Nil) → number17
nestinc(Cons(xs)) → nestinc(inc(Cons(xs)))
inc(Nil) → Cons(Nil)
number17 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil)))))))))))))))))
goal(x) → nestinc(x)

Types:
inc :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
nestinc :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number17 :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

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

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:
nestinc

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol nestinc.

(13) Obligation:

Innermost TRS:
Rules:
inc(Cons(xs)) → Cons(inc(xs))
nestinc(Nil) → number17
nestinc(Cons(xs)) → nestinc(inc(Cons(xs)))
inc(Nil) → Cons(Nil)
number17 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil)))))))))))))))))
goal(x) → nestinc(x)

Types:
inc :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
nestinc :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number17 :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
inc(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(+(1, n4_0)), 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.

(14) LowerBoundsProof (EQUIVALENT transformation)

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

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
inc(Cons(xs)) → Cons(inc(xs))
nestinc(Nil) → number17
nestinc(Cons(xs)) → nestinc(inc(Cons(xs)))
inc(Nil) → Cons(Nil)
number17 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil)))))))))))))))))
goal(x) → nestinc(x)

Types:
inc :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
nestinc :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
number17 :: Cons:Nil
goal :: Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
inc(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(+(1, n4_0)), 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.

(17) LowerBoundsProof (EQUIVALENT transformation)

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

(18) BOUNDS(n^1, INF)