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), 452 ms)
↳8 BEST
    ↳9 typed CpxTrs
        ↳10 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
        ↳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:

immatcopy(Cons(x, xs)) → Cons(Nil, immatcopy(xs))
nestimeql(Nil) → number42(Nil)
nestimeql(Cons(x, xs)) → nestimeql(immatcopy(Cons(x, xs)))
immatcopy(Nil) → 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) → nestimeql(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:

immatcopy(Cons(x, xs)) → Cons(Nil, immatcopy(xs))
nestimeql(Nil) → number42(Nil)
nestimeql(Cons(x, xs)) → nestimeql(immatcopy(Cons(x, xs)))
immatcopy(Nil) → 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) → nestimeql(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
immatcopy(Cons(x, xs)) → Cons(Nil, immatcopy(xs))
nestimeql(Nil) → number42(Nil)
nestimeql(Cons(x, xs)) → nestimeql(immatcopy(Cons(x, xs)))
immatcopy(Nil) → 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) → nestimeql(x)

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

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

Heuristically decided to analyse the following defined symbols:
immatcopy, nestimeql

They will be analysed ascendingly in the following order:
immatcopy < nestimeql

(6) Obligation:

Innermost TRS:
Rules:
immatcopy(Cons(x, xs)) → Cons(Nil, immatcopy(xs))
nestimeql(Nil) → number42(Nil)
nestimeql(Cons(x, xs)) → nestimeql(immatcopy(Cons(x, xs)))
immatcopy(Nil) → 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) → nestimeql(x)

Types:
immatcopy :: Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
nestimeql :: Cons:Nil → Cons:Nil
number42 :: Cons:Nil → 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(Nil, gen_Cons:Nil2_0(x))

The following defined symbols remain to be analysed:
immatcopy, nestimeql

They will be analysed ascendingly in the following order:
immatcopy < nestimeql

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

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

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

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

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
immatcopy(Cons(x, xs)) → Cons(Nil, immatcopy(xs))
nestimeql(Nil) → number42(Nil)
nestimeql(Cons(x, xs)) → nestimeql(immatcopy(Cons(x, xs)))
immatcopy(Nil) → 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) → nestimeql(x)

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

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

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

The following defined symbols remain to be analysed:
nestimeql

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
immatcopy(Cons(x, xs)) → Cons(Nil, immatcopy(xs))
nestimeql(Nil) → number42(Nil)
nestimeql(Cons(x, xs)) → nestimeql(immatcopy(Cons(x, xs)))
immatcopy(Nil) → 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) → nestimeql(x)

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

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

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, 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:
immatcopy(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(n4_0), rt ∈ Ω(1 + n40)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
immatcopy(Cons(x, xs)) → Cons(Nil, immatcopy(xs))
nestimeql(Nil) → number42(Nil)
nestimeql(Cons(x, xs)) → nestimeql(immatcopy(Cons(x, xs)))
immatcopy(Nil) → 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) → nestimeql(x)

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

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

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, 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:
immatcopy(gen_Cons:Nil2_0(n4_0)) → gen_Cons:Nil2_0(n4_0), rt ∈ Ω(1 + n40)

(16) BOUNDS(n^1, INF)