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

(0) Obligation:

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

from(X) → cons(X, from(s(X)))
length(nil) → 0
length(cons(X, Y)) → s(length1(Y))
length1(X) → length(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:

from(X) → cons(X, from(s(X)))
length(nil) → 0'
length(cons(X, Y)) → s(length1(Y))
length1(X) → length(X)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
length(nil) → 0'
length(cons(X, Y)) → s(length1(Y))
length1(X) → length(X)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
length :: cons:nil → s:0'
nil :: cons:nil
0' :: s:0'
length1 :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
from, length, length1

They will be analysed ascendingly in the following order:
length = length1

(6) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
length(nil) → 0'
length(cons(X, Y)) → s(length1(Y))
length1(X) → length(X)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
length :: cons:nil → s:0'
nil :: cons:nil
0' :: s:0'
length1 :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
from, length, length1

They will be analysed ascendingly in the following order:
length = length1

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
length(nil) → 0'
length(cons(X, Y)) → s(length1(Y))
length1(X) → length(X)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
length :: cons:nil → s:0'
nil :: cons:nil
0' :: s:0'
length1 :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
length1, length

They will be analysed ascendingly in the following order:
length = length1

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

Proved the following rewrite lemma:
length1(gen_cons:nil3_0(n57_0)) → gen_s:0'4_0(n57_0), rt ∈ Ω(1 + n57₀)

Induction Base:
length1(gen_cons:nil3_0(0)) →_R^Ω(1)
length(gen_cons:nil3_0(0)) →_R^Ω(1)
0'

Induction Step:
length1(gen_cons:nil3_0(+(n57_0, 1))) →_R^Ω(1)
length(gen_cons:nil3_0(+(n57_0, 1))) →_R^Ω(1)
s(length1(gen_cons:nil3_0(n57_0))) →_IH
s(gen_s:0'4_0(c58_0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
length(nil) → 0'
length(cons(X, Y)) → s(length1(Y))
length1(X) → length(X)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
length :: cons:nil → s:0'
nil :: cons:nil
0' :: s:0'
length1 :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
length1(gen_cons:nil3_0(n57_0)) → gen_s:0'4_0(n57_0), rt ∈ Ω(1 + n57₀)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
length

They will be analysed ascendingly in the following order:
length = length1

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

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

(13) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
length(nil) → 0'
length(cons(X, Y)) → s(length1(Y))
length1(X) → length(X)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
length :: cons:nil → s:0'
nil :: cons:nil
0' :: s:0'
length1 :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
length1(gen_cons:nil3_0(n57_0)) → gen_s:0'4_0(n57_0), rt ∈ Ω(1 + n57₀)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
length1(gen_cons:nil3_0(n57_0)) → gen_s:0'4_0(n57_0), rt ∈ Ω(1 + n57₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
length(nil) → 0'
length(cons(X, Y)) → s(length1(Y))
length1(X) → length(X)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
length :: cons:nil → s:0'
nil :: cons:nil
0' :: s:0'
length1 :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
length1(gen_cons:nil3_0(n57_0)) → gen_s:0'4_0(n57_0), rt ∈ Ω(1 + n57₀)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
length1(gen_cons:nil3_0(n57_0)) → gen_s:0'4_0(n57_0), rt ∈ Ω(1 + n57₀)

(18) BOUNDS(n^1, INF)