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) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
from/0
cons/0

(4) Obligation:

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

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

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

Types:
from :: cons:nil
cons :: cons:nil → cons:nil
length :: cons:nil → 0':s
nil :: cons:nil
0' :: 0':s
s :: 0':s → 0':s
length1 :: cons:nil → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
gen_cons:nil3_0 :: Nat → cons:nil
gen_0':s4_0 :: Nat → 0':s

(7) 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

(8) Obligation:

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

Types:
from :: cons:nil
cons :: cons:nil → cons:nil
length :: cons:nil → 0':s
nil :: cons:nil
0' :: 0':s
s :: 0':s → 0':s
length1 :: cons:nil → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
gen_cons:nil3_0 :: Nat → cons:nil
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(gen_cons:nil3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

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

Types:
from :: cons:nil
cons :: cons:nil → cons:nil
length :: cons:nil → 0':s
nil :: cons:nil
0' :: 0':s
s :: 0':s → 0':s
length1 :: cons:nil → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
gen_cons:nil3_0 :: Nat → cons:nil
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(gen_cons:nil3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

Proved the following rewrite lemma:
length1(gen_cons:nil3_0(n9_0)) → gen_0':s4_0(n9_0), rt ∈ Ω(1 + n9₀)

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(+(n9_0, 1))) →_R^Ω(1)
length(gen_cons:nil3_0(+(n9_0, 1))) →_R^Ω(1)
s(length1(gen_cons:nil3_0(n9_0))) →_IH
s(gen_0':s4_0(c10_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
from :: cons:nil
cons :: cons:nil → cons:nil
length :: cons:nil → 0':s
nil :: cons:nil
0' :: 0':s
s :: 0':s → 0':s
length1 :: cons:nil → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
gen_cons:nil3_0 :: Nat → cons:nil
gen_0':s4_0 :: Nat → 0':s

Lemmas:
length1(gen_cons:nil3_0(n9_0)) → gen_0':s4_0(n9_0), rt ∈ Ω(1 + n9₀)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(gen_cons:nil3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
length

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

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

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

Types:
from :: cons:nil
cons :: cons:nil → cons:nil
length :: cons:nil → 0':s
nil :: cons:nil
0' :: 0':s
s :: 0':s → 0':s
length1 :: cons:nil → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
gen_cons:nil3_0 :: Nat → cons:nil
gen_0':s4_0 :: Nat → 0':s

Lemmas:
length1(gen_cons:nil3_0(n9_0)) → gen_0':s4_0(n9_0), rt ∈ Ω(1 + n9₀)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(gen_cons:nil3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
length1(gen_cons:nil3_0(n9_0)) → gen_0':s4_0(n9_0), rt ∈ Ω(1 + n9₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

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

Types:
from :: cons:nil
cons :: cons:nil → cons:nil
length :: cons:nil → 0':s
nil :: cons:nil
0' :: 0':s
s :: 0':s → 0':s
length1 :: cons:nil → 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
gen_cons:nil3_0 :: Nat → cons:nil
gen_0':s4_0 :: Nat → 0':s

Lemmas:
length1(gen_cons:nil3_0(n9_0)) → gen_0':s4_0(n9_0), rt ∈ Ω(1 + n9₀)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(gen_cons:nil3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
length1(gen_cons:nil3_0(n9_0)) → gen_0':s4_0(n9_0), rt ∈ Ω(1 + n9₀)

(20) BOUNDS(n^1, INF)