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)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

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)))
first(0', Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0', cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

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)))
first(0', Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0', cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
0' :: s:0'
nil :: cons:nil
sel :: s:0' → 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, first, sel

(6) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
first(0', Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0', cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
0' :: s:0'
nil :: cons:nil
sel :: s:0' → 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, first, sel

(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)))
first(0', Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0', cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
0' :: s:0'
nil :: cons:nil
sel :: s:0' → 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:
first, sel

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

Proved the following rewrite lemma:
first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0)) → gen_cons:nil3_0(n111_0), rt ∈ Ω(1 + n111₀)

Induction Base:
first(gen_s:0'4_0(0), gen_cons:nil3_0(0)) →_R^Ω(1)
nil

Induction Step:
first(gen_s:0'4_0(+(n111_0, 1)), gen_cons:nil3_0(+(n111_0, 1))) →_R^Ω(1)
cons(0', first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0))) →_IH
cons(0', gen_cons:nil3_0(c112_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)))
first(0', Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0', cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
0' :: s:0'
nil :: cons:nil
sel :: s:0' → 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:
first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0)) → gen_cons:nil3_0(n111_0), rt ∈ Ω(1 + n111₀)

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:
sel

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sel(gen_s:0'4_0(n393_0), gen_cons:nil3_0(+(1, n393_0))) → gen_s:0'4_0(0), rt ∈ Ω(1 + n393₀)

Induction Base:
sel(gen_s:0'4_0(0), gen_cons:nil3_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
sel(gen_s:0'4_0(+(n393_0, 1)), gen_cons:nil3_0(+(1, +(n393_0, 1)))) →_R^Ω(1)
sel(gen_s:0'4_0(n393_0), gen_cons:nil3_0(+(1, n393_0))) →_IH
gen_s:0'4_0(0)

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
first(0', Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0', cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
0' :: s:0'
nil :: cons:nil
sel :: s:0' → 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:
first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0)) → gen_cons:nil3_0(n111_0), rt ∈ Ω(1 + n111₀)
sel(gen_s:0'4_0(n393_0), gen_cons:nil3_0(+(1, n393_0))) → gen_s:0'4_0(0), rt ∈ Ω(1 + n393₀)

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.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0)) → gen_cons:nil3_0(n111_0), rt ∈ Ω(1 + n111₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
first(0', Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0', cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
0' :: s:0'
nil :: cons:nil
sel :: s:0' → 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:
first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0)) → gen_cons:nil3_0(n111_0), rt ∈ Ω(1 + n111₀)
sel(gen_s:0'4_0(n393_0), gen_cons:nil3_0(+(1, n393_0))) → gen_s:0'4_0(0), rt ∈ Ω(1 + n393₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0)) → gen_cons:nil3_0(n111_0), rt ∈ Ω(1 + n111₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
from(X) → cons(X, from(s(X)))
first(0', Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
sel(0', cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
from :: s:0' → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
0' :: s:0'
nil :: cons:nil
sel :: s:0' → 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:
first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0)) → gen_cons:nil3_0(n111_0), rt ∈ Ω(1 + n111₀)

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.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
first(gen_s:0'4_0(n111_0), gen_cons:nil3_0(n111_0)) → gen_cons:nil3_0(n111_0), rt ∈ Ω(1 + n111₀)

(22) BOUNDS(n^1, INF)