The Runtime Complexity (full) 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)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

Rewrite Strategy: FULL

(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)))
after(0', XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

S is empty.
Rewrite Strategy: FULL

(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)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, XS)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
from → cons(from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, XS)

Types:
from :: cons
cons :: cons → cons
after :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
from, after

(8) Obligation:

TRS:
Rules:
from → cons(from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, XS)

Types:
from :: cons
cons :: cons → cons
after :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(gen_cons3_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, after

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

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

(10) Obligation:

TRS:
Rules:
from → cons(from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, XS)

Types:
from :: cons
cons :: cons → cons
after :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(gen_cons3_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:
after

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

Proved the following rewrite lemma:
after(gen_0':s4_0(n9_0), gen_cons3_0(n9_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n9₀)

Induction Base:
after(gen_0':s4_0(0), gen_cons3_0(0)) →_R^Ω(1)
gen_cons3_0(0)

Induction Step:
after(gen_0':s4_0(+(n9_0, 1)), gen_cons3_0(+(n9_0, 1))) →_R^Ω(1)
after(gen_0':s4_0(n9_0), gen_cons3_0(n9_0)) →_IH
gen_cons3_0(0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
from → cons(from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, XS)

Types:
from :: cons
cons :: cons → cons
after :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
after(gen_0':s4_0(n9_0), gen_cons3_0(n9_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n9₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(gen_cons3_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.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
after(gen_0':s4_0(n9_0), gen_cons3_0(n9_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n9₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
from → cons(from)
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, XS)

Types:
from :: cons
cons :: cons → cons
after :: 0':s → cons → cons
0' :: 0':s
s :: 0':s → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
after(gen_0':s4_0(n9_0), gen_cons3_0(n9_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n9₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(gen_cons3_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.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
after(gen_0':s4_0(n9_0), gen_cons3_0(n9_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n9₀)

(18) BOUNDS(n^1, INF)