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:

r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws)
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws)
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))

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:

r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws)
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws)
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws)
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws)
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))

Types:
r :: nil:cons → nil:cons → nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: succ → nil:cons → nil:cons
succ :: zero → succ
zero :: zero
hole_nil:cons1_0 :: nil:cons
hole_succ2_0 :: succ
hole_zero3_0 :: zero
gen_nil:cons4_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
r

(6) Obligation:

Innermost TRS:
Rules:
r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws)
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws)
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))

Types:
r :: nil:cons → nil:cons → nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: succ → nil:cons → nil:cons
succ :: zero → succ
zero :: zero
hole_nil:cons1_0 :: nil:cons
hole_succ2_0 :: succ
hole_zero3_0 :: zero
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(succ(zero), gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
r

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

Proved the following rewrite lemma:
r(gen_nil:cons4_0(0), gen_nil:cons4_0(0), gen_nil:cons4_0(c), gen_nil:cons4_0(n6_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n6₀)

Induction Base:
r(gen_nil:cons4_0(0), gen_nil:cons4_0(0), gen_nil:cons4_0(c), gen_nil:cons4_0(0)) →_R^Ω(1)
gen_nil:cons4_0(0)

Induction Step:
r(gen_nil:cons4_0(0), gen_nil:cons4_0(0), gen_nil:cons4_0(c), gen_nil:cons4_0(+(n6_0, 1))) →_R^Ω(1)
r(gen_nil:cons4_0(0), gen_nil:cons4_0(0), cons(succ(zero), gen_nil:cons4_0(c)), gen_nil:cons4_0(n6_0)) →_IH
gen_nil:cons4_0(0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws)
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws)
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))

Types:
r :: nil:cons → nil:cons → nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: succ → nil:cons → nil:cons
succ :: zero → succ
zero :: zero
hole_nil:cons1_0 :: nil:cons
hole_succ2_0 :: succ
hole_zero3_0 :: zero
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
r(gen_nil:cons4_0(0), gen_nil:cons4_0(0), gen_nil:cons4_0(c), gen_nil:cons4_0(n6_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(succ(zero), gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
r(gen_nil:cons4_0(0), gen_nil:cons4_0(0), gen_nil:cons4_0(c), gen_nil:cons4_0(n6_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n6₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
Rules:
r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws)
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws)
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))

Types:
r :: nil:cons → nil:cons → nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: succ → nil:cons → nil:cons
succ :: zero → succ
zero :: zero
hole_nil:cons1_0 :: nil:cons
hole_succ2_0 :: succ
hole_zero3_0 :: zero
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
r(gen_nil:cons4_0(0), gen_nil:cons4_0(0), gen_nil:cons4_0(c), gen_nil:cons4_0(n6_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(succ(zero), gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
r(gen_nil:cons4_0(0), gen_nil:cons4_0(0), gen_nil:cons4_0(c), gen_nil:cons4_0(n6_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n6₀)

(14) BOUNDS(n^1, INF)