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:

f(node(s(n), xs)) → f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) → ap
select(cons(ap, xs)) → select(xs)
addchild(node(y, ys), node(n, xs)) → node(y, cons(node(n, xs), ys))

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:

f(node(s(n), xs)) → f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) → ap
select(cons(ap, xs)) → select(xs)
addchild(node(y, ys), node(n, xs)) → node(y, cons(node(n, xs), ys))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
f(node(s(n), xs)) → f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) → ap
select(cons(ap, xs)) → select(xs)
addchild(node(y, ys), node(n, xs)) → node(y, cons(node(n, xs), ys))

Types:
f :: node → f
node :: s → cons → node
s :: s → s
addchild :: node → node → node
select :: cons → node
cons :: node → cons → cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat → s
gen_cons6_0 :: Nat → cons

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

Heuristically decided to analyse the following defined symbols:
f, select

They will be analysed ascendingly in the following order:
select < f

(6) Obligation:

Innermost TRS:
Rules:
f(node(s(n), xs)) → f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) → ap
select(cons(ap, xs)) → select(xs)
addchild(node(y, ys), node(n, xs)) → node(y, cons(node(n, xs), ys))

Types:
f :: node → f
node :: s → cons → node
s :: s → s
addchild :: node → node → node
select :: cons → node
cons :: node → cons → cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat → s
gen_cons6_0 :: Nat → cons

Generator Equations:
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))
gen_cons6_0(0) ⇔ hole_cons4_0
gen_cons6_0(+(x, 1)) ⇔ cons(node(hole_s3_0, hole_cons4_0), gen_cons6_0(x))

The following defined symbols remain to be analysed:
select, f

They will be analysed ascendingly in the following order:
select < f

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

Proved the following rewrite lemma:
select(gen_cons6_0(+(1, n8_0))) → *7_0, rt ∈ Ω(n8₀)

Induction Base:
select(gen_cons6_0(+(1, 0)))

Induction Step:
select(gen_cons6_0(+(1, +(n8_0, 1)))) →_R^Ω(1)
select(gen_cons6_0(+(1, n8_0))) →_IH
*7_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:
f(node(s(n), xs)) → f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) → ap
select(cons(ap, xs)) → select(xs)
addchild(node(y, ys), node(n, xs)) → node(y, cons(node(n, xs), ys))

Types:
f :: node → f
node :: s → cons → node
s :: s → s
addchild :: node → node → node
select :: cons → node
cons :: node → cons → cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat → s
gen_cons6_0 :: Nat → cons

Lemmas:
select(gen_cons6_0(+(1, n8_0))) → *7_0, rt ∈ Ω(n8₀)

Generator Equations:
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))
gen_cons6_0(0) ⇔ hole_cons4_0
gen_cons6_0(+(x, 1)) ⇔ cons(node(hole_s3_0, hole_cons4_0), gen_cons6_0(x))

The following defined symbols remain to be analysed:
f

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
f(node(s(n), xs)) → f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) → ap
select(cons(ap, xs)) → select(xs)
addchild(node(y, ys), node(n, xs)) → node(y, cons(node(n, xs), ys))

Types:
f :: node → f
node :: s → cons → node
s :: s → s
addchild :: node → node → node
select :: cons → node
cons :: node → cons → cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat → s
gen_cons6_0 :: Nat → cons

Lemmas:
select(gen_cons6_0(+(1, n8_0))) → *7_0, rt ∈ Ω(n8₀)

Generator Equations:
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))
gen_cons6_0(0) ⇔ hole_cons4_0
gen_cons6_0(+(x, 1)) ⇔ cons(node(hole_s3_0, hole_cons4_0), gen_cons6_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
select(gen_cons6_0(+(1, n8_0))) → *7_0, rt ∈ Ω(n8₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
f(node(s(n), xs)) → f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) → ap
select(cons(ap, xs)) → select(xs)
addchild(node(y, ys), node(n, xs)) → node(y, cons(node(n, xs), ys))

Types:
f :: node → f
node :: s → cons → node
s :: s → s
addchild :: node → node → node
select :: cons → node
cons :: node → cons → cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat → s
gen_cons6_0 :: Nat → cons

Lemmas:
select(gen_cons6_0(+(1, n8_0))) → *7_0, rt ∈ Ω(n8₀)

Generator Equations:
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))
gen_cons6_0(0) ⇔ hole_cons4_0
gen_cons6_0(+(x, 1)) ⇔ cons(node(hole_s3_0, hole_cons4_0), gen_cons6_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
select(gen_cons6_0(+(1, n8_0))) → *7_0, rt ∈ Ω(n8₀)

(16) BOUNDS(n^1, INF)