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:

app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))

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:

app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

Types:
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
app, sum, plus

They will be analysed ascendingly in the following order:
app < sum
plus < sum

(6) Obligation:

Innermost TRS:
Rules:
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

Types:
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil: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:
app, sum, plus

They will be analysed ascendingly in the following order:
app < sum
plus < sum

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

Proved the following rewrite lemma:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Induction Base:
app(gen_nil:cons3_0(0), gen_nil:cons3_0(b)) →_R^Ω(1)
gen_nil:cons3_0(b)

Induction Step:
app(gen_nil:cons3_0(+(n6_0, 1)), gen_nil:cons3_0(b)) →_R^Ω(1)
cons(0', app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b))) →_IH
cons(0', gen_nil:cons3_0(+(b, c7_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:
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

Types:
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil: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:
plus, sum

They will be analysed ascendingly in the following order:
plus < sum

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
plus(gen_0':s4_0(n696_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n696_0, b)), rt ∈ Ω(1 + n696₀)

Induction Base:
plus(gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
gen_0':s4_0(b)

Induction Step:
plus(gen_0':s4_0(+(n696_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
s(plus(gen_0':s4_0(n696_0), gen_0':s4_0(b))) →_IH
s(gen_0':s4_0(+(b, c697_0)))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

Types:
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
plus(gen_0':s4_0(n696_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n696_0, b)), rt ∈ Ω(1 + n696₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil: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:
sum

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_nil:cons3_0(+(1, n1407_0))) → gen_nil:cons3_0(1), rt ∈ Ω(1 + n1407₀)

Induction Base:
sum(gen_nil:cons3_0(+(1, 0))) →_R^Ω(1)
cons(0', nil)

Induction Step:
sum(gen_nil:cons3_0(+(1, +(n1407_0, 1)))) →_R^Ω(1)
sum(cons(plus(0', 0'), gen_nil:cons3_0(n1407_0))) →_L^Ω(1)
sum(cons(gen_0':s4_0(+(0, 0)), gen_nil:cons3_0(n1407_0))) →_IH
gen_nil:cons3_0(1)

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

Types:
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
plus(gen_0':s4_0(n696_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n696_0, b)), rt ∈ Ω(1 + n696₀)
sum(gen_nil:cons3_0(+(1, n1407_0))) → gen_nil:cons3_0(1), rt ∈ Ω(1 + n1407₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil: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.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

Types:
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
plus(gen_0':s4_0(n696_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n696_0, b)), rt ∈ Ω(1 + n696₀)
sum(gen_nil:cons3_0(+(1, n1407_0))) → gen_nil:cons3_0(1), rt ∈ Ω(1 + n1407₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil: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.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

Innermost TRS:
Rules:
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

Types:
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
plus(gen_0':s4_0(n696_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n696_0, b)), rt ∈ Ω(1 + n696₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil: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.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))

Types:
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil: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.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) → gen_nil:cons3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(26) BOUNDS(n^1, INF)