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:

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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
app(cons(x, l), k) →⁺ cons(x, app(l, k))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [l / cons(x, l)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) 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: FULL

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

Infered types.

(6) Obligation:

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

(7) 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

(8) Obligation:

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

(9) 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).

(10) Complex Obligation (BEST)

(11) Obligation:

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

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

Proved the following rewrite lemma:
plus(gen_0':s4_0(n636_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n636_0, b)), rt ∈ Ω(1 + n636₀)

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(+(n636_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
s(plus(gen_0':s4_0(n636_0), gen_0':s4_0(b))) →_IH
s(gen_0':s4_0(+(b, c637_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

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(n636_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n636_0, b)), rt ∈ Ω(1 + n636₀)

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

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_nil:cons3_0(+(1, n1327_0))) → gen_nil:cons3_0(1), rt ∈ Ω(1 + n1327₀)

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

Induction Step:
sum(gen_nil:cons3_0(+(1, +(n1327_0, 1)))) →_R^Ω(1)
sum(cons(plus(0', 0'), gen_nil:cons3_0(n1327_0))) →_L^Ω(1)
sum(cons(gen_0':s4_0(+(0, 0)), gen_nil:cons3_0(n1327_0))) →_IH
gen_nil:cons3_0(1)

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

(16) Complex Obligation (BEST)

(17) Obligation:

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(n636_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n636_0, b)), rt ∈ Ω(1 + n636₀)
sum(gen_nil:cons3_0(+(1, n1327_0))) → gen_nil:cons3_0(1), rt ∈ Ω(1 + n1327₀)

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.

(18) 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)

(19) BOUNDS(n^1, INF)

(20) Obligation:

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(n636_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n636_0, b)), rt ∈ Ω(1 + n636₀)
sum(gen_nil:cons3_0(+(1, n1327_0))) → gen_nil:cons3_0(1), rt ∈ Ω(1 + n1327₀)

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.

(21) 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)

(22) BOUNDS(n^1, INF)

(23) Obligation:

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(n636_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n636_0, b)), rt ∈ Ω(1 + n636₀)

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.

(24) 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)

(25) BOUNDS(n^1, INF)

(26) Obligation:

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.

(27) 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)

(28) BOUNDS(n^1, INF)