The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0, x), y) → sum(x, y)
sum(nil, y) → y
weight(cons(n, cons(m, x))) → weight(sum(cons(n, cons(m, x)), cons(0, x)))
weight(cons(n, nil)) → n

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:

sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
weight(cons(n, cons(m, x))) → weight(sum(cons(n, cons(m, x)), cons(0', x)))
weight(cons(n, nil)) → n

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
weight(cons(n, cons(m, x))) → weight(sum(cons(n, cons(m, x)), cons(0', x)))
weight(cons(n, nil)) → n

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
0' :: s:0'
nil :: cons:nil
weight :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
sum, weight

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

(6) Obligation:

Innermost TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
weight(cons(n, cons(m, x))) → weight(sum(cons(n, cons(m, x)), cons(0', x)))
weight(cons(n, nil)) → n

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
0' :: s:0'
nil :: cons:nil
weight :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
sum, weight

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

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

Proved the following rewrite lemma:
sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) → gen_cons:nil3_0(b), rt ∈ Ω(1 + n6₀)

Induction Base:
sum(gen_cons:nil3_0(0), gen_cons:nil3_0(b)) →_R^Ω(1)
gen_cons:nil3_0(b)

Induction Step:
sum(gen_cons:nil3_0(+(n6_0, 1)), gen_cons:nil3_0(b)) →_R^Ω(1)
sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) →_IH
gen_cons:nil3_0(b)

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:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
weight(cons(n, cons(m, x))) → weight(sum(cons(n, cons(m, x)), cons(0', x)))
weight(cons(n, nil)) → n

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
0' :: s:0'
nil :: cons:nil
weight :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) → gen_cons:nil3_0(b), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
weight

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

Proved the following rewrite lemma:
weight(gen_cons:nil3_0(+(1, n559_0))) → gen_s:0'4_0(0), rt ∈ Ω(1 + n559₀ + n559₀²)

Induction Base:
weight(gen_cons:nil3_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
weight(gen_cons:nil3_0(+(1, +(n559_0, 1)))) →_R^Ω(1)
weight(sum(cons(0', cons(0', gen_cons:nil3_0(n559_0))), cons(0', gen_cons:nil3_0(n559_0)))) →_L^{Ω(3 + n559₀)}
weight(gen_cons:nil3_0(+(n559_0, 1))) →_IH
gen_s:0'4_0(0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
weight(cons(n, cons(m, x))) → weight(sum(cons(n, cons(m, x)), cons(0', x)))
weight(cons(n, nil)) → n

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
0' :: s:0'
nil :: cons:nil
weight :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) → gen_cons:nil3_0(b), rt ∈ Ω(1 + n6₀)
weight(gen_cons:nil3_0(+(1, n559_0))) → gen_s:0'4_0(0), rt ∈ Ω(1 + n559₀ + n559₀²)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
weight(gen_cons:nil3_0(+(1, n559_0))) → gen_s:0'4_0(0), rt ∈ Ω(1 + n559₀ + n559₀²)

(14) BOUNDS(n^2, INF)

(15) Obligation:

Innermost TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
weight(cons(n, cons(m, x))) → weight(sum(cons(n, cons(m, x)), cons(0', x)))
weight(cons(n, nil)) → n

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
0' :: s:0'
nil :: cons:nil
weight :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) → gen_cons:nil3_0(b), rt ∈ Ω(1 + n6₀)
weight(gen_cons:nil3_0(+(1, n559_0))) → gen_s:0'4_0(0), rt ∈ Ω(1 + n559₀ + n559₀²)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
weight(gen_cons:nil3_0(+(1, n559_0))) → gen_s:0'4_0(0), rt ∈ Ω(1 + n559₀ + n559₀²)

(17) BOUNDS(n^2, INF)

(18) Obligation:

Innermost TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
weight(cons(n, cons(m, x))) → weight(sum(cons(n, cons(m, x)), cons(0', x)))
weight(cons(n, nil)) → n

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0' → cons:nil → cons:nil
s :: s:0' → s:0'
0' :: s:0'
nil :: cons:nil
weight :: cons:nil → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
gen_cons:nil3_0 :: Nat → cons:nil
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) → gen_cons:nil3_0(b), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_cons:nil3_0(0) ⇔ nil
gen_cons:nil3_0(+(x, 1)) ⇔ cons(0', gen_cons:nil3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) → gen_cons:nil3_0(b), rt ∈ Ω(1 + n6₀)

(20) BOUNDS(n^1, INF)