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:

sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))

Rewrite Strategy: FULL

(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(0') → 0'
sum(s(x)) → +'(sum(x), s(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x), s(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
sum :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
sum, +'

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

(6) Obligation:

TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x), s(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
sum :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
+', sum

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

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

Proved the following rewrite lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Induction Base:
+'(gen_0':s2_0(a), gen_0':s2_0(0)) →_R^Ω(1)
gen_0':s2_0(a)

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) →_R^Ω(1)
s(+'(gen_0':s2_0(a), gen_0':s2_0(n4_0))) →_IH
s(gen_0':s2_0(+(a, c5_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x), s(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
sum :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
sum

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

Proved the following rewrite lemma:
sum(gen_0':s2_0(+(1, n403_0))) → *3_0, rt ∈ Ω(n403₀)

Induction Base:
sum(gen_0':s2_0(+(1, 0)))

Induction Step:
sum(gen_0':s2_0(+(1, +(n403_0, 1)))) →_R^Ω(1)
+'(sum(gen_0':s2_0(+(1, n403_0))), s(gen_0':s2_0(+(1, n403_0)))) →_IH
+'(*3_0, s(gen_0':s2_0(+(1, n403_0))))

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

(11) Complex Obligation (BEST)

(12) Obligation:

TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x), s(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
sum :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
sum(gen_0':s2_0(+(1, n403_0))) → *3_0, rt ∈ Ω(n403₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(14) BOUNDS(n^1, INF)

(15) Obligation:

TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x), s(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
sum :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
sum(gen_0':s2_0(+(1, n403_0))) → *3_0, rt ∈ Ω(n403₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x), s(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
sum :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(20) BOUNDS(n^1, INF)