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:

sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))

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(0') → 0'
sum(s(x)) → +'(sum(x), s(x))
sum1(0') → 0'
sum1(s(x)) → s(+'(sum1(x), +'(x, x)))

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
+'/1

(4) Obligation:

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

sum(0') → 0'
sum(s(x)) → +'(sum(x))
sum1(0') → 0'
sum1(s(x)) → s(+'(sum1(x)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x))
sum1(0') → 0'
sum1(s(x)) → s(+'(sum1(x)))

Types:
sum :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
sum1 :: 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
sum, sum1

(8) Obligation:

Innermost TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x))
sum1(0') → 0'
sum1(s(x)) → s(+'(sum1(x)))

Types:
sum :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
sum1 :: 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
sum, sum1

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_0':s:+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
sum(gen_0':s:+'2_0(+(1, 0)))

Induction Step:
sum(gen_0':s:+'2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
+'(sum(gen_0':s:+'2_0(+(1, n4_0)))) →_IH
+'(*3_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x))
sum1(0') → 0'
sum1(s(x)) → s(+'(sum1(x)))

Types:
sum :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
sum1 :: 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
sum(gen_0':s:+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
sum1

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

Proved the following rewrite lemma:
sum1(gen_0':s:+'2_0(+(1, n831_0))) → *3_0, rt ∈ Ω(n831₀)

Induction Base:
sum1(gen_0':s:+'2_0(+(1, 0)))

Induction Step:
sum1(gen_0':s:+'2_0(+(1, +(n831_0, 1)))) →_R^Ω(1)
s(+'(sum1(gen_0':s:+'2_0(+(1, n831_0))))) →_IH
s(+'(*3_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x))
sum1(0') → 0'
sum1(s(x)) → s(+'(sum1(x)))

Types:
sum :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
sum1 :: 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
sum(gen_0':s:+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)
sum1(gen_0':s:+'2_0(+(1, n831_0))) → *3_0, rt ∈ Ω(n831₀)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sum(gen_0':s:+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x))
sum1(0') → 0'
sum1(s(x)) → s(+'(sum1(x)))

Types:
sum :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
sum1 :: 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
sum(gen_0':s:+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)
sum1(gen_0':s:+'2_0(+(1, n831_0))) → *3_0, rt ∈ Ω(n831₀)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sum(gen_0':s:+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
sum(0') → 0'
sum(s(x)) → +'(sum(x))
sum1(0') → 0'
sum1(s(x)) → s(+'(sum1(x)))

Types:
sum :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
sum1 :: 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
sum(gen_0':s:+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sum(gen_0':s:+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(22) BOUNDS(n^1, INF)