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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
sum(s(x)) →⁺ +(sum(x), s(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x)].
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:

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: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
+'/1

(6) 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: FULL

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

Infered types.

(8) Obligation:

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:+'

(9) OrderProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

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

(11) 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).

(12) Complex Obligation (BEST)

(13) Obligation:

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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum1(gen_0':s:+'2_0(+(1, n819_0))) → *3_0, rt ∈ Ω(n819₀)

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

Induction Step:
sum1(gen_0':s:+'2_0(+(1, +(n819_0, 1)))) →_R^Ω(1)
s(+'(sum1(gen_0':s:+'2_0(+(1, n819_0))))) →_IH
s(+'(*3_0))

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

(15) Complex Obligation (BEST)

(16) Obligation:

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, n819_0))) → *3_0, rt ∈ Ω(n819₀)

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.

(17) 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)

(18) BOUNDS(n^1, INF)

(19) Obligation:

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, n819_0))) → *3_0, rt ∈ Ω(n819₀)

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.

(20) 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)

(21) BOUNDS(n^1, INF)

(22) Obligation:

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.

(23) 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)

(24) BOUNDS(n^1, INF)