The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

div2(S(S(x))) → +(S(0), div2(x))
div2(S(0)) → 0
div2(0) → 0

The (relative) TRS S consists of the following rules:

+(x, S(0)) → S(x)
+(S(0), y) → S(y)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
div2(S(S(x))) →⁺ +(S(0), div2(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / S(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:

div2(S(S(x))) → +'(S(0'), div2(x))
div2(S(0')) → 0'
div2(0') → 0'

The (relative) TRS S consists of the following rules:

+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
div2(S(S(x))) → +'(S(0'), div2(x))
div2(S(0')) → 0'
div2(0') → 0'
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
div2 :: S:0' → S:0'
S :: S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
div2

(8) Obligation:

Innermost TRS:
Rules:
div2(S(S(x))) → +'(S(0'), div2(x))
div2(S(0')) → 0'
div2(0') → 0'
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
div2 :: S:0' → S:0'
S :: S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

The following defined symbols remain to be analysed:
div2

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

Proved the following rewrite lemma:
div2(gen_S:0'2_0(+(1, *(2, n4_0)))) → gen_S:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
div2(gen_S:0'2_0(+(1, *(2, 0)))) →_R^Ω(1)
0'

Induction Step:
div2(gen_S:0'2_0(+(1, *(2, +(n4_0, 1))))) →_R^Ω(1)
+'(S(0'), div2(gen_S:0'2_0(+(1, *(2, n4_0))))) →_IH
+'(S(0'), gen_S:0'2_0(c5_0)) →_R^Ω(0)
S(gen_S:0'2_0(n4_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:
div2(S(S(x))) → +'(S(0'), div2(x))
div2(S(0')) → 0'
div2(0') → 0'
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
div2 :: S:0' → S:0'
S :: S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
div2(gen_S:0'2_0(+(1, *(2, n4_0)))) → gen_S:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
div2(gen_S:0'2_0(+(1, *(2, n4_0)))) → gen_S:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
div2(S(S(x))) → +'(S(0'), div2(x))
div2(S(0')) → 0'
div2(0') → 0'
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
div2 :: S:0' → S:0'
S :: S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
div2(gen_S:0'2_0(+(1, *(2, n4_0)))) → gen_S:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
div2(gen_S:0'2_0(+(1, *(2, n4_0)))) → gen_S:0'2_0(n4_0), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)