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:

add0(S(x), x2) → +(S(0), add0(x2, x))
add0(0, x2) → x2

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
add0(S(x), S(x11_1)) →⁺ +(S(0), +(S(0), add0(x, x11_1)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [x / S(x), x11_1 / S(x11_1)].
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:

add0(S(x), x2) → +'(S(0'), add0(x2, x))
add0(0', x2) → x2

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:
add0(S(x), x2) → +'(S(0'), add0(x2, x))
add0(0', x2) → x2
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
add0 :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
add0

(8) Obligation:

Innermost TRS:
Rules:
add0(S(x), x2) → +'(S(0'), add0(x2, x))
add0(0', x2) → x2
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
add0 :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

The following defined symbols remain to be analysed:
add0

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol add0.

(10) Obligation:

Innermost TRS:
Rules:
add0(S(x), x2) → +'(S(0'), add0(x2, x))
add0(0', x2) → x2
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
add0 :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

No more defined symbols left to analyse.