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:

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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
+(s(x), s(s(x14_1))) →⁺ s(+(s(x), s(x14_1)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x14_1 / s(x14_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:

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

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

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

Heuristically decided to analyse the following defined symbols:
+'

(8) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
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:
+'

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

Could not prove a rewrite lemma for the defined symbol +'.

(10) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
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))

No more defined symbols left to analyse.