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:

f(x, 0, 0) → s(x)
f(0, y, 0) → s(y)
f(0, 0, z) → s(z)
f(s(0), y, z) → f(0, s(y), s(z))
f(s(x), s(y), 0) → f(x, y, s(0))
f(s(x), 0, s(z)) → f(x, s(0), z)
f(0, s(0), s(0)) → s(s(0))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0, s(s(y)), s(0)) → f(0, y, s(0))
f(0, s(0), s(s(z))) → f(0, s(0), z)
f(0, s(s(y)), s(s(z))) → f(0, y, f(0, s(s(y)), s(z)))

Rewrite Strategy: FULL

(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:

f(x, 0', 0') → s(x)
f(0', y, 0') → s(y)
f(0', 0', z) → s(z)
f(s(0'), y, z) → f(0', s(y), s(z))
f(s(x), s(y), 0') → f(x, y, s(0'))
f(s(x), 0', s(z)) → f(x, s(0'), z)
f(0', s(0'), s(0')) → s(s(0'))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0', s(s(y)), s(0')) → f(0', y, s(0'))
f(0', s(0'), s(s(z))) → f(0', s(0'), z)
f(0', s(s(y)), s(s(z))) → f(0', y, f(0', s(s(y)), s(z)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

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

Types:
f :: 0':s → 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

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f

(6) Obligation:

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

Types:
f :: 0':s → 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:
f

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, n4_0))), gen_0':s2_0(1)) → gen_0':s2_0(2), rt ∈ Ω(1 + n4₀)

Induction Base:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, 0))), gen_0':s2_0(1)) →_R^Ω(1)
s(s(0'))

Induction Step:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, +(n4_0, 1)))), gen_0':s2_0(1)) →_R^Ω(1)
f(0', gen_0':s2_0(+(1, *(2, n4_0))), s(0')) →_IH
gen_0':s2_0(2)

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

(8) Complex Obligation (BEST)

(9) Obligation:

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

Types:
f :: 0':s → 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

Lemmas:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, n4_0))), gen_0':s2_0(1)) → gen_0':s2_0(2), rt ∈ Ω(1 + n4₀)

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.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, n4_0))), gen_0':s2_0(1)) → gen_0':s2_0(2), rt ∈ Ω(1 + n4₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

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

Types:
f :: 0':s → 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

Lemmas:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, n4_0))), gen_0':s2_0(1)) → gen_0':s2_0(2), rt ∈ Ω(1 + n4₀)

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.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, n4_0))), gen_0':s2_0(1)) → gen_0':s2_0(2), rt ∈ Ω(1 + n4₀)

(14) BOUNDS(n^1, INF)