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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
g(s(x)) →⁺ s(g(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:

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s → f
g :: 0':s → 0':s
s :: 0':s → 0':s
0' :: 0':s
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
f, g

They will be analysed ascendingly in the following order:
g < f

(8) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s → f
g :: 0':s → 0':s
s :: 0':s → 0':s
0' :: 0':s
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
g, f

They will be analysed ascendingly in the following order:
g < f

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

Proved the following rewrite lemma:
g(gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

Induction Base:
g(gen_0':s3_0(0)) →_R^Ω(1)
0'

Induction Step:
g(gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
s(g(gen_0':s3_0(n5_0))) →_IH
s(gen_0':s3_0(c6_0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s → f
g :: 0':s → 0':s
s :: 0':s → 0':s
0' :: 0':s
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
g(gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
f

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s → f
g :: 0':s → 0':s
s :: 0':s → 0':s
0' :: 0':s
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
g(gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s → f
g :: 0':s → 0':s
s :: 0':s → 0':s
0' :: 0':s
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
g(gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)