The Runtime Complexity (innermost) 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(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

Rewrite Strategy: INNERMOST

(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(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0') → t
g(s(x), s(y)) → g(x, y)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0') → t
g(s(x), s(y)) → g(x, y)

Types:
f :: t → s:0' → s:0' → f
t :: t
g :: s:0' → s:0' → t
s :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

(5) 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

(6) Obligation:

Innermost TRS:
Rules:
f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0') → t
g(s(x), s(y)) → g(x, y)

Types:
f :: t → s:0' → s:0' → f
t :: t
g :: s:0' → s:0' → t
s :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

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

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

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

Proved the following rewrite lemma:
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) → t, rt ∈ Ω(1 + n6₀)

Induction Base:
g(gen_s:0'4_0(+(1, 0)), gen_s:0'4_0(0)) →_R^Ω(1)
t

Induction Step:
g(gen_s:0'4_0(+(1, +(n6_0, 1))), gen_s:0'4_0(+(n6_0, 1))) →_R^Ω(1)
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) →_IH
t

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0') → t
g(s(x), s(y)) → g(x, y)

Types:
f :: t → s:0' → s:0' → f
t :: t
g :: s:0' → s:0' → t
s :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) → t, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
f

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0') → t
g(s(x), s(y)) → g(x, y)

Types:
f :: t → s:0' → s:0' → f
t :: t
g :: s:0' → s:0' → t
s :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) → t, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) → t, rt ∈ Ω(1 + n6₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0') → t
g(s(x), s(y)) → g(x, y)

Types:
f :: t → s:0' → s:0' → f
t :: t
g :: s:0' → s:0' → t
s :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) → t, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) → t, rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)