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

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

(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:
f < g

(6) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
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:
f, g

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

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

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

Induction Base:
f(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
s(0')

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

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

Lemmas:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, 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:
g

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

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

(11) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

Lemmas:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, 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.

(12) LowerBoundsProof (EQUIVALENT transformation)

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

(13) BOUNDS(n^1, INF)

(14) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

Lemmas:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, 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.

(15) LowerBoundsProof (EQUIVALENT transformation)

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

(16) BOUNDS(n^1, INF)