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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(s(x)) →⁺ f(x)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
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(f(x)) → f(x)
f(s(x)) → f(x)
g(s(0')) → g(f(s(0')))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

(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:
f < g

(8) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

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

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

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

Proved the following rewrite lemma:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
f(gen_s:0'3_0(+(1, 0)))

Induction Step:
f(gen_s:0'3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
f(gen_s:0'3_0(+(1, n5_0))) →_IH
*4_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(f(x)) → f(x)
f(s(x)) → f(x)
g(s(0')) → g(f(s(0')))

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
g

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

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

(13) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(18) BOUNDS(n^1, INF)