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(s(x), x) → f(s(x), round(s(x)))
round(0) → 0
round(0) → s(0)
round(s(0)) → s(0)
round(s(s(x))) → s(s(round(x)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

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

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

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

(8) Obligation:

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

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
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:
round, f

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

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

Proved the following rewrite lemma:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

Induction Base:
round(gen_s:0'3_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
round(gen_s:0'3_0(*(2, +(n5_0, 1)))) →_R^Ω(1)
s(s(round(gen_s:0'3_0(*(2, n5_0))))) →_IH
s(s(gen_s:0'3_0(*(2, 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(s(x), x) → f(s(x), round(s(x)))
round(0') → 0'
round(0') → s(0')
round(s(0')) → s(0')
round(s(s(x))) → s(s(round(x)))

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + 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:
f

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

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

(13) Obligation:

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

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + 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:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + 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:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)