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(0, y) → 0
f(s(x), y) → f(f(x, y), y)

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

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
f/1

(6) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

f(0') → 0'
f(s(x)) → f(f(x))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

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

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f

(10) Obligation:

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

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
f

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

Induction Base:
f(gen_0':s2_0(0)) →_R^Ω(1)
0'

Induction Step:
f(gen_0':s2_0(+(n4_0, 1))) →_R^Ω(1)
f(f(gen_0':s2_0(n4_0))) →_IH
f(gen_0':s2_0(0)) →_R^Ω(1)
0'

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
f(gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_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_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
f :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
f(gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_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_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)