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:

average(s(x), y) → average(x, s(y))
average(x, s(s(s(y)))) → s(average(s(x), y))
average(0, 0) → 0
average(0, s(0)) → 0
average(0, s(s(0))) → 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
average(s(x), y) →⁺ average(x, s(y))
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 [y / s(y)].

(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:

average(s(x), y) → average(x, s(y))
average(x, s(s(s(y)))) → s(average(s(x), y))
average(0', 0') → 0'
average(0', s(0')) → 0'
average(0', s(s(0'))) → s(0')

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
average(s(x), y) → average(x, s(y))
average(x, s(s(s(y)))) → s(average(s(x), y))
average(0', 0') → 0'
average(0', s(0')) → 0'
average(0', s(s(0'))) → s(0')

Types:
average :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
average

(8) Obligation:

TRS:
Rules:
average(s(x), y) → average(x, s(y))
average(x, s(s(s(y)))) → s(average(s(x), y))
average(0', 0') → 0'
average(0', s(0')) → 0'
average(0', s(s(0'))) → s(0')

Types:
average :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
average

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

Proved the following rewrite lemma:
average(gen_s:0'2_0(+(1, n4_0)), gen_s:0'2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
average(gen_s:0'2_0(+(1, 0)), gen_s:0'2_0(b))

Induction Step:
average(gen_s:0'2_0(+(1, +(n4_0, 1))), gen_s:0'2_0(b)) →_R^Ω(1)
average(gen_s:0'2_0(+(1, n4_0)), s(gen_s:0'2_0(b))) →_IH
*3_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:
average(s(x), y) → average(x, s(y))
average(x, s(s(s(y)))) → s(average(s(x), y))
average(0', 0') → 0'
average(0', s(0')) → 0'
average(0', s(s(0'))) → s(0')

Types:
average :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
average(gen_s:0'2_0(+(1, n4_0)), gen_s:0'2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
average(gen_s:0'2_0(+(1, n4_0)), gen_s:0'2_0(b)) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
average(s(x), y) → average(x, s(y))
average(x, s(s(s(y)))) → s(average(s(x), y))
average(0', 0') → 0'
average(0', s(0')) → 0'
average(0', s(s(0'))) → s(0')

Types:
average :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
average(gen_s:0'2_0(+(1, n4_0)), gen_s:0'2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
average(gen_s:0'2_0(+(1, n4_0)), gen_s:0'2_0(b)) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)