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:

dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))

Rewrite Strategy: FULL

(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:

dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))

Types:
dfib :: s → dfib → dfib
s :: s → s
hole_dfib1_0 :: dfib
hole_s2_0 :: s
gen_s3_0 :: Nat → s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
dfib

(6) Obligation:

TRS:
Rules:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))

Types:
dfib :: s → dfib → dfib
s :: s → s
hole_dfib1_0 :: dfib
hole_s2_0 :: s
gen_s3_0 :: Nat → s

Generator Equations:
gen_s3_0(0) ⇔ hole_s2_0
gen_s3_0(+(x, 1)) ⇔ s(gen_s3_0(x))

The following defined symbols remain to be analysed:
dfib

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

Proved the following rewrite lemma:
dfib(gen_s3_0(+(2, *(2, n5_0))), hole_dfib1_0) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
dfib(gen_s3_0(+(2, *(2, 0))), hole_dfib1_0)

Induction Step:
dfib(gen_s3_0(+(2, *(2, +(n5_0, 1)))), hole_dfib1_0) →_R^Ω(1)
dfib(s(gen_s3_0(+(2, *(2, n5_0)))), dfib(gen_s3_0(+(2, *(2, n5_0))), hole_dfib1_0)) →_IH
dfib(s(gen_s3_0(+(2, *(2, n5_0)))), *4_0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))

Types:
dfib :: s → dfib → dfib
s :: s → s
hole_dfib1_0 :: dfib
hole_s2_0 :: s
gen_s3_0 :: Nat → s

Lemmas:
dfib(gen_s3_0(+(2, *(2, n5_0))), hole_dfib1_0) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_s3_0(0) ⇔ hole_s2_0
gen_s3_0(+(x, 1)) ⇔ s(gen_s3_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dfib(gen_s3_0(+(2, *(2, n5_0))), hole_dfib1_0) → *4_0, rt ∈ Ω(n5₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

TRS:
Rules:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))

Types:
dfib :: s → dfib → dfib
s :: s → s
hole_dfib1_0 :: dfib
hole_s2_0 :: s
gen_s3_0 :: Nat → s

Lemmas:
dfib(gen_s3_0(+(2, *(2, n5_0))), hole_dfib1_0) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_s3_0(0) ⇔ hole_s2_0
gen_s3_0(+(x, 1)) ⇔ s(gen_s3_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dfib(gen_s3_0(+(2, *(2, n5_0))), hole_dfib1_0) → *4_0, rt ∈ Ω(n5₀)

(14) BOUNDS(n^1, INF)