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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
dfib(s(s(x)), y) →⁺ dfib(s(x), dfib(x, 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 / dfib(x, 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:

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(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

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

Heuristically decided to analyse the following defined symbols:
dfib

(8) 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

(9) 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).

(10) Complex Obligation (BEST)

(11) 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.

(12) 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)

(13) BOUNDS(n^1, INF)

(14) 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.

(15) 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)

(16) BOUNDS(n^1, INF)