### (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)].

### (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

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) →RΩ(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 ircR ∈ Ω(n).

### (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)

### (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)