The Runtime Complexity (innermost) 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:

fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(x))

Rewrite Strategy: INNERMOST

(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:

fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(x))) → +'(fib(s(x)), fib(x))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(x))) → +'(fib(s(x)), fib(x))

Types:
fib :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

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

Heuristically decided to analyse the following defined symbols:
fib

(6) Obligation:

Innermost TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(x))) → +'(fib(s(x)), fib(x))

Types:
fib :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
fib

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

Proved the following rewrite lemma:
fib(gen_0':s:+'2_0(+(2, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
fib(gen_0':s:+'2_0(+(2, 0)))

Induction Step:
fib(gen_0':s:+'2_0(+(2, +(n4_0, 1)))) →_R^Ω(1)
+'(fib(s(gen_0':s:+'2_0(+(1, n4_0)))), fib(gen_0':s:+'2_0(+(1, n4_0)))) →_IH
+'(*3_0, fib(gen_0':s:+'2_0(+(1, n4_0))))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(x))) → +'(fib(s(x)), fib(x))

Types:
fib :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
fib(gen_0':s:+'2_0(+(2, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
fib(gen_0':s:+'2_0(+(2, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(x))) → +'(fib(s(x)), fib(x))

Types:
fib :: 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
fib(gen_0':s:+'2_0(+(2, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
fib(gen_0':s:+'2_0(+(2, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(14) BOUNDS(n^1, INF)