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:

fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(0))) → s(0)
fib(s(s(x))) → sp(g(x))
g(0) → pair(s(0), 0)
g(s(0)) → pair(s(0), s(0))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +(x, y)
np(pair(x, y)) → pair(+(x, y), x)
+(x, 0) → x
+(x, s(y)) → s(+(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
g(s(x)) →⁺ np(g(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x)].
The result substitution is [ ].

(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:

fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(0'))) → s(0')
fib(s(s(x))) → sp(g(x))
g(0') → pair(s(0'), 0')
g(s(0')) → pair(s(0'), s(0'))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +'(x, y)
np(pair(x, y)) → pair(+'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(0'))) → s(0')
fib(s(s(x))) → sp(g(x))
g(0') → pair(s(0'), 0')
g(s(0')) → pair(s(0'), s(0'))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +'(x, y)
np(pair(x, y)) → pair(+'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
fib :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sp :: pair → 0':s
g :: 0':s → pair
pair :: 0':s → 0':s → pair
np :: pair → pair
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
g, +'

(8) Obligation:

TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(0'))) → s(0')
fib(s(s(x))) → sp(g(x))
g(0') → pair(s(0'), 0')
g(s(0')) → pair(s(0'), s(0'))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +'(x, y)
np(pair(x, y)) → pair(+'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
fib :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sp :: pair → 0':s
g :: 0':s → pair
pair :: 0':s → 0':s → pair
np :: pair → pair
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
g, +'

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol g.

(10) Obligation:

TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(0'))) → s(0')
fib(s(s(x))) → sp(g(x))
g(0') → pair(s(0'), 0')
g(s(0')) → pair(s(0'), s(0'))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +'(x, y)
np(pair(x, y)) → pair(+'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
fib :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sp :: pair → 0':s
g :: 0':s → pair
pair :: 0':s → 0':s → pair
np :: pair → pair
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
+'

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n4793_0)) → gen_0':s3_0(+(n4793_0, a)), rt ∈ Ω(1 + n4793₀)

Induction Base:
+'(gen_0':s3_0(a), gen_0':s3_0(0)) →_R^Ω(1)
gen_0':s3_0(a)

Induction Step:
+'(gen_0':s3_0(a), gen_0':s3_0(+(n4793_0, 1))) →_R^Ω(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n4793_0))) →_IH
s(gen_0':s3_0(+(a, c4794_0)))

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(0'))) → s(0')
fib(s(s(x))) → sp(g(x))
g(0') → pair(s(0'), 0')
g(s(0')) → pair(s(0'), s(0'))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +'(x, y)
np(pair(x, y)) → pair(+'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
fib :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sp :: pair → 0':s
g :: 0':s → pair
pair :: 0':s → 0':s → pair
np :: pair → pair
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n4793_0)) → gen_0':s3_0(+(n4793_0, a)), rt ∈ Ω(1 + n4793₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n4793_0)) → gen_0':s3_0(+(n4793_0, a)), rt ∈ Ω(1 + n4793₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
fib(0') → 0'
fib(s(0')) → s(0')
fib(s(s(0'))) → s(0')
fib(s(s(x))) → sp(g(x))
g(0') → pair(s(0'), 0')
g(s(0')) → pair(s(0'), s(0'))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +'(x, y)
np(pair(x, y)) → pair(+'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
fib :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sp :: pair → 0':s
g :: 0':s → pair
pair :: 0':s → 0':s → pair
np :: pair → pair
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n4793_0)) → gen_0':s3_0(+(n4793_0, a)), rt ∈ Ω(1 + n4793₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n4793_0)) → gen_0':s3_0(+(n4793_0, a)), rt ∈ Ω(1 + n4793₀)

(18) BOUNDS(n^1, INF)