### (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 [ ].

### (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

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) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
g(gen_0':s3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Induction Base:
g(gen_0':s3_0(+(1, 0)))

Induction Step:
g(gen_0':s3_0(+(1, +(n5_0, 1)))) →RΩ(1)
np(g(gen_0':s3_0(+(1, n5_0)))) →IH
np(*4_0)

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

### (11) 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:
g(gen_0':s3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

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:
+'

### (12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n4858_0)) → gen_0':s3_0(+(n4858_0, a)), rt ∈ Ω(1 + n48580)

Induction Base:
+'(gen_0':s3_0(a), gen_0':s3_0(0)) →RΩ(1)
gen_0':s3_0(a)

Induction Step:
+'(gen_0':s3_0(a), gen_0':s3_0(+(n4858_0, 1))) →RΩ(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n4858_0))) →IH
s(gen_0':s3_0(+(a, c4859_0)))

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

### (14) 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:
g(gen_0':s3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
+'(gen_0':s3_0(a), gen_0':s3_0(n4858_0)) → gen_0':s3_0(+(n4858_0, a)), rt ∈ Ω(1 + n48580)

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.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
g(gen_0':s3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (17) 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:
g(gen_0':s3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
+'(gen_0':s3_0(a), gen_0':s3_0(n4858_0)) → gen_0':s3_0(+(n4858_0, a)), rt ∈ Ω(1 + n48580)

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.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
g(gen_0':s3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (20) 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:
g(gen_0':s3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

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.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
g(gen_0':s3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)