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:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fibo(0) → fib(0)
fibo(s(0)) → fib(s(0))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0) → s(0)
fib(s(0)) → s(0)
fib(s(s(x))) → if(true, 0, s(s(x)), 0, 0)
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))

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:

lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
lt, fibo, fib, sum, if

They will be analysed ascendingly in the following order:
lt < if
fibo = fib
sum < fibo
fibo = if
fib = if
sum < if

(6) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
lt, fibo, fib, sum, if

They will be analysed ascendingly in the following order:
lt < if
fibo = fib
sum < fibo
fibo = if
fib = if
sum < if

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

Proved the following rewrite lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

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

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) →_IH
true

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:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + 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:
sum, fibo, fib, if

They will be analysed ascendingly in the following order:
fibo = fib
sum < fibo
fibo = if
fib = if
sum < if

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_0':s3_0(a), gen_0':s3_0(n354_0)) → gen_0':s3_0(+(n354_0, a)), rt ∈ Ω(1 + n354₀)

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

Induction Step:
sum(gen_0':s3_0(a), gen_0':s3_0(+(n354_0, 1))) →_R^Ω(1)
s(sum(gen_0':s3_0(a), gen_0':s3_0(n354_0))) →_IH
s(gen_0':s3_0(+(a, c355_0)))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
sum(gen_0':s3_0(a), gen_0':s3_0(n354_0)) → gen_0':s3_0(+(n354_0, a)), rt ∈ Ω(1 + n354₀)

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:
fib, fibo, if

They will be analysed ascendingly in the following order:
fibo = fib
fibo = if
fib = if

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
sum(gen_0':s3_0(a), gen_0':s3_0(n354_0)) → gen_0':s3_0(+(n354_0, a)), rt ∈ Ω(1 + n354₀)

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:
if, fibo

They will be analysed ascendingly in the following order:
fibo = fib
fibo = if
fib = if

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
sum(gen_0':s3_0(a), gen_0':s3_0(n354_0)) → gen_0':s3_0(+(n354_0, a)), rt ∈ Ω(1 + n354₀)

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:
fibo

They will be analysed ascendingly in the following order:
fibo = fib
fibo = if
fib = if

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
sum(gen_0':s3_0(a), gen_0':s3_0(n354_0)) → gen_0':s3_0(+(n354_0, a)), rt ∈ Ω(1 + n354₀)

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.

(19) LowerBoundsProof (EQUIVALENT transformation)

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

(20) BOUNDS(n^1, INF)

(21) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
sum(gen_0':s3_0(a), gen_0':s3_0(n354_0)) → gen_0':s3_0(+(n354_0, a)), rt ∈ Ω(1 + n354₀)

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.

(22) LowerBoundsProof (EQUIVALENT transformation)

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

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
fibo(0') → fib(0')
fibo(s(0')) → fib(s(0'))
fibo(s(s(x))) → sum(fibo(s(x)), fibo(x))
fib(0') → s(0')
fib(s(0')) → s(0')
fib(s(s(x))) → if(true, 0', s(s(x)), 0', 0')
if(true, c, s(s(x)), a, b) → if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
if(false, c, s(s(x)), a, b) → sum(fibo(a), fibo(b))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
fibo :: 0':s → 0':s
fib :: 0':s → 0':s
sum :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + 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.

(25) LowerBoundsProof (EQUIVALENT transformation)

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

(26) BOUNDS(n^1, INF)