Runtime Complexity TRS:
The TRS R consists of the following rules:

plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fib(x) → fibiter(x, 0, 0, s(0))
fibiter(b, c, x, y) → if(lt(c, b), b, c, x, y)
if(false, b, c, x, y) → x
if(true, b, c, x, y) → fibiter(b, s(c), y, plus(x, y))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
The TRS R consists of the following rules:

plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
lt'(0', s'(y)) → true'
lt'(x, 0') → false'
lt'(s'(x), s'(y)) → lt'(x, y)
fib'(x) → fibiter'(x, 0', 0', s'(0'))
fibiter'(b, c, x, y) → if'(lt'(c, b), b, c, x, y)
if'(false', b, c, x, y) → x
if'(true', b, c, x, y) → fibiter'(b, s'(c), y, plus'(x, y))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
lt'(0', s'(y)) → true'
lt'(x, 0') → false'
lt'(s'(x), s'(y)) → lt'(x, y)
fib'(x) → fibiter'(x, 0', 0', s'(0'))
fibiter'(b, c, x, y) → if'(lt'(c, b), b, c, x, y)
if'(false', b, c, x, y) → x
if'(true', b, c, x, y) → fibiter'(b, s'(c), y, plus'(x, y))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
fib' :: 0':s' → 0':s'
fibiter' :: 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
plus', lt', fibiter'

They will be analysed ascendingly in the following order:
plus' < fibiter'
lt' < fibiter'

Rules:
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
lt'(0', s'(y)) → true'
lt'(x, 0') → false'
lt'(s'(x), s'(y)) → lt'(x, y)
fib'(x) → fibiter'(x, 0', 0', s'(0'))
fibiter'(b, c, x, y) → if'(lt'(c, b), b, c, x, y)
if'(false', b, c, x, y) → x
if'(true', b, c, x, y) → fibiter'(b, s'(c), y, plus'(x, y))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
fib' :: 0':s' → 0':s'
fibiter' :: 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
plus', lt', fibiter'

They will be analysed ascendingly in the following order:
plus' < fibiter'
lt' < fibiter'

Proved the following rewrite lemma:
plus'(_gen_0':s'3(_n5), _gen_0':s'3(b)) → _gen_0':s'3(+(_n5, b)), rt ∈ Ω(1 + n5)

Induction Base:
plus'(_gen_0':s'3(0), _gen_0':s'3(b)) →RΩ(1)
_gen_0':s'3(b)

Induction Step:
plus'(_gen_0':s'3(+(_\$n6, 1)), _gen_0':s'3(_b138)) →RΩ(1)
s'(plus'(_gen_0':s'3(_\$n6), _gen_0':s'3(_b138))) →IH
s'(_gen_0':s'3(+(_\$n6, _b138)))

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

Rules:
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
lt'(0', s'(y)) → true'
lt'(x, 0') → false'
lt'(s'(x), s'(y)) → lt'(x, y)
fib'(x) → fibiter'(x, 0', 0', s'(0'))
fibiter'(b, c, x, y) → if'(lt'(c, b), b, c, x, y)
if'(false', b, c, x, y) → x
if'(true', b, c, x, y) → fibiter'(b, s'(c), y, plus'(x, y))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
fib' :: 0':s' → 0':s'
fibiter' :: 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
plus'(_gen_0':s'3(_n5), _gen_0':s'3(b)) → _gen_0':s'3(+(_n5, b)), rt ∈ Ω(1 + n5)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
lt', fibiter'

They will be analysed ascendingly in the following order:
lt' < fibiter'

Proved the following rewrite lemma:
lt'(_gen_0':s'3(_n759), _gen_0':s'3(+(1, _n759))) → true', rt ∈ Ω(1 + n759)

Induction Base:
lt'(_gen_0':s'3(0), _gen_0':s'3(+(1, 0))) →RΩ(1)
true'

Induction Step:
lt'(_gen_0':s'3(+(_\$n760, 1)), _gen_0':s'3(+(1, +(_\$n760, 1)))) →RΩ(1)
lt'(_gen_0':s'3(_\$n760), _gen_0':s'3(+(1, _\$n760))) →IH
true'

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

Rules:
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
lt'(0', s'(y)) → true'
lt'(x, 0') → false'
lt'(s'(x), s'(y)) → lt'(x, y)
fib'(x) → fibiter'(x, 0', 0', s'(0'))
fibiter'(b, c, x, y) → if'(lt'(c, b), b, c, x, y)
if'(false', b, c, x, y) → x
if'(true', b, c, x, y) → fibiter'(b, s'(c), y, plus'(x, y))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
fib' :: 0':s' → 0':s'
fibiter' :: 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
plus'(_gen_0':s'3(_n5), _gen_0':s'3(b)) → _gen_0':s'3(+(_n5, b)), rt ∈ Ω(1 + n5)
lt'(_gen_0':s'3(_n759), _gen_0':s'3(+(1, _n759))) → true', rt ∈ Ω(1 + n759)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
fibiter'

Could not prove a rewrite lemma for the defined symbol fibiter'.

Rules:
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
lt'(0', s'(y)) → true'
lt'(x, 0') → false'
lt'(s'(x), s'(y)) → lt'(x, y)
fib'(x) → fibiter'(x, 0', 0', s'(0'))
fibiter'(b, c, x, y) → if'(lt'(c, b), b, c, x, y)
if'(false', b, c, x, y) → x
if'(true', b, c, x, y) → fibiter'(b, s'(c), y, plus'(x, y))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
fib' :: 0':s' → 0':s'
fibiter' :: 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
plus'(_gen_0':s'3(_n5), _gen_0':s'3(b)) → _gen_0':s'3(+(_n5, b)), rt ∈ Ω(1 + n5)
lt'(_gen_0':s'3(_n759), _gen_0':s'3(+(1, _n759))) → true', rt ∈ Ω(1 + n759)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
plus'(_gen_0':s'3(_n5), _gen_0':s'3(b)) → _gen_0':s'3(+(_n5, b)), rt ∈ Ω(1 + n5)