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))
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))
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)