dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
dfib'(s'(s'(x)), y) → dfib'(s'(x), dfib'(x, y))
Infered types.
Rules:
dfib'(s'(s'(x)), y) → dfib'(s'(x), dfib'(x, y))
Types:
dfib' :: s' → dfib' → dfib'
s' :: s' → s'
_hole_dfib'1 :: dfib'
_hole_s'2 :: s'
_gen_s'3 :: Nat → s'
Heuristically decided to analyse the following defined symbols:
dfib'
Rules:
dfib'(s'(s'(x)), y) → dfib'(s'(x), dfib'(x, y))
Types:
dfib' :: s' → dfib' → dfib'
s' :: s' → s'
_hole_dfib'1 :: dfib'
_hole_s'2 :: s'
_gen_s'3 :: Nat → s'
Generator Equations:
_gen_s'3(0) ⇔ _hole_s'2
_gen_s'3(+(x, 1)) ⇔ s'(_gen_s'3(x))
The following defined symbols remain to be analysed:
dfib'
Proved the following rewrite lemma:
dfib'(_gen_s'3(+(2, *(2, _n5))), _hole_dfib'1) → _*4, rt ∈ Ω(n5)
Induction Base:
dfib'(_gen_s'3(+(2, *(2, 0))), _hole_dfib'1)
Induction Step:
dfib'(_gen_s'3(+(2, *(2, +(_$n6, 1)))), _hole_dfib'1) →RΩ(1)
dfib'(s'(_gen_s'3(+(2, *(2, _$n6)))), dfib'(_gen_s'3(+(2, *(2, _$n6))), _hole_dfib'1)) →IH
dfib'(s'(_gen_s'3(+(2, *(2, _$n6)))), _*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
dfib'(s'(s'(x)), y) → dfib'(s'(x), dfib'(x, y))
Types:
dfib' :: s' → dfib' → dfib'
s' :: s' → s'
_hole_dfib'1 :: dfib'
_hole_s'2 :: s'
_gen_s'3 :: Nat → s'
Lemmas:
dfib'(_gen_s'3(+(2, *(2, _n5))), _hole_dfib'1) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_s'3(0) ⇔ _hole_s'2
_gen_s'3(+(x, 1)) ⇔ s'(_gen_s'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
dfib'(_gen_s'3(+(2, *(2, _n5))), _hole_dfib'1) → _*4, rt ∈ Ω(n5)