*(x, +(y, z)) → +(*(x, y), *(x, z))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
*'(x, +'(y, z)) → +'(*'(x, y), *'(x, z))
Sliced the following arguments:
*'/0
Runtime Complexity TRS:
The TRS R consists of the following rules:
*'(+'(y, z)) → +'(*'(y), *'(z))
Infered types.
Rules:
*'(+'(y, z)) → +'(*'(y), *'(z))
Types:
*' :: +' → +'
+' :: +' → +' → +'
_hole_+'1 :: +'
_gen_+'2 :: Nat → +'
Heuristically decided to analyse the following defined symbols:
*'
Rules:
*'(+'(y, z)) → +'(*'(y), *'(z))
Types:
*' :: +' → +'
+' :: +' → +' → +'
_hole_+'1 :: +'
_gen_+'2 :: Nat → +'
Generator Equations:
_gen_+'2(0) ⇔ _hole_+'1
_gen_+'2(+(x, 1)) ⇔ +'(_hole_+'1, _gen_+'2(x))
The following defined symbols remain to be analysed:
*'
Proved the following rewrite lemma:
*'(_gen_+'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Induction Base:
*'(_gen_+'2(+(1, 0)))
Induction Step:
*'(_gen_+'2(+(1, +(_$n5, 1)))) →RΩ(1)
+'(*'(_hole_+'1), *'(_gen_+'2(+(1, _$n5)))) →IH
+'(*'(_hole_+'1), _*3)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
*'(+'(y, z)) → +'(*'(y), *'(z))
Types:
*' :: +' → +'
+' :: +' → +' → +'
_hole_+'1 :: +'
_gen_+'2 :: Nat → +'
Lemmas:
*'(_gen_+'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_+'2(0) ⇔ _hole_+'1
_gen_+'2(+(x, 1)) ⇔ +'(_hole_+'1, _gen_+'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
*'(_gen_+'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)