+(0, y) → y
+(s(x), y) → s(+(x, y))
+(s(x), y) → +(x, s(y))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
+'(0', y) → y
+'(s'(x), y) → s'(+'(x, y))
+'(s'(x), y) → +'(x, s'(y))
Infered types.
Rules:
+'(0', y) → y
+'(s'(x), y) → s'(+'(x, y))
+'(s'(x), y) → +'(x, s'(y))
Types:
+' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
+'
Rules:
+'(0', y) → y
+'(s'(x), y) → s'(+'(x, y))
+'(s'(x), y) → +'(x, s'(y))
Types:
+' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
The following defined symbols remain to be analysed:
+'
Proved the following rewrite lemma:
+'(_gen_0':s'2(_n4), _gen_0':s'2(b)) → _gen_0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)
Induction Base:
+'(_gen_0':s'2(0), _gen_0':s'2(b)) →RΩ(1)
_gen_0':s'2(b)
Induction Step:
+'(_gen_0':s'2(+(_$n5, 1)), _gen_0':s'2(_b169)) →RΩ(1)
s'(+'(_gen_0':s'2(_$n5), _gen_0':s'2(_b169))) →IH
s'(_gen_0':s'2(+(_$n5, _b169)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
+'(0', y) → y
+'(s'(x), y) → s'(+'(x, y))
+'(s'(x), y) → +'(x, s'(y))
Types:
+' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
+'(_gen_0':s'2(_n4), _gen_0':s'2(b)) → _gen_0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
+'(_gen_0':s'2(_n4), _gen_0':s'2(b)) → _gen_0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)