+(+(x, y), z) → +(x, +(y, z))
+(f(x), f(y)) → f(+(x, y))
+(f(x), +(f(y), z)) → +(f(+(x, y)), 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, z))
+'(f'(x), f'(y)) → f'(+'(x, y))
+'(f'(x), +'(f'(y), z)) → +'(f'(+'(x, y)), z)
Infered types.
Rules:
+'(+'(x, y), z) → +'(x, +'(y, z))
+'(f'(x), f'(y)) → f'(+'(x, y))
+'(f'(x), +'(f'(y), z)) → +'(f'(+'(x, y)), z)
Types:
+' :: f' → f' → f'
f' :: f' → f'
_hole_f'1 :: f'
_gen_f'2 :: Nat → f'
Heuristically decided to analyse the following defined symbols:
+'
Rules:
+'(+'(x, y), z) → +'(x, +'(y, z))
+'(f'(x), f'(y)) → f'(+'(x, y))
+'(f'(x), +'(f'(y), z)) → +'(f'(+'(x, y)), z)
Types:
+' :: f' → f' → f'
f' :: f' → f'
_hole_f'1 :: f'
_gen_f'2 :: Nat → f'
Generator Equations:
_gen_f'2(0) ⇔ _hole_f'1
_gen_f'2(+(x, 1)) ⇔ f'(_gen_f'2(x))
The following defined symbols remain to be analysed:
+'
Proved the following rewrite lemma:
+'(_gen_f'2(+(1, _n4)), _gen_f'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Induction Base:
+'(_gen_f'2(+(1, 0)), _gen_f'2(+(1, 0)))
Induction Step:
+'(_gen_f'2(+(1, +(_$n5, 1))), _gen_f'2(+(1, +(_$n5, 1)))) →RΩ(1)
f'(+'(_gen_f'2(+(1, _$n5)), _gen_f'2(+(1, _$n5)))) →IH
f'(_*3)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
+'(+'(x, y), z) → +'(x, +'(y, z))
+'(f'(x), f'(y)) → f'(+'(x, y))
+'(f'(x), +'(f'(y), z)) → +'(f'(+'(x, y)), z)
Types:
+' :: f' → f' → f'
f' :: f' → f'
_hole_f'1 :: f'
_gen_f'2 :: Nat → f'
Lemmas:
+'(_gen_f'2(+(1, _n4)), _gen_f'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_f'2(0) ⇔ _hole_f'1
_gen_f'2(+(x, 1)) ⇔ f'(_gen_f'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
+'(_gen_f'2(+(1, _n4)), _gen_f'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)