a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) → a(f(x, y))
f(b(x), b(y)) → b(f(x, y))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
a'(a'(f'(x, y))) → f'(a'(b'(a'(b'(a'(x))))), a'(b'(a'(b'(a'(y))))))
f'(a'(x), a'(y)) → a'(f'(x, y))
f'(b'(x), b'(y)) → b'(f'(x, y))
Infered types.
Rules:
a'(a'(f'(x, y))) → f'(a'(b'(a'(b'(a'(x))))), a'(b'(a'(b'(a'(y))))))
f'(a'(x), a'(y)) → a'(f'(x, y))
f'(b'(x), b'(y)) → b'(f'(x, y))
Types:
a' :: b' → b'
f' :: b' → b' → b'
b' :: b' → b'
_hole_b'1 :: b'
_gen_b'2 :: Nat → b'
Heuristically decided to analyse the following defined symbols:
a', f'
They will be analysed ascendingly in the following order:
a' = f'
Rules:
a'(a'(f'(x, y))) → f'(a'(b'(a'(b'(a'(x))))), a'(b'(a'(b'(a'(y))))))
f'(a'(x), a'(y)) → a'(f'(x, y))
f'(b'(x), b'(y)) → b'(f'(x, y))
Types:
a' :: b' → b'
f' :: b' → b' → b'
b' :: b' → b'
_hole_b'1 :: b'
_gen_b'2 :: Nat → b'
Generator Equations:
_gen_b'2(0) ⇔ _hole_b'1
_gen_b'2(+(x, 1)) ⇔ b'(_gen_b'2(x))
The following defined symbols remain to be analysed:
f', a'
They will be analysed ascendingly in the following order:
a' = f'
Proved the following rewrite lemma:
f'(_gen_b'2(+(1, _n4)), _gen_b'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Induction Base:
f'(_gen_b'2(+(1, 0)), _gen_b'2(+(1, 0)))
Induction Step:
f'(_gen_b'2(+(1, +(_$n5, 1))), _gen_b'2(+(1, +(_$n5, 1)))) →RΩ(1)
b'(f'(_gen_b'2(+(1, _$n5)), _gen_b'2(+(1, _$n5)))) →IH
b'(_*3)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
a'(a'(f'(x, y))) → f'(a'(b'(a'(b'(a'(x))))), a'(b'(a'(b'(a'(y))))))
f'(a'(x), a'(y)) → a'(f'(x, y))
f'(b'(x), b'(y)) → b'(f'(x, y))
Types:
a' :: b' → b'
f' :: b' → b' → b'
b' :: b' → b'
_hole_b'1 :: b'
_gen_b'2 :: Nat → b'
Lemmas:
f'(_gen_b'2(+(1, _n4)), _gen_b'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_b'2(0) ⇔ _hole_b'1
_gen_b'2(+(x, 1)) ⇔ b'(_gen_b'2(x))
The following defined symbols remain to be analysed:
a'
They will be analysed ascendingly in the following order:
a' = f'
Could not prove a rewrite lemma for the defined symbol a'.
Rules:
a'(a'(f'(x, y))) → f'(a'(b'(a'(b'(a'(x))))), a'(b'(a'(b'(a'(y))))))
f'(a'(x), a'(y)) → a'(f'(x, y))
f'(b'(x), b'(y)) → b'(f'(x, y))
Types:
a' :: b' → b'
f' :: b' → b' → b'
b' :: b' → b'
_hole_b'1 :: b'
_gen_b'2 :: Nat → b'
Lemmas:
f'(_gen_b'2(+(1, _n4)), _gen_b'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_b'2(0) ⇔ _hole_b'1
_gen_b'2(+(x, 1)) ⇔ b'(_gen_b'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_b'2(+(1, _n4)), _gen_b'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)