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