f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(f'(0', x), 1') → f'(g'(f'(x, x)), x)
f'(g'(x), y) → g'(f'(x, y))
Infered types.
Rules:
f'(f'(0', x), 1') → f'(g'(f'(x, x)), x)
f'(g'(x), y) → g'(f'(x, y))
Types:
f' :: 0':1':g' → 0':1':g' → 0':1':g'
0' :: 0':1':g'
1' :: 0':1':g'
g' :: 0':1':g' → 0':1':g'
_hole_0':1':g'1 :: 0':1':g'
_gen_0':1':g'2 :: Nat → 0':1':g'
Heuristically decided to analyse the following defined symbols:
f'
Rules:
f'(f'(0', x), 1') → f'(g'(f'(x, x)), x)
f'(g'(x), y) → g'(f'(x, y))
Types:
f' :: 0':1':g' → 0':1':g' → 0':1':g'
0' :: 0':1':g'
1' :: 0':1':g'
g' :: 0':1':g' → 0':1':g'
_hole_0':1':g'1 :: 0':1':g'
_gen_0':1':g'2 :: Nat → 0':1':g'
Generator Equations:
_gen_0':1':g'2(0) ⇔ 1'
_gen_0':1':g'2(+(x, 1)) ⇔ g'(_gen_0':1':g'2(x))
The following defined symbols remain to be analysed:
f'
Proved the following rewrite lemma:
f'(_gen_0':1':g'2(+(1, _n4)), _gen_0':1':g'2(b)) → _*3, rt ∈ Ω(n4)
Induction Base:
f'(_gen_0':1':g'2(+(1, 0)), _gen_0':1':g'2(b))
Induction Step:
f'(_gen_0':1':g'2(+(1, +(_$n5, 1))), _gen_0':1':g'2(_b555)) →RΩ(1)
g'(f'(_gen_0':1':g'2(+(1, _$n5)), _gen_0':1':g'2(_b555))) →IH
g'(_*3)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(f'(0', x), 1') → f'(g'(f'(x, x)), x)
f'(g'(x), y) → g'(f'(x, y))
Types:
f' :: 0':1':g' → 0':1':g' → 0':1':g'
0' :: 0':1':g'
1' :: 0':1':g'
g' :: 0':1':g' → 0':1':g'
_hole_0':1':g'1 :: 0':1':g'
_gen_0':1':g'2 :: Nat → 0':1':g'
Lemmas:
f'(_gen_0':1':g'2(+(1, _n4)), _gen_0':1':g'2(b)) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_0':1':g'2(0) ⇔ 1'
_gen_0':1':g'2(+(x, 1)) ⇔ g'(_gen_0':1':g'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_0':1':g'2(+(1, _n4)), _gen_0':1':g'2(b)) → _*3, rt ∈ Ω(n4)