f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(0', 1', x) → f'(s'(x), x, x)
f'(x, y, s'(z)) → s'(f'(0', 1', z))
Infered types.
Rules:
f'(0', 1', x) → f'(s'(x), x, x)
f'(x, y, s'(z)) → s'(f'(0', 1', z))
Types:
f' :: 0':1':s' → 0':1':s' → 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
_hole_0':1':s'1 :: 0':1':s'
_gen_0':1':s'2 :: Nat → 0':1':s'
Heuristically decided to analyse the following defined symbols:
f'
Rules:
f'(0', 1', x) → f'(s'(x), x, x)
f'(x, y, s'(z)) → s'(f'(0', 1', z))
Types:
f' :: 0':1':s' → 0':1':s' → 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
_hole_0':1':s'1 :: 0':1':s'
_gen_0':1':s'2 :: Nat → 0':1':s'
Generator Equations:
_gen_0':1':s'2(0) ⇔ 1'
_gen_0':1':s'2(+(x, 1)) ⇔ s'(_gen_0':1':s'2(x))
The following defined symbols remain to be analysed:
f'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
f'(0', 1', x) → f'(s'(x), x, x)
f'(x, y, s'(z)) → s'(f'(0', 1', z))
Types:
f' :: 0':1':s' → 0':1':s' → 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
_hole_0':1':s'1 :: 0':1':s'
_gen_0':1':s'2 :: Nat → 0':1':s'
Generator Equations:
_gen_0':1':s'2(0) ⇔ 1'
_gen_0':1':s'2(+(x, 1)) ⇔ s'(_gen_0':1':s'2(x))
No more defined symbols left to analyse.