f(s(x1), x2, x3, x4, x5, x6, x7, x8, x9, x10) → f(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f(0, s(x2), x3, x4, x5, x6, x7, x8, x9, x10) → f(x2, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f(0, 0, s(x3), x4, x5, x6, x7, x8, x9, x10) → f(x3, x3, x3, x4, x5, x6, x7, x8, x9, x10)
f(0, 0, 0, s(x4), x5, x6, x7, x8, x9, x10) → f(x4, x4, x4, x4, x5, x6, x7, x8, x9, x10)
f(0, 0, 0, 0, s(x5), x6, x7, x8, x9, x10) → f(x5, x5, x5, x5, x5, x6, x7, x8, x9, x10)
f(0, 0, 0, 0, 0, s(x6), x7, x8, x9, x10) → f(x6, x6, x6, x6, x6, x6, x7, x8, x9, x10)
f(0, 0, 0, 0, 0, 0, s(x7), x8, x9, x10) → f(x7, x7, x7, x7, x7, x7, x7, x8, x9, x10)
f(0, 0, 0, 0, 0, 0, 0, s(x8), x9, x10) → f(x8, x8, x8, x8, x8, x8, x8, x8, x9, x10)
f(0, 0, 0, 0, 0, 0, 0, 0, s(x9), x10) → f(x9, x9, x9, x9, x9, x9, x9, x9, x9, x10)
f(0, 0, 0, 0, 0, 0, 0, 0, 0, s(x10)) → f(x10, x10, x10, x10, x10, x10, x10, x10, x10, x10)
f(0, 0, 0, 0, 0, 0, 0, 0, 0, 0) → 0
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(s'(x1), x2, x3, x4, x5, x6, x7, x8, x9, x10) → f'(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', s'(x2), x3, x4, x5, x6, x7, x8, x9, x10) → f'(x2, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', s'(x3), x4, x5, x6, x7, x8, x9, x10) → f'(x3, x3, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', s'(x4), x5, x6, x7, x8, x9, x10) → f'(x4, x4, x4, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', s'(x5), x6, x7, x8, x9, x10) → f'(x5, x5, x5, x5, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', s'(x6), x7, x8, x9, x10) → f'(x6, x6, x6, x6, x6, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', s'(x7), x8, x9, x10) → f'(x7, x7, x7, x7, x7, x7, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', s'(x8), x9, x10) → f'(x8, x8, x8, x8, x8, x8, x8, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', s'(x9), x10) → f'(x9, x9, x9, x9, x9, x9, x9, x9, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', 0', s'(x10)) → f'(x10, x10, x10, x10, x10, x10, x10, x10, x10, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', 0', 0') → 0'
Infered types.
Rules:
f'(s'(x1), x2, x3, x4, x5, x6, x7, x8, x9, x10) → f'(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', s'(x2), x3, x4, x5, x6, x7, x8, x9, x10) → f'(x2, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', s'(x3), x4, x5, x6, x7, x8, x9, x10) → f'(x3, x3, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', s'(x4), x5, x6, x7, x8, x9, x10) → f'(x4, x4, x4, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', s'(x5), x6, x7, x8, x9, x10) → f'(x5, x5, x5, x5, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', s'(x6), x7, x8, x9, x10) → f'(x6, x6, x6, x6, x6, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', s'(x7), x8, x9, x10) → f'(x7, x7, x7, x7, x7, x7, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', s'(x8), x9, x10) → f'(x8, x8, x8, x8, x8, x8, x8, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', s'(x9), x10) → f'(x9, x9, x9, x9, x9, x9, x9, x9, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', 0', s'(x10)) → f'(x10, x10, x10, x10, x10, x10, x10, x10, x10, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', 0', 0') → 0'
Types:
f' :: s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
_hole_s':0'1 :: s':0'
_gen_s':0'2 :: Nat → s':0'
Heuristically decided to analyse the following defined symbols:
f'
Rules:
f'(s'(x1), x2, x3, x4, x5, x6, x7, x8, x9, x10) → f'(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', s'(x2), x3, x4, x5, x6, x7, x8, x9, x10) → f'(x2, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', s'(x3), x4, x5, x6, x7, x8, x9, x10) → f'(x3, x3, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', s'(x4), x5, x6, x7, x8, x9, x10) → f'(x4, x4, x4, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', s'(x5), x6, x7, x8, x9, x10) → f'(x5, x5, x5, x5, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', s'(x6), x7, x8, x9, x10) → f'(x6, x6, x6, x6, x6, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', s'(x7), x8, x9, x10) → f'(x7, x7, x7, x7, x7, x7, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', s'(x8), x9, x10) → f'(x8, x8, x8, x8, x8, x8, x8, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', s'(x9), x10) → f'(x9, x9, x9, x9, x9, x9, x9, x9, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', 0', s'(x10)) → f'(x10, x10, x10, x10, x10, x10, x10, x10, x10, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', 0', 0') → 0'
Types:
f' :: s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
_hole_s':0'1 :: s':0'
_gen_s':0'2 :: Nat → s':0'
Generator Equations:
_gen_s':0'2(0) ⇔ 0'
_gen_s':0'2(+(x, 1)) ⇔ s'(_gen_s':0'2(x))
The following defined symbols remain to be analysed:
f'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
f'(s'(x1), x2, x3, x4, x5, x6, x7, x8, x9, x10) → f'(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', s'(x2), x3, x4, x5, x6, x7, x8, x9, x10) → f'(x2, x2, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', s'(x3), x4, x5, x6, x7, x8, x9, x10) → f'(x3, x3, x3, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', s'(x4), x5, x6, x7, x8, x9, x10) → f'(x4, x4, x4, x4, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', s'(x5), x6, x7, x8, x9, x10) → f'(x5, x5, x5, x5, x5, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', s'(x6), x7, x8, x9, x10) → f'(x6, x6, x6, x6, x6, x6, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', s'(x7), x8, x9, x10) → f'(x7, x7, x7, x7, x7, x7, x7, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', s'(x8), x9, x10) → f'(x8, x8, x8, x8, x8, x8, x8, x8, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', s'(x9), x10) → f'(x9, x9, x9, x9, x9, x9, x9, x9, x9, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', 0', s'(x10)) → f'(x10, x10, x10, x10, x10, x10, x10, x10, x10, x10)
f'(0', 0', 0', 0', 0', 0', 0', 0', 0', 0') → 0'
Types:
f' :: s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
_hole_s':0'1 :: s':0'
_gen_s':0'2 :: Nat → s':0'
Generator Equations:
_gen_s':0'2(0) ⇔ 0'
_gen_s':0'2(+(x, 1)) ⇔ s'(_gen_s':0'2(x))
No more defined symbols left to analyse.