g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(f'(x)) → f'(d'(f'(x)))
f'(x) → x
Infered types.
Rules:
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(f'(x)) → f'(d'(f'(x)))
f'(x) → x
Types:
g' :: c':d' → g'
c' :: s' → s' → c':d'
s' :: s' → s'
f' :: c':d' → c':d'
d' :: c':d' → c':d'
_hole_g'1 :: g'
_hole_c':d'2 :: c':d'
_hole_s'3 :: s'
_gen_c':d'4 :: Nat → c':d'
_gen_s'5 :: Nat → s'
Heuristically decided to analyse the following defined symbols:
g', f'
Rules:
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(f'(x)) → f'(d'(f'(x)))
f'(x) → x
Types:
g' :: c':d' → g'
c' :: s' → s' → c':d'
s' :: s' → s'
f' :: c':d' → c':d'
d' :: c':d' → c':d'
_hole_g'1 :: g'
_hole_c':d'2 :: c':d'
_hole_s'3 :: s'
_gen_c':d'4 :: Nat → c':d'
_gen_s'5 :: Nat → s'
Generator Equations:
_gen_c':d'4(0) ⇔ c'(_hole_s'3, _hole_s'3)
_gen_c':d'4(+(x, 1)) ⇔ d'(_gen_c':d'4(x))
_gen_s'5(0) ⇔ _hole_s'3
_gen_s'5(+(x, 1)) ⇔ s'(_gen_s'5(x))
The following defined symbols remain to be analysed:
g', f'
Could not prove a rewrite lemma for the defined symbol g'.
Rules:
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(f'(x)) → f'(d'(f'(x)))
f'(x) → x
Types:
g' :: c':d' → g'
c' :: s' → s' → c':d'
s' :: s' → s'
f' :: c':d' → c':d'
d' :: c':d' → c':d'
_hole_g'1 :: g'
_hole_c':d'2 :: c':d'
_hole_s'3 :: s'
_gen_c':d'4 :: Nat → c':d'
_gen_s'5 :: Nat → s'
Generator Equations:
_gen_c':d'4(0) ⇔ c'(_hole_s'3, _hole_s'3)
_gen_c':d'4(+(x, 1)) ⇔ d'(_gen_c':d'4(x))
_gen_s'5(0) ⇔ _hole_s'3
_gen_s'5(+(x, 1)) ⇔ s'(_gen_s'5(x))
The following defined symbols remain to be analysed:
f'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(f'(x)) → f'(d'(f'(x)))
f'(x) → x
Types:
g' :: c':d' → g'
c' :: s' → s' → c':d'
s' :: s' → s'
f' :: c':d' → c':d'
d' :: c':d' → c':d'
_hole_g'1 :: g'
_hole_c':d'2 :: c':d'
_hole_s'3 :: s'
_gen_c':d'4 :: Nat → c':d'
_gen_s'5 :: Nat → s'
Generator Equations:
_gen_c':d'4(0) ⇔ c'(_hole_s'3, _hole_s'3)
_gen_c':d'4(+(x, 1)) ⇔ d'(_gen_c':d'4(x))
_gen_s'5(0) ⇔ _hole_s'3
_gen_s'5(+(x, 1)) ⇔ s'(_gen_s'5(x))
No more defined symbols left to analyse.