f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(x, g'(x)) → x
f'(x, h'(y)) → f'(h'(x), y)
Infered types.
Rules:
f'(x, g'(x)) → x
f'(x, h'(y)) → f'(h'(x), y)
Types:
f' :: g':h' → g':h' → g':h'
g' :: g':h' → g':h'
h' :: g':h' → g':h'
_hole_g':h'1 :: g':h'
_gen_g':h'2 :: Nat → g':h'
Heuristically decided to analyse the following defined symbols:
f'
Rules:
f'(x, g'(x)) → x
f'(x, h'(y)) → f'(h'(x), y)
Types:
f' :: g':h' → g':h' → g':h'
g' :: g':h' → g':h'
h' :: g':h' → g':h'
_hole_g':h'1 :: g':h'
_gen_g':h'2 :: Nat → g':h'
Generator Equations:
_gen_g':h'2(0) ⇔ _hole_g':h'1
_gen_g':h'2(+(x, 1)) ⇔ h'(_gen_g':h'2(x))
The following defined symbols remain to be analysed:
f'
Could not prove a rewrite lemma for the defined symbol f'.
The following conjecture could not be proven:
f'(_gen_g':h'2(a), _gen_g':h'2(+(1, _n4))) →? _*3
Rules:
f'(x, g'(x)) → x
f'(x, h'(y)) → f'(h'(x), y)
Types:
f' :: g':h' → g':h' → g':h'
g' :: g':h' → g':h'
h' :: g':h' → g':h'
_hole_g':h'1 :: g':h'
_gen_g':h'2 :: Nat → g':h'
Generator Equations:
_gen_g':h'2(0) ⇔ _hole_g':h'1
_gen_g':h'2(+(x, 1)) ⇔ h'(_gen_g':h'2(x))
No more defined symbols left to analyse.