f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(a, empty') → g'(a, empty')
f'(a, cons'(x, k)) → f'(cons'(x, a), k)
g'(empty', d) → d
g'(cons'(x, k), d) → g'(k, cons'(x, d))
Sliced the following arguments:
cons'/0
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(a, empty') → g'(a, empty')
f'(a, cons'(k)) → f'(cons'(a), k)
g'(empty', d) → d
g'(cons'(k), d) → g'(k, cons'(d))
Infered types.
Rules:
f'(a, empty') → g'(a, empty')
f'(a, cons'(k)) → f'(cons'(a), k)
g'(empty', d) → d
g'(cons'(k), d) → g'(k, cons'(d))
Types:
f' :: empty':cons' → empty':cons' → empty':cons'
empty' :: empty':cons'
g' :: empty':cons' → empty':cons' → empty':cons'
cons' :: empty':cons' → empty':cons'
_hole_empty':cons'1 :: empty':cons'
_gen_empty':cons'2 :: Nat → empty':cons'
Heuristically decided to analyse the following defined symbols:
f', g'
They will be analysed ascendingly in the following order:
g' < f'
Rules:
f'(a, empty') → g'(a, empty')
f'(a, cons'(k)) → f'(cons'(a), k)
g'(empty', d) → d
g'(cons'(k), d) → g'(k, cons'(d))
Types:
f' :: empty':cons' → empty':cons' → empty':cons'
empty' :: empty':cons'
g' :: empty':cons' → empty':cons' → empty':cons'
cons' :: empty':cons' → empty':cons'
_hole_empty':cons'1 :: empty':cons'
_gen_empty':cons'2 :: Nat → empty':cons'
Generator Equations:
_gen_empty':cons'2(0) ⇔ empty'
_gen_empty':cons'2(+(x, 1)) ⇔ cons'(_gen_empty':cons'2(x))
The following defined symbols remain to be analysed:
g', f'
They will be analysed ascendingly in the following order:
g' < f'
Could not prove a rewrite lemma for the defined symbol g'.
The following conjecture could not be proven:
g'(_gen_empty':cons'2(_n4), _gen_empty':cons'2(b)) →? _gen_empty':cons'2(+(_n4, b))
Rules:
f'(a, empty') → g'(a, empty')
f'(a, cons'(k)) → f'(cons'(a), k)
g'(empty', d) → d
g'(cons'(k), d) → g'(k, cons'(d))
Types:
f' :: empty':cons' → empty':cons' → empty':cons'
empty' :: empty':cons'
g' :: empty':cons' → empty':cons' → empty':cons'
cons' :: empty':cons' → empty':cons'
_hole_empty':cons'1 :: empty':cons'
_gen_empty':cons'2 :: Nat → empty':cons'
Generator Equations:
_gen_empty':cons'2(0) ⇔ empty'
_gen_empty':cons'2(+(x, 1)) ⇔ cons'(_gen_empty':cons'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_empty':cons'2(a), _gen_empty':cons'2(+(1, _n794))) →? _*3
Rules:
f'(a, empty') → g'(a, empty')
f'(a, cons'(k)) → f'(cons'(a), k)
g'(empty', d) → d
g'(cons'(k), d) → g'(k, cons'(d))
Types:
f' :: empty':cons' → empty':cons' → empty':cons'
empty' :: empty':cons'
g' :: empty':cons' → empty':cons' → empty':cons'
cons' :: empty':cons' → empty':cons'
_hole_empty':cons'1 :: empty':cons'
_gen_empty':cons'2 :: Nat → empty':cons'
Generator Equations:
_gen_empty':cons'2(0) ⇔ empty'
_gen_empty':cons'2(+(x, 1)) ⇔ cons'(_gen_empty':cons'2(x))
No more defined symbols left to analyse.