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