a__f(f(a)) → a__f(g(f(a)))
mark(f(X)) → a__f(X)
mark(a) → a
mark(g(X)) → g(mark(X))
a__f(X) → f(X)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__f'(f'(a')) → a__f'(g'(f'(a')))
mark'(f'(X)) → a__f'(X)
mark'(a') → a'
mark'(g'(X)) → g'(mark'(X))
a__f'(X) → f'(X)
Infered types.
Rules:
a__f'(f'(a')) → a__f'(g'(f'(a')))
mark'(f'(X)) → a__f'(X)
mark'(a') → a'
mark'(g'(X)) → g'(mark'(X))
a__f'(X) → f'(X)
Types:
a__f' :: a':f':g' → a':f':g'
f' :: a':f':g' → a':f':g'
a' :: a':f':g'
g' :: a':f':g' → a':f':g'
mark' :: a':f':g' → a':f':g'
_hole_a':f':g'1 :: a':f':g'
_gen_a':f':g'2 :: Nat → a':f':g'
Heuristically decided to analyse the following defined symbols:
a__f', mark'
They will be analysed ascendingly in the following order:
a__f' < mark'
Rules:
a__f'(f'(a')) → a__f'(g'(f'(a')))
mark'(f'(X)) → a__f'(X)
mark'(a') → a'
mark'(g'(X)) → g'(mark'(X))
a__f'(X) → f'(X)
Types:
a__f' :: a':f':g' → a':f':g'
f' :: a':f':g' → a':f':g'
a' :: a':f':g'
g' :: a':f':g' → a':f':g'
mark' :: a':f':g' → a':f':g'
_hole_a':f':g'1 :: a':f':g'
_gen_a':f':g'2 :: Nat → a':f':g'
Generator Equations:
_gen_a':f':g'2(0) ⇔ a'
_gen_a':f':g'2(+(x, 1)) ⇔ f'(_gen_a':f':g'2(x))
The following defined symbols remain to be analysed:
a__f', mark'
They will be analysed ascendingly in the following order:
a__f' < mark'
Could not prove a rewrite lemma for the defined symbol a__f'.
Rules:
a__f'(f'(a')) → a__f'(g'(f'(a')))
mark'(f'(X)) → a__f'(X)
mark'(a') → a'
mark'(g'(X)) → g'(mark'(X))
a__f'(X) → f'(X)
Types:
a__f' :: a':f':g' → a':f':g'
f' :: a':f':g' → a':f':g'
a' :: a':f':g'
g' :: a':f':g' → a':f':g'
mark' :: a':f':g' → a':f':g'
_hole_a':f':g'1 :: a':f':g'
_gen_a':f':g'2 :: Nat → a':f':g'
Generator Equations:
_gen_a':f':g'2(0) ⇔ a'
_gen_a':f':g'2(+(x, 1)) ⇔ f'(_gen_a':f':g'2(x))
The following defined symbols remain to be analysed:
mark'
Could not prove a rewrite lemma for the defined symbol mark'.
Rules:
a__f'(f'(a')) → a__f'(g'(f'(a')))
mark'(f'(X)) → a__f'(X)
mark'(a') → a'
mark'(g'(X)) → g'(mark'(X))
a__f'(X) → f'(X)
Types:
a__f' :: a':f':g' → a':f':g'
f' :: a':f':g' → a':f':g'
a' :: a':f':g'
g' :: a':f':g' → a':f':g'
mark' :: a':f':g' → a':f':g'
_hole_a':f':g'1 :: a':f':g'
_gen_a':f':g'2 :: Nat → a':f':g'
Generator Equations:
_gen_a':f':g'2(0) ⇔ a'
_gen_a':f':g'2(+(x, 1)) ⇔ f'(_gen_a':f':g'2(x))
No more defined symbols left to analyse.