from(X) → cons(X, n__from(n__s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
from'(X) → cons'(X, n__from'(n__s'(X)))
sel'(0', cons'(X, Y)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X
Infered types.
Rules:
from'(X) → cons'(X, n__from'(n__s'(X)))
sel'(0', cons'(X, Y)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X
Types:
from' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
cons' :: n__s':n__from':cons':0' → n__s':n__from':cons':0' → n__s':n__from':cons':0'
n__from' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
n__s' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
sel' :: n__s':n__from':cons':0' → n__s':n__from':cons':0' → n__s':n__from':cons':0'
0' :: n__s':n__from':cons':0'
s' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
activate' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
_hole_n__s':n__from':cons':0'1 :: n__s':n__from':cons':0'
_gen_n__s':n__from':cons':0'2 :: Nat → n__s':n__from':cons':0'
Heuristically decided to analyse the following defined symbols:
sel', activate'
They will be analysed ascendingly in the following order:
activate' < sel'
Rules:
from'(X) → cons'(X, n__from'(n__s'(X)))
sel'(0', cons'(X, Y)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X
Types:
from' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
cons' :: n__s':n__from':cons':0' → n__s':n__from':cons':0' → n__s':n__from':cons':0'
n__from' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
n__s' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
sel' :: n__s':n__from':cons':0' → n__s':n__from':cons':0' → n__s':n__from':cons':0'
0' :: n__s':n__from':cons':0'
s' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
activate' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
_hole_n__s':n__from':cons':0'1 :: n__s':n__from':cons':0'
_gen_n__s':n__from':cons':0'2 :: Nat → n__s':n__from':cons':0'
Generator Equations:
_gen_n__s':n__from':cons':0'2(0) ⇔ 0'
_gen_n__s':n__from':cons':0'2(+(x, 1)) ⇔ cons'(0', _gen_n__s':n__from':cons':0'2(x))
The following defined symbols remain to be analysed:
activate', sel'
They will be analysed ascendingly in the following order:
activate' < sel'
Could not prove a rewrite lemma for the defined symbol activate'.
Rules:
from'(X) → cons'(X, n__from'(n__s'(X)))
sel'(0', cons'(X, Y)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X
Types:
from' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
cons' :: n__s':n__from':cons':0' → n__s':n__from':cons':0' → n__s':n__from':cons':0'
n__from' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
n__s' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
sel' :: n__s':n__from':cons':0' → n__s':n__from':cons':0' → n__s':n__from':cons':0'
0' :: n__s':n__from':cons':0'
s' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
activate' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
_hole_n__s':n__from':cons':0'1 :: n__s':n__from':cons':0'
_gen_n__s':n__from':cons':0'2 :: Nat → n__s':n__from':cons':0'
Generator Equations:
_gen_n__s':n__from':cons':0'2(0) ⇔ 0'
_gen_n__s':n__from':cons':0'2(+(x, 1)) ⇔ cons'(0', _gen_n__s':n__from':cons':0'2(x))
The following defined symbols remain to be analysed:
sel'
Could not prove a rewrite lemma for the defined symbol sel'.
Rules:
from'(X) → cons'(X, n__from'(n__s'(X)))
sel'(0', cons'(X, Y)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
s'(X) → n__s'(X)
activate'(n__from'(X)) → from'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(X) → X
Types:
from' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
cons' :: n__s':n__from':cons':0' → n__s':n__from':cons':0' → n__s':n__from':cons':0'
n__from' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
n__s' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
sel' :: n__s':n__from':cons':0' → n__s':n__from':cons':0' → n__s':n__from':cons':0'
0' :: n__s':n__from':cons':0'
s' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
activate' :: n__s':n__from':cons':0' → n__s':n__from':cons':0'
_hole_n__s':n__from':cons':0'1 :: n__s':n__from':cons':0'
_gen_n__s':n__from':cons':0'2 :: Nat → n__s':n__from':cons':0'
Generator Equations:
_gen_n__s':n__from':cons':0'2(0) ⇔ 0'
_gen_n__s':n__from':cons':0'2(+(x, 1)) ⇔ cons'(0', _gen_n__s':n__from':cons':0'2(x))
No more defined symbols left to analyse.