(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
from(
x1) =
from(
x1)
cons(
x1,
x2) =
cons(
x1,
x2)
n__from(
x1) =
x1
s(
x1) =
s(
x1)
sel(
x1,
x2) =
sel(
x1,
x2)
0 =
0
activate(
x1) =
activate(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
sel2 > [from1, activate1] > cons2
sel2 > [from1, activate1] > s1
Status:
from1: multiset
cons2: [2,1]
s1: [1]
sel2: [1,2]
0: multiset
activate1: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
from(X) → cons(X, n__from(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
activate(X) → X
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
activate(n__from(X)) → from(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
activate1 > from1 > nfrom1
and weight map:
activate_1=1
n__from_1=1
from_1=2
The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
activate(n__from(X)) → from(X)
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES