(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
and(not(not(x)), y, not(z)) → and(y, band(x, z), x)
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
and(not(not(x)), y, not(z)) → and(y, band(x, z), x)
The signature Sigma is {
and}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
and(not(not(x)), y, not(z)) → and(y, band(x, z), x)
The set Q consists of the following terms:
and(not(not(x0)), x1, not(x2))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
AND(not(not(x)), y, not(z)) → AND(y, band(x, z), x)
The TRS R consists of the following rules:
and(not(not(x)), y, not(z)) → and(y, band(x, z), x)
The set Q consists of the following terms:
and(not(not(x0)), x1, not(x2))
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
AND(not(not(x)), y, not(z)) → AND(y, band(x, z), x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(
x1,
x2,
x3) =
AND(
x1,
x2,
x3)
not(
x1) =
not(
x1)
band(
x1,
x2) =
band(
x2)
and(
x1,
x2,
x3) =
and
Recursive path order with status [RPO].
Quasi-Precedence:
and > [AND3, not1, band1]
Status:
AND3: multiset
not1: multiset
band1: multiset
and: multiset
The following usable rules [FROCOS05] were oriented:
and(not(not(x)), y, not(z)) → and(y, band(x, z), x)
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
and(not(not(x)), y, not(z)) → and(y, band(x, z), x)
The set Q consists of the following terms:
and(not(not(x0)), x1, not(x2))
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE