(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)  =  x1
and(x1, x2, x3)  =  and

Recursive path order with status [RPO].
Precedence:
not1 > AND3
and > AND3

Status:
AND3: multiset
not1: 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