Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

or(x, x) → x
and(x, x) → x
not(not(x)) → x
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NOT(and(x, y)) → OR(not(x), not(y))
NOT(and(x, y)) → NOT(x)
NOT(and(x, y)) → NOT(y)
NOT(or(x, y)) → AND(not(x), not(y))
NOT(or(x, y)) → NOT(x)
NOT(or(x, y)) → NOT(y)

The TRS R consists of the following rules:

or(x, x) → x
and(x, x) → x
not(not(x)) → x
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NOT(and(x, y)) → NOT(y)
NOT(and(x, y)) → NOT(x)
NOT(or(x, y)) → NOT(x)
NOT(or(x, y)) → NOT(y)

The TRS R consists of the following rules:

or(x, x) → x
and(x, x) → x
not(not(x)) → x
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))

Q is empty.
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.

NOT(or(x, y)) → NOT(x)
NOT(or(x, y)) → NOT(y)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
NOT(x0, x1)  =  NOT(x0, x1)

Tags:
NOT has argument tags [1,1] and root tag 0

Comparison: DMS
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(NOT(x₁)) = 1   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(or(x₁, x₂)) = 1 + x₁ + x₂   

The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NOT(and(x, y)) → NOT(y)
NOT(and(x, y)) → NOT(x)

The TRS R consists of the following rules:

or(x, x) → x
and(x, x) → x
not(not(x)) → x
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NOT(and(x, y)) → NOT(y)
NOT(and(x, y)) → NOT(x)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
NOT(x0, x1)  =  NOT(x1)

Tags:
NOT has argument tags [1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(NOT(x₁)) = 1   
POL(and(x₁, x₂)) = 1 + x₁ + x₂   

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

or(x, x) → x
and(x, x) → x
not(not(x)) → x
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(9) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(10) TRUE