(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))
and(x, or(y, z)) → or(and(x, y), and(x, z))
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)) → 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)
AND(x, or(y, z)) → AND(x, y)
AND(x, or(y, z)) → AND(x, z)
The TRS R consists of the following rules:
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))
and(x, or(y, z)) → or(and(x, y), and(x, z))
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 2 SCCs with 1 less node.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
AND(x, or(y, z)) → AND(x, z)
AND(x, or(y, z)) → AND(x, y)
The TRS R consists of the following rules:
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))
and(x, or(y, z)) → or(and(x, y), and(x, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
AND(x, or(y, z)) → AND(x, z)
AND(x, or(y, z)) → AND(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
AND(
x0,
x1,
x2) =
AND(
x0,
x1)
Tags:
AND has argument tags [0,0,0] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial
Status:
AND2: [1,2]
or2: [1,2]
The following usable rules [FROCOS05] were oriented:
none
(7) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))
and(x, or(y, z)) → or(and(x, y), and(x, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) 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:
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))
and(x, or(y, z)) → or(and(x, y), and(x, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) 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)
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(
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:
Combined order from the following AFS and order.
NOT(
x1) =
NOT
and(
x1,
x2) =
and(
x1,
x2)
or(
x1,
x2) =
or(
x1,
x2)
Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial
Status:
NOT: []
and2: [1,2]
or2: [2,1]
The following usable rules [FROCOS05] were oriented:
none
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))
and(x, or(y, z)) → or(and(x, y), and(x, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) TRUE