(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
not(x) → xor(x, true)
or(x, y) → xor(and(x, y), xor(x, y))
implies(x, y) → xor(and(x, y), xor(x, true))
and(x, true) → x
and(x, false) → false
and(x, x) → x
xor(x, false) → x
xor(x, x) → false
and(xor(x, y), z) → xor(and(x, z), and(y, 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(x) → XOR(x, true)
OR(x, y) → XOR(and(x, y), xor(x, y))
OR(x, y) → AND(x, y)
OR(x, y) → XOR(x, y)
IMPLIES(x, y) → XOR(and(x, y), xor(x, true))
IMPLIES(x, y) → AND(x, y)
IMPLIES(x, y) → XOR(x, true)
AND(xor(x, y), z) → XOR(and(x, z), and(y, z))
AND(xor(x, y), z) → AND(x, z)
AND(xor(x, y), z) → AND(y, z)
The TRS R consists of the following rules:
not(x) → xor(x, true)
or(x, y) → xor(and(x, y), xor(x, y))
implies(x, y) → xor(and(x, y), xor(x, true))
and(x, true) → x
and(x, false) → false
and(x, x) → x
xor(x, false) → x
xor(x, x) → false
and(xor(x, y), z) → xor(and(x, z), and(y, 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 1 SCC with 8 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
AND(xor(x, y), z) → AND(y, z)
AND(xor(x, y), z) → AND(x, z)
The TRS R consists of the following rules:
not(x) → xor(x, true)
or(x, y) → xor(and(x, y), xor(x, y))
implies(x, y) → xor(and(x, y), xor(x, true))
and(x, true) → x
and(x, false) → false
and(x, x) → x
xor(x, false) → x
xor(x, x) → false
and(xor(x, y), z) → xor(and(x, z), and(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
AND(xor(x, y), z) → AND(y, z)
AND(xor(x, y), z) → AND(x, z)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- AND(xor(x, y), z) → AND(y, z)
The graph contains the following edges 1 > 1, 2 >= 2
- AND(xor(x, y), z) → AND(x, z)
The graph contains the following edges 1 > 1, 2 >= 2
(8) TRUE