(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
and(x, false) → false
and(x, not(false)) → x
not(not(x)) → x
implies(false, y) → not(false)
implies(x, false) → not(x)
implies(not(x), not(y)) → implies(y, and(x, 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:
IMPLIES(false, y) → NOT(false)
IMPLIES(x, false) → NOT(x)
IMPLIES(not(x), not(y)) → IMPLIES(y, and(x, y))
IMPLIES(not(x), not(y)) → AND(x, y)
The TRS R consists of the following rules:
and(x, false) → false
and(x, not(false)) → x
not(not(x)) → x
implies(false, y) → not(false)
implies(x, false) → not(x)
implies(not(x), not(y)) → implies(y, and(x, 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 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
IMPLIES(not(x), not(y)) → IMPLIES(y, and(x, y))
The TRS R consists of the following rules:
and(x, false) → false
and(x, not(false)) → x
not(not(x)) → x
implies(false, y) → not(false)
implies(x, false) → not(x)
implies(not(x), not(y)) → implies(y, and(x, 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.
IMPLIES(not(x), not(y)) → IMPLIES(y, 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:
IMPLIES(
x1,
x2) =
IMPLIES(
x1,
x2)
Tags:
IMPLIES has tags [0,0]
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(
x1)
and(
x1,
x2) =
x1
false =
false
Lexicographic path order with status [LPO].
Quasi-Precedence:
not1 > false
Status:
not1: [1]
false: []
The following usable rules [FROCOS05] were oriented:
and(x, false) → false
and(x, not(false)) → x
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
and(x, false) → false
and(x, not(false)) → x
not(not(x)) → x
implies(false, y) → not(false)
implies(x, false) → not(x)
implies(not(x), not(y)) → implies(y, and(x, y))
Q is empty.
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