(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(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:
IMPLIES(not(x), or(y, z)) → IMPLIES(y, or(x, z))
IMPLIES(x, or(y, z)) → IMPLIES(x, z)
The TRS R consists of the following rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
IMPLIES(x, or(y, z)) → IMPLIES(x, z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
IMPLIES(
x0,
x1,
x2) =
IMPLIES(
x0,
x2)
Tags:
IMPLIES has argument tags [1,0,3] 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.
IMPLIES(
x1,
x2) =
IMPLIES(
x2)
not(
x1) =
not
or(
x1,
x2) =
or(
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
[IMPLIES1, not, or1]
Status:
IMPLIES1: multiset
not: []
or1: multiset
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
IMPLIES(not(x), or(y, z)) → IMPLIES(y, or(x, z))
The TRS R consists of the following rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))
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), or(y, z)) → IMPLIES(y, or(x, z))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
IMPLIES(
x0,
x1,
x2) =
IMPLIES(
x1,
x2)
Tags:
IMPLIES has argument tags [1,2,2] and root tag 0
Comparison: DMS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
IMPLIES(
x1,
x2) =
IMPLIES
not(
x1) =
not(
x1)
or(
x1,
x2) =
or(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
[IMPLIES, not1, or1]
Status:
IMPLIES: multiset
not1: [1]
or1: [1]
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))
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