(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: Combined order from the following AFS and order.
IMPLIES(x1, x2)  =  IMPLIES(x2)
not(x1)  =  not(x1)
or(x1, x2)  =  or(x2)
implies(x1, x2)  =  implies(x2)

Recursive path order with status [RPO].
Precedence:
not1 > IMPLIES1
not1 > or1
implies1 > or1

Status:
IMPLIES1: multiset
not1: multiset
or1: [1]
implies1: [1]

The following usable rules [FROCOS05] were oriented:

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))

(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: Combined order from the following AFS and order.
IMPLIES(x1, x2)  =  IMPLIES(x1, x2)
not(x1)  =  not(x1)
or(x1, x2)  =  x1
implies(x1, x2)  =  implies(x1, x2)

Recursive path order with status [RPO].
Precedence:
not1 > IMPLIES2
not1 > implies2

Status:
IMPLIES2: multiset
not1: multiset
implies2: multiset

The following usable rules [FROCOS05] were oriented:

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))

(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