(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)  =  x2
or(x1, x2)  =  or(x2)
implies(x1, x2)  =  implies(x2)
not(x1)  =  not

Lexicographic Path Order [LPO].
Precedence:
implies1 > or1


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.