(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
The set Q consists of the following terms:
-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
-1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(-(x, y), -(x, y))
-1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(x, y)
The TRS R consists of the following rules:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
The set Q consists of the following terms:
-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))
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.
-1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
-1(
x1,
x2) =
-1(
x2)
-(
x1,
x2) =
-(
x2)
neg(
x1) =
x1
Lexicographic Path Order [LPO].
Precedence:
trivial
The following usable rules [FROCOS05] were oriented:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
-1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(-(x, y), -(x, y))
The TRS R consists of the following rules:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
The set Q consists of the following terms:
-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
-1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(-(x, y), -(x, y))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
-1(
x1,
x2) =
-1(
x2)
-(
x1,
x2) =
-(
x2)
neg(
x1) =
neg(
x1)
Lexicographic Path Order [LPO].
Precedence:
-^11 > -1
neg1 > -1
The following usable rules [FROCOS05] were oriented:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
The set Q consists of the following terms:
-(-(neg(x0), neg(x0)), -(neg(x1), neg(x1)))
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE