(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, 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:
++1(.(x, y), z) → ++1(y, z)
++1(++(x, y), z) → ++1(x, ++(y, z))
++1(++(x, y), z) → ++1(y, z)
The TRS R consists of the following rules:
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, 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.
++1(++(x, y), z) → ++1(x, ++(y, z))
++1(++(x, y), z) → ++1(y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
++1(
x1,
x2) =
++1(
x1)
Tags:
++1 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.
.(
x1,
x2) =
x2
++(
x1,
x2) =
++(
x1,
x2)
nil =
nil
Recursive path order with status [RPO].
Quasi-Precedence:
nil > ++2
Status:
++2: multiset
nil: multiset
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
++1(.(x, y), z) → ++1(y, z)
The TRS R consists of the following rules:
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, 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.
++1(.(x, y), z) → ++1(y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
++1(
x1,
x2) =
++1(
x1)
Tags:
++1 has tags [1,0]
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
trivial
Status:
.2: multiset
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, 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