(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
minus(0) → 0
+(x, 0) → x
+(0, y) → y
+(minus(1), 1) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
+(minus(+(x, 1)), 1) → minus(x)
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, minus(y)) → MINUS(+(minus(x), y))
+1(x, minus(y)) → +1(minus(x), y)
+1(x, minus(y)) → MINUS(x)
+1(x, +(y, z)) → +1(+(x, y), z)
+1(x, +(y, z)) → +1(x, y)
+1(minus(+(x, 1)), 1) → MINUS(x)
The TRS R consists of the following rules:
minus(0) → 0
+(x, 0) → x
+(0, y) → y
+(minus(1), 1) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
+(minus(+(x, 1)), 1) → minus(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(x, +(y, z)) → +1(+(x, y), z)
+1(x, minus(y)) → +1(minus(x), y)
+1(x, +(y, z)) → +1(x, y)
The TRS R consists of the following rules:
minus(0) → 0
+(x, 0) → x
+(0, y) → y
+(minus(1), 1) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
+(minus(+(x, 1)), 1) → minus(x)
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(+(x, y), z)
+1(x, +(y, z)) → +1(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
+1(
x0,
x1,
x2) =
+1(
x0,
x2)
Tags:
+1 has argument tags [2,3,0] 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.
+1(
x1,
x2) =
+1
+(
x1,
x2) =
+(
x1,
x2)
minus(
x1) =
x1
0 =
0
1 =
1
Recursive path order with status [RPO].
Quasi-Precedence:
[+^1, +2]
Status:
+^1: multiset
+2: multiset
0: multiset
1: multiset
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(x, minus(y)) → +1(minus(x), y)
The TRS R consists of the following rules:
minus(0) → 0
+(x, 0) → x
+(0, y) → y
+(minus(1), 1) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
+(minus(+(x, 1)), 1) → minus(x)
Q is empty.
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(x, minus(y)) → +1(minus(x), y)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
+1(
x0,
x1,
x2) =
+1(
x2)
Tags:
+1 has argument tags [3,3,0] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
[+^12, minus1]
Status:
+^12: multiset
minus1: [1]
0: multiset
The following usable rules [FROCOS05] were oriented:
minus(0) → 0
minus(minus(x)) → x
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
minus(0) → 0
+(x, 0) → x
+(0, y) → y
+(minus(1), 1) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
+(minus(+(x, 1)), 1) → minus(x)
Q is empty.
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