(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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
minus(
x1) =
x1
0 =
0
+(
x1,
x2) =
+(
x1,
x2)
1 =
1
Recursive path order with status [RPO].
Quasi-Precedence:
[0, 1]
Status:
0: multiset
+2: [2,1]
1: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
+(x, 0) → x
+(0, y) → y
+(minus(1), 1) → 0
+(x, +(y, z)) → +(+(x, y), z)
+(minus(+(x, 1)), 1) → minus(x)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
minus(0) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(+(x1, x2)) = x1 + 2·x2
POL(0) = 1
POL(minus(x1)) = 2 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
minus(0) → 0
minus(minus(x)) → x
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(x, minus(y)) → minus(+(minus(x), y))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
+2 > minus1
Status:
+2: [2,1]
minus1: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
+(x, minus(y)) → minus(+(minus(x), y))
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) YES