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