(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
i(0) → 0
+(0, y) → y
+(x, 0) → x
i(i(x)) → x
+(i(x), x) → 0
+(x, i(x)) → 0
i(+(x, y)) → +(i(x), i(y))
+(x, +(y, z)) → +(+(x, y), z)
+(+(x, i(y)), y) → x
+(+(x, y), i(y)) → x
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(+(x1, x2)) = 2 + x1 + 2·x2
POL(0) = 2
POL(i(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
i(0) → 0
+(0, y) → y
+(x, 0) → x
i(+(x, y)) → +(i(x), i(y))
+(x, +(y, z)) → +(+(x, y), z)
+(+(x, i(y)), y) → x
+(+(x, y), i(y)) → x
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
i(i(x)) → x
+(i(x), x) → 0
+(x, i(x)) → 0
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
+2 > 0 > i1
and weight map:
0=3
i_1=1
+_2=0
The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
i(i(x)) → x
+(i(x), x) → 0
+(x, i(x)) → 0
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES