(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
flatten(nil) → nil
flatten(unit(x)) → flatten(x)
flatten(++(x, y)) → ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y))
flatten(flatten(x)) → flatten(x)
rev(nil) → nil
rev(unit(x)) → unit(x)
rev(++(x, y)) → ++(rev(y), rev(x))
rev(rev(x)) → x
++(x, nil) → x
++(nil, y) → y
++(++(x, y), z) → ++(x, ++(y, z))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(++(x1, x2)) = 2 + x1 + x2
POL(flatten(x1)) = 2 + 2·x1
POL(nil) = 2
POL(rev(x1)) = 2 + 2·x1
POL(unit(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
flatten(nil) → nil
flatten(unit(x)) → flatten(x)
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y))
flatten(flatten(x)) → flatten(x)
rev(nil) → nil
rev(unit(x)) → unit(x)
rev(rev(x)) → x
++(x, nil) → x
++(nil, y) → y
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
flatten(++(x, y)) → ++(flatten(x), flatten(y))
rev(++(x, y)) → ++(rev(y), rev(x))
++(++(x, y), z) → ++(x, ++(y, z))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(++(x1, x2)) = 1 + x1 + x2
POL(flatten(x1)) = 2·x1
POL(rev(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
flatten(++(x, y)) → ++(flatten(x), flatten(y))
rev(++(x, y)) → ++(rev(y), rev(x))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
++(++(x, y), z) → ++(x, ++(y, z))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
trivial
and weight map:
++_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:
++(++(x, y), z) → ++(x, ++(y, z))
(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