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