(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:
Recursive path order with status [RPO].
Quasi-Precedence:
rev1 > [nil, .2, ++2, true, false]
car1 > [nil, .2, ++2, true, false]
cdr1 > [nil, .2, ++2, true, false]
null1 > [nil, .2, ++2, true, false]
Status:
rev1: [1]
nil: multiset
.2: [1,2]
++2: [1,2]
car1: multiset
cdr1: multiset
null1: multiset
true: multiset
false: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
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))
(2) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(3) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(4) TRUE