(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
merge(nil, y) → y
merge(x, nil) → x
merge(.(x, y), .(u, v)) → if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
++(nil, y) → y
++(.(x, y), z) → .(x, ++(y, z))
if(true, x, y) → x
if(false, x, y) → x
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Precedence:
merge2 > .2 > if3 > <2
nil > <2
++2 > .2 > if3 > <2
true > <2
false > <2
Status:
++2: multiset
if3: multiset
<2: multiset
merge2: multiset
true: multiset
false: multiset
.2: [1,2]
nil: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
merge(nil, y) → y
merge(x, nil) → x
merge(.(x, y), .(u, v)) → if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
++(nil, y) → y
++(.(x, y), z) → .(x, ++(y, z))
if(true, x, y) → x
if(false, x, y) → x
(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