(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].
Quasi-Precedence:
[merge2, <2] > [.2, if3]
nil > [.2, if3]
++2 > [.2, if3]
true > [.2, if3]
false > [.2, if3]
Status:
merge2: [1,2]
nil: multiset
.2: [1,2]
if3: [2,1,3]
<2: [1,2]
++2: 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:
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) YES