(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:
Lexicographic path order with status [LPO].
Quasi-Precedence:
merge2 > [.2, ++2] > [if3, <2]
nil > [if3, <2]
true > [if3, <2]
false > [if3, <2]
Status:
merge2: [1,2]
nil: []
.2: [1,2]
if3: [1,3,2]
<2: [1,2]
++2: [1,2]
true: []
false: []
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