(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

\(x, x) → e
/(x, x) → e
.(e, x) → x
.(x, e) → x
\(e, x) → x
/(x, e) → x
.(x, \(x, y)) → y
.(/(y, x), x) → y
\(x, .(x, y)) → y
/(.(y, x), x) → y
/(x, \(y, x)) → y
\(/(x, y), x) → y

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
[\2, e, /2]

Status:
\2: [2,1]
e: []
/2: [2,1]
.2: [1,2]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

\(x, x) → e
/(x, x) → e
.(e, x) → x
.(x, e) → x
\(e, x) → x
/(x, e) → x
.(x, \(x, y)) → y
.(/(y, x), x) → y
\(x, .(x, y)) → y
/(.(y, x), x) → y
/(x, \(y, x)) → y
\(/(x, y), x) → y


(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