Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
  ↳7 RisEmptyProof (⇔)
    ↳8 TRUE
  ↳9 RisEmptyProof (⇔)
    ↳10 TRUE

(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:
Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + x2   
POL(/(x1, x2)) = x1 + x2   
POL(\(x1, x2)) = x1 + x2   
POL(e) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(2) 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
/(x, \(y, x)) → y
\(/(x, y), x) → y

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(/(x1, x2)) = 2 + x1 + x2   
POL(\(x1, x2)) = 1 + x1 + x2   
POL(e) = 1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(4) Obligation:

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

\(x, x) → e

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(\(x1, x2)) = 1 + x1 + x2   
POL(e) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

\(x, x) → e


(6) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(10) TRUE