YES Termination proof of rt1-4.trs

Termination of the given RelTRS could be proven:

0 RelTRS
↳1 RelTRSRRRProof (⇔, 41 ms)
↳2 RelTRS
↳3 RIsEmptyProof (⇔, 0 ms)
↳4 YES

(0) Obligation:

Relative term rewrite system:
The relative TRS consists of the following R rules:

f(el(x), y) → f(x, el(y))

The relative TRS consists of the following S rules:

l(el(x)) → el(l(x))
f(x, y) → f(l(x), y)
el(r(x)) → r(el(x))
f(x, y) → f(x, r(y))

(1) RelTRSRRRProof (EQUIVALENT transformation)

We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(f(x1, x2)) =
/0\
\1/
 +
/11\
\00/
·x1 +
/10\
\00/
·x2

POL(el(x1)) =
/0\
\1/
 +
/10\
\01/
·x1

POL(l(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(r(x1)) =
/0\
\0/
 +
/10\
\00/
·x1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

f(el(x), y) → f(x, el(y))
Rules from S:
none


(2) Obligation:

Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

l(el(x)) → el(l(x))
f(x, y) → f(l(x), y)
el(r(x)) → r(el(x))
f(x, y) → f(x, r(y))

(3) RIsEmptyProof (EQUIVALENT transformation)

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

(4) YES