YES Termination proof of rt3-1.trs

Termination of the given RelTRS could be proven:

0 RelTRS
↳1 RelTRSRRRProof (⇔, 42 ms)
↳2 RelTRS
↳3 RelTRSRRRProof (⇔, 0 ms)
↳4 RelTRS
↳5 RIsEmptyProof (⇔, 0 ms)
↳6 YES

(0) Obligation:

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

f(g(x), y, z) → f(x, y, g(z))

The relative TRS consists of the following S rules:

f(x, y, z) → f(x, y, g(z))
f(x, a, z) → f(x, g(a), z)
f(x, y, g(z)) → f(x, g(y), z)

(1) RelTRSRRRProof (EQUIVALENT transformation)

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

POL(f(x1, x2, x3)) =
/0\
\0/
 +
/10\
\10/
·x1 +
/11\
\00/
·x2 +
/10\
\00/
·x3

POL(g(x1)) =
/0\
\0/
 +
/10\
\00/
·x1

POL(a) =
/0\
\1/
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

f(x, a, z) → f(x, g(a), z)


(2) Obligation:

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

f(g(x), y, z) → f(x, y, g(z))

The relative TRS consists of the following S rules:

f(x, y, z) → f(x, y, g(z))
f(x, y, g(z)) → f(x, g(y), z)

(3) RelTRSRRRProof (EQUIVALENT transformation)

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

POL(f(x1, x2, x3)) =
/1\
\0/
 +
/11\
\11/
·x1 +
/10\
\00/
·x2 +
/10\
\00/
·x3

POL(g(x1)) =
/0\
\1/
 +
/10\
\11/
·x1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

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


(4) Obligation:

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

f(x, y, z) → f(x, y, g(z))
f(x, y, g(z)) → f(x, g(y), z)

(5) RIsEmptyProof (EQUIVALENT transformation)

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

(6) YES