YES Termination proof of rt2-3.trs

Termination of the given RelTRS could be proven:

0 RelTRS
↳1 RelTRSRRRProof (⇔, 45 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:

R(x, B2) → B2
W(x, B2) → B2

The relative TRS consists of the following S rules:

B1 → W(T, B1)
B1 → R(T, B1)

(1) RelTRSRRRProof (EQUIVALENT transformation)

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

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

POL(B2) =
/1\
\1/

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

POL(B1) =
/0\
\0/

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

R(x, B2) → B2
Rules from S:
none


(2) Obligation:

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

W(x, B2) → B2

The relative TRS consists of the following S rules:

B1 → W(T, B1)
B1 → R(T, B1)

(3) RelTRSRRRProof (EQUIVALENT transformation)

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

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

POL(B2) =
/1\
\1/

POL(B1) =
/0\
\0/

POL(T) =
/0\
\1/

POL(R(x1, x2)) =
/0\
\0/
 +
/10\
\00/
·x1 +
/10\
\00/
·x2
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

W(x, B2) → B2
Rules from S:
none


(4) Obligation:

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

B1 → W(T, B1)
B1 → R(T, B1)

(5) RIsEmptyProof (EQUIVALENT transformation)

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

(6) YES