YES Termination proof of ex4.trs

Termination of the given RelTRS could be proven:

0 RelTRS
↳1 RelTRSRRRProof (⇔, 50 ms)
↳2 RelTRS
↳3 RIsEmptyProof (⇔, 0 ms)
↳4 YES

(0) Obligation:

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

a → b

The relative TRS consists of the following S rules:

c(x, c(y, z)) → c(y, c(x, z))
f(s(x)) → c(x, f(x))

(1) RelTRSRRRProof (EQUIVALENT transformation)

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

POL(a) =
/1\
\1/

POL(b) =
/0\
\0/

POL(c(x1, x2)) =
/0\
\0/
 +
/11\
\00/
·x1 +
/11\
\01/
·x2

POL(y) =
/0\
\1/

POL(z) =
/0\
\1/

POL(f(x1)) =
/0\
\0/
 +
/11\
\00/
·x1

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

a → b
Rules from S:
none


(2) Obligation:

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

c(x, c(y, z)) → c(y, c(x, z))
f(s(x)) → c(x, f(x))

(3) RIsEmptyProof (EQUIVALENT transformation)

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

(4) YES