Certification Problem

Input (TPDB TRS_Relative/Relative_05/rt3-1)

The relative rewrite relation R/S is considered where R is the following TRS

f(g(x),y,z) f(x,y,g(z)) (1)

and S is the following TRS.

f(x,y,g(z)) f(x,g(y),z) (2)
f(x,a,z) f(x,g(a),z) (3)
f(x,y,z) f(x,y,g(z)) (4)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals
[f(x1, x2, x3)] =
1 0 0 1
0 0 0 0
1 0 0 0
0 0 0 0
· x1 +
1 1 1 0
0 0 0 0
0 1 1 0
0 0 1 0
· x2 +
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
· x3 +
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
[g(x1)] =
1 0 0 0
0 1 1 0
0 0 0 0
0 0 0 1
· x1 +
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
[a] =
0 0 0 0
1 0 0 0
1 0 0 0
0 0 0 0
all of the following rules can be deleted.
f(g(x),y,z) f(x,y,g(z)) (1)

1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.