Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-145)

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

a(b(c(x1))) → c(b(c(x1))) (1)
c(b(c(x1))) → a(c(b(x1))) (2)
b(c(c(x1))) → b(b(b(x1))) (3)
b(b(a(x1))) → b(c(a(x1))) (4)

and S is the following TRS.

a(c(a(x1))) → a(b(a(x1))) (5)
a(b(b(x1))) → c(c(c(x1))) (6)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 3 with strict dimension 1 over the integers
[a(x1)] =
0
1
0
+
2 0 2
0 0 0
0 0 2
· x1
[b(x1)] =
0
0
2
+
2 0 0
0 0 0
0 0 0
· x1
[c(x1)] =
0
1
0
+
2 0 0
0 0 0
0 2 0
· x1
all of the following rules can be deleted.
a(b(c(x1))) → c(b(c(x1))) (1)
a(b(b(x1))) → c(c(c(x1))) (6)

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[c(x1)] = 1 + 1 · x1
[b(x1)] = 1 + 1 · x1
[a(x1)] = 1 · x1
all of the following rules can be deleted.
c(b(c(x1))) → a(c(b(x1))) (2)

1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[b(x1)] =
0
0
+
1 0
0 0
· x1
[c(x1)] =
0
2
+
1 2
0 0
· x1
[a(x1)] =
0
0
+
1 0
0 0
· x1
all of the following rules can be deleted.
b(c(c(x1))) → b(b(b(x1))) (3)

1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[b(x1)] =
0
2
+
2 2
0 0
· x1
[a(x1)] =
0
0
+
1 0
1 0
· x1
[c(x1)] =
0
0
+
2 2
0 0
· x1
all of the following rules can be deleted.
b(b(a(x1))) → b(c(a(x1))) (4)

1.1.1.1.1 R is empty

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