Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-45)

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

c(c(c(x1))) → c(a(a(x1))) (1)
b(a(b(x1))) → b(b(c(x1))) (2)
c(a(a(x1))) → c(b(a(x1))) (3)
b(a(a(x1))) → a(b(c(x1))) (4)
b(b(c(x1))) → a(a(c(x1))) (5)

and S is the following TRS.

a(c(c(x1))) → b(a(c(x1))) (6)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
c(c(c(x1))) → a(a(c(x1))) (7)
b(a(b(x1))) → c(b(b(x1))) (8)
a(a(c(x1))) → a(b(c(x1))) (9)
a(a(b(x1))) → c(b(a(x1))) (10)
c(b(b(x1))) → c(a(a(x1))) (11)
c(c(a(x1))) → c(a(b(x1))) (12)

1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[c(x1)] =
1
1
+
2 0
2 0
· x1
[a(x1)] =
0
1
+
1 1
1 1
· x1
[b(x1)] =
0
1
+
1 1
1 1
· x1
all of the following rules can be deleted.
c(c(c(x1))) → a(a(c(x1))) (7)

1.1.1 Rule Removal

Using the matrix interpretations of dimension 6 with strict dimension 1 over the integers
[b(x1)] =
0
0
0
0
0
1
+
1 1 0 0 0 0
0 0 0 0 0 0
0 0 1 0 1 0
0 0 1 1 0 0
0 0 1 1 1 0
0 0 0 0 0 0
· x1
[a(x1)] =
0
0
0
0
0
0
+
1 1 0 0 0 0
0 0 1 0 1 0
0 0 1 0 1 0
0 0 0 0 0 0
0 0 1 0 1 1
0 0 1 0 0 0
· x1
[c(x1)] =
0
0
1
0
1
0
+
1 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
· x1
all of the following rules can be deleted.
a(a(c(x1))) → a(b(c(x1))) (9)

1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[b(x1)] =
1
0
+
2 0
2 0
· x1
[a(x1)] =
0
2
+
2 0
2 0
· x1
[c(x1)] =
0
0
+
1 1
0 2
· x1
all of the following rules can be deleted.
a(a(b(x1))) → c(b(a(x1))) (10)
c(b(b(x1))) → c(a(a(x1))) (11)

1.1.1.1.1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(b) = 2 weight(b) = 1
prec(a) = 0 weight(a) = 1
prec(c) = 1 weight(c) = 1
all of the following rules can be deleted.
b(a(b(x1))) → c(b(b(x1))) (8)
c(c(a(x1))) → c(a(b(x1))) (12)

1.1.1.1.1.1 R is empty

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