Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z082)

The rewrite relation of the following TRS is considered.

a(c(a(x1))) → c(a(c(x1))) (1)
a(a(b(x1))) → a(d(b(x1))) (2)
a(b(x1)) → b(a(a(x1))) (3)
d(d(x1)) → a(d(b(x1))) (4)
b(b(x1)) → b(c(x1)) (5)
a(d(c(x1))) → c(a(x1)) (6)
b(c(x1)) → a(a(a(x1))) (7)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(c(a(x1))) → c(a(c(x1))) (1)
b(a(a(x1))) → b(d(a(x1))) (8)
b(a(x1)) → a(a(b(x1))) (9)
d(d(x1)) → b(d(a(x1))) (10)
b(b(x1)) → c(b(x1)) (11)
c(d(a(x1))) → a(c(x1)) (12)
c(b(x1)) → a(a(a(x1))) (13)

1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[c(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[d(x1)] =
1 1 0
1 0 1
0 1 1
· x1 +
0 0 0
1 0 0
0 0 0
[a(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[b(x1)] =
1 0 1
1 0 0
1 0 1
· x1 +
0 0 0
0 0 0
1 0 0
all of the following rules can be deleted.
d(d(x1)) → b(d(a(x1))) (10)
b(b(x1)) → c(b(x1)) (11)

1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[c(x1)] = 0 · x1 + -∞
[d(x1)] = 0 · x1 + -∞
[a(x1)] = 0 · x1 + -∞
[b(x1)] = 2 · x1 + -∞
all of the following rules can be deleted.
c(b(x1)) → a(a(a(x1))) (13)

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[c(x1)] =
1 0 1
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[d(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
1 0 0
[a(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[b(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
1 0 0
0 0 0
all of the following rules can be deleted.
c(d(a(x1))) → a(c(x1)) (12)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[c(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[d(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
1 0 0
[a(x1)] =
1 0 0
0 0 1
0 1 0
· x1 +
0 0 0
1 0 0
0 0 0
[b(x1)] =
1 1 1
0 1 1
0 1 1
· x1 +
0 0 0
0 0 0
1 0 0
all of the following rules can be deleted.
b(a(x1)) → a(a(b(x1))) (9)

1.1.1.1.1.1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(d) = 0 weight(d) = 2
prec(b) = 1 weight(b) = 2
prec(c) = 2 weight(c) = 2
prec(a) = 3 weight(a) = 2
all of the following rules can be deleted.
a(c(a(x1))) → c(a(c(x1))) (1)
b(a(a(x1))) → b(d(a(x1))) (8)

1.1.1.1.1.1.1 R is empty

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