Certification Problem

Input (TPDB SRS_Standard/Zantema_06/06)

The rewrite relation of the following TRS is considered.

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

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
b(a(b(x1))) → a(b(a(x1))) (1)
a(b(b(x1))) → b(b(b(x1))) (5)
a(c(x1)) → c(b(a(x1))) (6)
b(c(x1)) → c(a(b(x1))) (7)

1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[a(x1)] =
1 0 1
0 0 1
0 1 0
· x1 +
0 0 0
0 0 0
0 0 0
[b(x1)] =
1 0 1
0 0 1
0 1 0
· x1 +
0 0 0
0 0 0
0 0 0
[c(x1)] =
1 0 0
0 1 1
0 1 1
· x1 +
0 0 0
1 0 0
1 0 0
all of the following rules can be deleted.
a(c(x1)) → c(b(a(x1))) (6)
b(c(x1)) → c(a(b(x1))) (7)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b#(a(b(x1))) → a#(x1) (8)
b#(a(b(x1))) → b#(a(x1)) (9)
b#(a(b(x1))) → a#(b(a(x1))) (10)
a#(b(b(x1))) → b#(b(b(x1))) (11)

1.1.1.1 Subterm Criterion Processor

We use the projection to multisets
π(a#) = { 1, 1 }
π(b#) = { 1, 1 }
π(a) = { 1, 1 }
π(b) = { 1, 1 }
to remove the pairs:
b#(a(b(x1))) → a#(x1) (8)
b#(a(b(x1))) → b#(a(x1)) (9)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers
[a#(x1)] =
-∞ -∞ 1
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[a(x1)] =
0 0 1
-∞ 0 0
-∞ -∞ 0
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[b#(x1)] =
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[b(x1)] =
0 0 0
0 0 0
-∞ 0 0
· x1 +
0 -∞ -∞
0 -∞ -∞
-∞ -∞ -∞
together with the usable rules
b(a(b(x1))) → a(b(a(x1))) (1)
a(b(b(x1))) → b(b(b(x1))) (5)
(w.r.t. the implicit argument filter of the reduction pair), the pair
a#(b(b(x1))) → b#(b(b(x1))) (11)
could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.