Certification Problem

Input (TPDB SRS_Standard/Bouchare_06/06)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(a(a(x1))) → b#(b(a(x1))) (4)
a#(a(a(x1))) → b#(a(x1)) (5)
a#(b(a(x1))) → b#(b(a(x1))) (6)
b#(a(b(x1))) → a#(a(b(x1))) (7)

1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals
[a(x1)] = 1 + 2 · x1
[b(x1)] = 1 + 2 · x1
[a#(x1)] = 1 · x1
[b#(x1)] = 1 · x1
the pair
a#(a(a(x1))) → b#(a(x1)) (5)
and no rules could be deleted.

1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
[a#(x1)] =
0
-∞
-∞
+
1 1 -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
-∞
+
-∞ 1 0
0 0 -∞
0 -∞ 0
· x1
[b#(x1)] =
-∞
-∞
-∞
+
0 1 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
0
-∞
-∞
+
-∞ 1 0
-∞ 0 -∞
-∞ 1 -∞
· x1
the pairs
a#(a(a(x1))) → b#(b(a(x1))) (4)
a#(b(a(x1))) → b#(b(a(x1))) (6)
could be deleted.

1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

b#(a(b(x1))) → a#(a(b(x1))) (7)
1 ≥ 1

As there is no critical graph in the transitive closure, there are no infinite chains.