Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-2-num-6)

The rewrite relation of the following TRS is considered.

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

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#(b(b(b(x1)))) → a#(b(a(x1))) (3)
a#(b(b(b(x1)))) → a#(x1) (4)

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
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
0
0
0
+
-∞ 0 0
0 -∞ -∞
0 1 -∞
· x1
[a(x1)] =
0
0
0
+
1 0 0
0 -∞ -∞
0 -∞ -∞
· x1
the pair
a#(b(b(b(x1)))) → a#(x1) (4)
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 0 1
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
0
0
0
+
-∞ 1 0
-∞ -∞ 0
0 0 -∞
· x1
[a(x1)] =
0
0
0
+
-∞ -∞ 0
-∞ -∞ 0
0 0 1
· x1
the pair
a#(b(b(b(x1)))) → a#(b(a(x1))) (3)
could be deleted.

1.1.1.1 P is empty

There are no pairs anymore.