Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-49)

The rewrite relation of the following TRS is considered.

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

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
-∞
1
+
-∞ 1 -∞
0 0 0
0 -∞ -∞
· x1
[a(x1)] =
0
0
1
+
0 1 0
-∞ 0 -∞
0 1 0
· x1
[c(x1)] =
0
-∞
0
+
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
a#(b(b(x1))) → a#(x1) (5)
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)] =
-∞
-∞
-∞
+
1 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
-∞
1
-∞
+
0 0 -∞
1 0 0
0 -∞ -∞
· x1
[a(x1)] =
0
1
0
+
0 -∞ -∞
1 0 0
0 -∞ 0
· x1
[c(x1)] =
0
0
0
+
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
a#(b(b(x1))) → a#(a(x1)) (4)
could be deleted.

1.1.1.1 P is empty

There are no pairs anymore.