Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/3)

The rewrite relation of the following TRS is considered.

a(a(a(b(x1)))) → b(a(b(a(x1)))) (1)
b(b(a(x1))) → a(a(a(b(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#(a(a(b(x1)))) → b#(a(b(a(x1)))) (3)
a#(a(a(b(x1)))) → a#(b(a(x1))) (4)
a#(a(a(b(x1)))) → b#(a(x1)) (5)
a#(a(a(b(x1)))) → a#(x1) (6)
b#(b(a(x1))) → a#(a(a(b(x1)))) (7)
b#(b(a(x1))) → a#(a(b(x1))) (8)
b#(b(a(x1))) → a#(b(x1)) (9)
b#(b(a(x1))) → b#(x1) (10)

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
[a(x1)] =
0
0
0
+
-∞ 0 -∞
-∞ -∞ 0
0 -∞ -∞
· x1
[b(x1)] =
0
1
1
+
-∞ -∞ -∞
0 0 0
0 0 0
· x1
[b#(x1)] =
0
-∞
-∞
+
-∞ -∞ 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
a#(a(a(b(x1)))) → b#(a(b(a(x1)))) (3)
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 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
0
+
-∞ 0 -∞
-∞ -∞ 0
0 -∞ -∞
· x1
[b(x1)] =
0
1
1
+
-∞ -∞ -∞
0 0 0
0 0 0
· x1
[b#(x1)] =
1
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pairs
a#(a(a(b(x1)))) → a#(b(a(x1))) (4)
b#(b(a(x1))) → a#(b(x1)) (9)
could be deleted.

1.1.1.1 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 -∞ 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
0
+
-∞ -∞ 0
0 -∞ -∞
-∞ 0 -∞
· x1
[b(x1)] =
1
0
0
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b#(x1)] =
0
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
b#(b(a(x1))) → a#(a(b(x1))) (8)
could be deleted.

1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 6 with strict dimension 1 over the integers
[a#(x1)] =
0
0
0
0
0
0
+
0 0 0 0 1 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
· x1
[a(x1)] =
0
0
0
0
0
0
+
0 0 0 0 1 1
0 0 1 0 0 0
0 0 0 0 1 0
0 0 0 0 1 1
0 1 0 0 0 0
0 0 0 0 1 0
· x1
[b(x1)] =
1
0
1
0
0
0
+
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 1 1
0 0 1 1 0 0
0 0 0 0 0 0
1 0 0 0 0 0
· x1
[b#(x1)] =
1
0
0
0
0
0
+
0 0 0 0 0 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
· x1
the pair
a#(a(a(b(x1)))) → a#(x1) (6)
could be deleted.

1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 5 with strict dimension 1 over the arctic semiring over the naturals
[a#(x1)] =
-∞
-∞
-∞
-∞
-∞
+
0 0 0 0 0
-∞ -∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞ -∞
· x1
[a(x1)] =
0
0
0
-∞
1
+
-∞ -∞ -∞ 0 -∞
-∞ 0 -∞ 0 0
0 -∞ -∞ 0 -∞
-∞ -∞ -∞ -∞ 0
-∞ 0 -∞ 0 0
· x1
[b(x1)] =
0
0
0
0
0
+
1 -∞ 0 -∞ -∞
0 -∞ -∞ 1 -∞
0 -∞ 0 0 -∞
0 -∞ -∞ -∞ -∞
0 -∞ -∞ -∞ -∞
· x1
[b#(x1)] =
0
-∞
-∞
-∞
-∞
+
0 -∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞ -∞
· x1
the pair
a#(a(a(b(x1)))) → b#(a(x1)) (5)
could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.