Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-86)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(x1) → x1 (1)
a(a(x1)) → b(c(a(b(x1)))) (5)
b(x1) → c(x1) (3)
c(b(x1)) → a(x1) (6)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(a(x1)) → b#(c(a(b(x1)))) (7)
a#(a(x1)) → c#(a(b(x1))) (8)
a#(a(x1)) → a#(b(x1)) (9)
a#(a(x1)) → b#(x1) (10)
b#(x1) → c#(x1) (11)
c#(b(x1)) → a#(x1) (12)

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
+
1 0 -∞
0 1 1
1 1 0
· x1
[b#(x1)] =
0
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[c(x1)] =
-∞
-∞
0
+
0 -∞ 0
-∞ -∞ 0
0 -∞ 0
· x1
[b(x1)] =
-∞
-∞
0
+
0 0 0
0 -∞ 0
1 1 1
· x1
[c#(x1)] =
0
-∞
-∞
+
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
a#(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 1 1
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
-∞
0
0
+
0 1 0
1 1 1
-∞ -∞ 0
· x1
[b#(x1)] =
1
-∞
-∞
+
1 1 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[c(x1)] =
-∞
-∞
0
+
0 -∞ 0
0 -∞ 0
-∞ -∞ 0
· x1
[b(x1)] =
-∞
-∞
0
+
1 1 1
0 0 0
-∞ -∞ 0
· x1
[c#(x1)] =
0
-∞
-∞
+
0 0 -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
b#(x1) → c#(x1) (11)
could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.