Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-309)

The rewrite relation of the following TRS is considered.

a(x1) → x1 (1)
a(b(b(a(x1)))) → a(c(c(b(x1)))) (2)
c(x1) → 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#(b(b(a(x1)))) → a#(c(c(b(x1)))) (4)
a#(b(b(a(x1)))) → c#(c(b(x1))) (5)
a#(b(b(a(x1)))) → c#(b(x1)) (6)
c#(x1) → a#(b(x1)) (7)

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
[b(x1)] =
0
-∞
-∞
+
-∞ 0 -∞
-∞ -∞ 0
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
1
+
0 0 0
0 0 0
1 0 1
· x1
[c(x1)] =
0
0
1
+
-∞ 0 0
-∞ 0 0
0 1 0
· x1
[c#(x1)] =
1
-∞
-∞
+
0 1 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
c#(x1) → a#(b(x1)) (7)
could be deleted.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.