Certification Problem

Input (TPDB SRS_Standard/Bouchare_06/18)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(b(a(x1))) → b#(b(a(x1))) (4)
b#(b(b(x1))) → a#(x1) (5)
b#(b(b(x1))) → b#(a(x1)) (6)
b#(b(x1)) → a#(x1) (7)
b#(b(x1)) → a#(a(x1)) (8)
b#(b(x1)) → a#(a(a(x1))) (9)

1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 0
1 0
· x1 +
0 -∞
0 -∞
[b#(x1)] =
0 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 -∞
1 0
· x1 +
-∞ -∞
-∞ -∞
[a#(x1)] =
0 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
together with the usable rules
a(b(a(x1))) → b(b(a(x1))) (1)
b(b(b(x1))) → b(a(x1)) (2)
b(b(x1)) → a(a(a(x1))) (3)
(w.r.t. the implicit argument filter of the reduction pair), the pair
b#(b(b(x1))) → a#(x1) (5)
could be deleted.

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 1
0 0
· x1 +
0 -∞
0 -∞
[b#(x1)] =
2 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 -∞
0 0
· x1 +
0 -∞
0 -∞
[a#(x1)] =
2 2
-∞ -∞
· x1 +
0 -∞
-∞ -∞
together with the usable rules
a(b(a(x1))) → b(b(a(x1))) (1)
b(b(b(x1))) → b(a(x1)) (2)
b(b(x1)) → a(a(a(x1))) (3)
(w.r.t. the implicit argument filter of the reduction pair), the pair
b#(b(b(x1))) → b#(a(x1)) (6)
could be deleted.

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 0
1 0
· x1 +
0 -∞
1 -∞
[b#(x1)] =
1 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 0
-∞ 0
· x1 +
-∞ -∞
0 -∞
[a#(x1)] =
-∞ 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
together with the usable rules
a(b(a(x1))) → b(b(a(x1))) (1)
b(b(b(x1))) → b(a(x1)) (2)
b(b(x1)) → a(a(a(x1))) (3)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b#(b(x1)) → a#(x1) (7)
b#(b(x1)) → a#(a(x1)) (8)
b#(b(x1)) → a#(a(a(x1))) (9)
could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.