Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-519)

The rewrite relation of the following TRS is considered.

b(b(b(a(x1)))) b(b(a(a(x1)))) (1)
b(a(b(a(x1)))) b(a(b(b(x1)))) (2)
b(a(a(a(x1)))) a(a(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.
b#(b(b(a(x1)))) b#(b(a(a(x1)))) (4)
b#(b(b(a(x1)))) b#(a(a(x1))) (5)
b#(a(b(a(x1)))) b#(a(b(b(x1)))) (6)
b#(a(b(a(x1)))) b#(b(x1)) (7)
b#(a(b(a(x1)))) b#(x1) (8)
b#(a(a(a(x1)))) b#(x1) (9)

1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals
[b#(x1)] = 1 · x1
[b(x1)] = 1 + 1 · x1
[a(x1)] = 1 + 1 · x1
the pairs
b#(b(b(a(x1)))) b#(a(a(x1))) (5)
b#(a(b(a(x1)))) b#(b(x1)) (7)
b#(a(b(a(x1)))) b#(x1) (8)
b#(a(a(a(x1)))) b#(x1) (9)
could be deleted.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.