Certification Problem

Input (TPDB SRS_Standard/Zantema_06/06)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
b(a(b(x1))) → a(b(a(x1))) (1)
a(b(b(x1))) → b(b(b(x1))) (5)
a(c(x1)) → c(b(a(x1))) (6)
b(c(x1)) → c(a(b(x1))) (7)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b#(c(x1)) → b#(x1) (8)
b#(c(x1)) → a#(b(x1)) (9)
b#(a(b(x1))) → b#(a(x1)) (10)
b#(a(b(x1))) → a#(x1) (11)
b#(a(b(x1))) → a#(b(a(x1))) (12)
a#(c(x1)) → b#(a(x1)) (13)
a#(c(x1)) → a#(x1) (14)
a#(b(b(x1))) → b#(b(b(x1))) (15)

1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c(x1)] = x1 +
1
[b(x1)] = x1 +
0
[a(x1)] = x1 +
0
[b#(x1)] = x1 +
0
[a#(x1)] = x1 +
0
together with the usable rules
b(a(b(x1))) → a(b(a(x1))) (1)
a(b(b(x1))) → b(b(b(x1))) (5)
a(c(x1)) → c(b(a(x1))) (6)
b(c(x1)) → c(a(b(x1))) (7)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b#(c(x1)) → b#(x1) (8)
b#(c(x1)) → a#(b(x1)) (9)
a#(c(x1)) → b#(a(x1)) (13)
a#(c(x1)) → a#(x1) (14)
and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.