Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/multum6)

The rewrite relation of the following TRS is considered.

c(c(b(x1))) → a(c(b(x1))) (1)
a(c(b(a(x1)))) → b(c(c(x1))) (2)
b(a(c(x1))) → a(b(c(a(x1)))) (3)
b(c(a(x1))) → c(a(b(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(c(c(x1))) → b(c(a(x1))) (5)
a(b(c(a(x1)))) → c(c(b(x1))) (6)
c(a(b(x1))) → a(c(b(a(x1)))) (7)
a(c(b(x1))) → b(a(c(x1))) (8)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
c#(a(b(x1))) → c#(b(a(x1))) (9)
c#(a(b(x1))) → b#(a(x1)) (10)
c#(a(b(x1))) → a#(x1) (11)
c#(a(b(x1))) → a#(c(b(a(x1)))) (12)
b#(c(c(x1))) → c#(a(x1)) (13)
b#(c(c(x1))) → b#(c(a(x1))) (14)
b#(c(c(x1))) → a#(x1) (15)
a#(c(b(x1))) → c#(x1) (16)
a#(c(b(x1))) → b#(a(c(x1))) (17)
a#(c(b(x1))) → a#(c(x1)) (18)
a#(b(c(a(x1)))) → c#(c(b(x1))) (19)
a#(b(c(a(x1)))) → c#(b(x1)) (20)
a#(b(c(a(x1)))) → b#(x1) (21)

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 +
0
[b(x1)] = x1 +
2
[a(x1)] = x1 +
0
[c#(x1)] = x1 +
0
[b#(x1)] = x1 +
1
[a#(x1)] = x1 +
0
together with the usable rules
b(c(c(x1))) → b(c(a(x1))) (5)
a(b(c(a(x1)))) → c(c(b(x1))) (6)
c(a(b(x1))) → a(c(b(a(x1)))) (7)
a(c(b(x1))) → b(a(c(x1))) (8)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
c#(a(b(x1))) → b#(a(x1)) (10)
c#(a(b(x1))) → a#(x1) (11)
b#(c(c(x1))) → c#(a(x1)) (13)
b#(c(c(x1))) → a#(x1) (15)
a#(c(b(x1))) → c#(x1) (16)
a#(c(b(x1))) → b#(a(c(x1))) (17)
a#(c(b(x1))) → a#(c(x1)) (18)
a#(b(c(a(x1)))) → b#(x1) (21)
and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.