Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/num-515)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by matchbox @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(a(a(x1))) → b(x1) (1)
c(b(x1)) → a(a(a(a(c(c(x1)))))) (3)

1.1 Loop

The following loop proves nontermination.

t0 = c(a(a(a(a(a(a(a(a(a(a(a(a(x1)))))))))))))
→ c(b(a(a(a(a(a(a(a(a(a(x1)))))))))))
→ a(a(a(a(c(c(a(a(a(a(a(a(a(a(a(x1)))))))))))))))
→ a(a(a(a(c(c(b(a(a(a(a(a(a(x1)))))))))))))
→ a(a(a(a(c(a(a(a(a(c(c(a(a(a(a(a(a(x1)))))))))))))))))
→ a(a(a(a(c(a(a(a(a(c(c(b(a(a(a(x1)))))))))))))))
→ a(a(a(a(c(a(a(a(a(c(a(a(a(a(c(c(a(a(a(x1)))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(c(b(a(c(c(a(a(a(x1)))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(c(c(a(c(c(a(a(a(x1)))))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(c(c(a(c(c(b(x1)))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(c(c(a(c(a(a(a(a(c(c(x1)))))))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(c(c(a(c(b(a(c(c(x1)))))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(c(c(a(a(a(a(a(c(c(a(c(c(x1)))))))))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(c(c(b(a(a(c(c(a(c(c(x1)))))))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(c(a(a(a(a(c(c(a(a(c(c(a(c(c(x1)))))))))))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(c(b(a(c(c(a(a(c(c(a(c(c(x1)))))))))))))))))))))))))
→ a(a(a(a(c(a(a(a(a(a(a(a(a(a(a(a(a(c(c(a(c(c(a(a(c(c(a(c(c(x1)))))))))))))))))))))))))))))
= t16
where t16 = C[t0σ] and σ = {x1/c(c(a(c(c(a(a(c(c(a(c(c(x1))))))))))))} and C = a(a(a(a(☐))))