Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-216)

The relative rewrite relation R/S is considered where R is the following TRS

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

and S is the following TRS.

b(a(b(x1))) → c(a(b(x1))) (4)
b(b(c(x1))) → c(c(a(x1))) (5)
a(c(a(x1))) → a(b(b(x1))) (6)

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by matchbox @ termCOMP 2023)

1 Loop

The following loop proves that R/S is not relative terminating.

t0 = a(a(c(a(c(a(a(a(a(b(b(b(b(x1)))))))))))))
→R a(a(c(a(c(a(a(c(b(a(b(b(b(x1)))))))))))))
→S a(a(c(a(b(b(a(c(b(a(b(b(b(x1)))))))))))))
→R a(a(c(a(b(a(a(a(b(a(b(b(b(x1)))))))))))))
→R a(a(c(a(b(a(c(b(a(a(b(b(b(x1)))))))))))))
→R a(a(c(a(a(a(a(b(a(a(b(b(b(x1)))))))))))))
→R a(a(c(a(a(c(b(a(a(a(b(b(b(x1)))))))))))))
→S a(a(b(b(a(c(b(a(a(a(b(b(b(x1)))))))))))))
→R a(a(b(b(a(c(b(a(c(b(a(b(b(x1)))))))))))))
→R a(a(b(b(a(c(a(a(a(b(a(b(b(x1)))))))))))))
→S a(a(b(b(a(b(b(a(a(b(a(b(b(x1)))))))))))))
→S a(a(b(b(a(b(b(a(a(c(a(b(b(x1)))))))))))))
→S a(a(b(b(a(b(b(a(a(b(b(b(b(x1)))))))))))))
→R a(a(b(b(a(b(b(c(b(a(b(b(b(x1)))))))))))))
→S a(a(b(b(a(b(b(c(c(a(b(b(b(x1)))))))))))))
→S a(a(b(b(a(c(c(a(c(a(b(b(b(x1)))))))))))))
→R a(a(b(a(a(a(c(a(c(a(b(b(b(x1)))))))))))))
→S a(a(b(a(a(a(b(b(c(a(b(b(b(x1)))))))))))))
→R a(a(b(a(c(b(a(b(c(a(b(b(b(x1)))))))))))))
→S a(a(b(a(c(c(a(b(c(a(b(b(b(x1)))))))))))))
→R a(a(a(a(a(c(a(b(c(a(b(b(b(x1)))))))))))))
→S a(a(a(a(a(b(b(b(c(a(b(b(b(x1)))))))))))))
→S a(a(a(a(a(b(c(c(a(a(b(b(b(x1)))))))))))))
→R a(a(a(c(b(a(c(c(a(a(b(b(b(x1)))))))))))))
→R a(a(a(c(a(a(a(c(a(a(b(b(b(x1)))))))))))))
→S a(a(a(c(a(a(a(b(b(a(b(b(b(x1)))))))))))))
→S a(a(a(c(a(a(a(b(c(a(b(b(b(x1)))))))))))))
→R a(a(a(c(a(c(b(a(c(a(b(b(b(x1)))))))))))))
→R a(a(a(c(a(c(a(a(a(a(b(b(b(x1)))))))))))))
→R a(a(a(c(a(c(a(a(c(b(a(b(b(x1)))))))))))))
→S a(a(a(c(a(b(b(a(c(b(a(b(b(x1)))))))))))))
→R a(a(a(c(a(b(a(a(a(b(a(b(b(x1)))))))))))))
→R a(a(a(c(a(b(a(c(b(a(a(b(b(x1)))))))))))))
→R a(a(a(c(a(b(a(c(b(c(b(a(b(x1)))))))))))))
→R a(a(a(c(a(a(a(a(b(c(b(a(b(x1)))))))))))))
→R a(a(a(c(a(a(c(b(a(c(b(a(b(x1)))))))))))))
→R a(a(a(c(a(a(c(a(a(a(b(a(b(x1)))))))))))))
→S a(a(a(c(a(a(b(b(a(a(b(a(b(x1)))))))))))))
→S a(a(a(b(b(a(b(b(a(a(b(a(b(x1)))))))))))))
→S a(a(a(b(c(a(b(b(a(a(b(a(b(x1)))))))))))))
→R a(c(b(a(c(a(b(b(a(a(b(a(b(x1)))))))))))))
→R a(c(a(a(a(a(b(b(a(a(b(a(b(x1)))))))))))))
→R a(c(a(a(c(b(a(b(a(a(b(a(b(x1)))))))))))))
→S a(c(a(a(c(c(a(b(a(a(b(a(b(x1)))))))))))))
→S a(b(b(a(c(c(a(b(a(a(b(a(b(x1)))))))))))))
→R a(b(a(a(a(c(a(b(a(a(b(a(b(x1)))))))))))))
→S a(b(a(a(a(b(b(b(a(a(b(a(b(x1)))))))))))))
→R a(b(a(c(b(a(b(b(a(a(b(a(b(x1)))))))))))))
→S a(b(a(c(b(a(b(b(a(a(c(a(b(x1)))))))))))))
→S a(b(a(c(b(a(b(b(a(a(b(b(b(x1)))))))))))))
→R a(b(a(c(b(a(b(b(c(b(a(b(b(x1)))))))))))))
→S a(b(a(c(b(a(b(b(c(c(a(b(b(x1)))))))))))))
→S a(b(a(c(b(a(c(c(a(c(a(b(b(x1)))))))))))))
→R a(b(a(c(a(a(a(c(a(c(a(b(b(x1)))))))))))))
→S a(b(a(c(a(a(a(b(b(c(a(b(b(x1)))))))))))))
→R a(b(a(c(a(c(b(a(b(c(a(b(b(x1)))))))))))))
→S a(b(a(c(a(c(c(a(b(c(a(b(b(x1)))))))))))))
→S a(b(a(b(b(c(c(a(b(c(a(b(b(x1)))))))))))))
→S a(b(a(c(c(a(c(a(b(c(a(b(b(x1)))))))))))))
→S a(b(a(c(c(a(b(b(b(c(a(b(b(x1)))))))))))))
→S a(b(a(c(c(a(b(c(c(a(a(b(b(x1)))))))))))))
→R a(a(a(a(c(a(b(c(c(a(a(b(b(x1)))))))))))))
→S a(a(a(a(b(b(b(c(c(a(a(b(b(x1)))))))))))))
→S a(a(a(a(b(c(c(a(c(a(a(b(b(x1)))))))))))))
→S a(a(a(a(b(c(c(a(b(b(a(b(b(x1)))))))))))))
→S a(a(a(a(b(c(c(a(b(c(a(b(b(x1)))))))))))))
→R a(a(c(b(a(c(c(a(b(c(a(b(b(x1)))))))))))))
→R a(a(c(a(a(a(c(a(b(c(a(b(b(x1)))))))))))))
→S a(a(c(a(a(a(b(b(b(c(a(b(b(x1)))))))))))))
→S a(a(c(a(a(a(b(c(c(a(a(b(b(x1)))))))))))))
→R a(a(c(a(c(b(a(c(c(a(a(b(b(x1)))))))))))))
→R a(a(c(a(c(a(a(a(c(a(a(b(b(x1)))))))))))))
→S a(a(c(a(c(a(a(a(b(b(a(b(b(x1)))))))))))))
→S a(a(c(a(c(a(a(a(b(c(a(b(b(x1)))))))))))))
→R a(a(c(a(c(a(c(b(a(c(a(b(b(x1)))))))))))))
→R a(a(c(a(c(a(c(a(a(a(a(b(b(x1)))))))))))))
→R a(a(c(a(c(a(c(a(a(c(b(a(b(x1)))))))))))))
→S a(a(c(a(c(a(c(a(a(c(c(a(b(x1)))))))))))))
→S a(a(c(a(c(a(b(b(a(c(c(a(b(x1)))))))))))))
→R a(a(c(a(c(a(b(a(a(a(c(a(b(x1)))))))))))))
→S a(a(c(a(c(a(b(a(a(a(b(b(b(x1)))))))))))))
→R a(a(c(a(c(a(b(a(c(b(a(b(b(x1)))))))))))))
→S a(a(c(a(c(a(b(a(c(c(a(b(b(x1)))))))))))))
→R a(a(c(a(c(a(a(a(a(c(a(b(b(x1)))))))))))))
→S a(a(c(a(c(a(a(a(a(b(b(b(b(x1)))))))))))))
= t84
where t84 = t0σ and σ = {x1/x1}