Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-177)

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

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

and S is the following TRS.

a(b(a(x1))) → a(c(c(x1))) (5)
b(a(b(x1))) → b(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 = b(a(a(a(b(a(a(a(b(c(c(b(c(x1)))))))))))))
→S b(a(a(a(c(c(a(a(b(c(c(b(c(x1)))))))))))))
→R b(a(a(a(c(a(a(b(b(c(c(b(c(x1)))))))))))))
→R b(a(a(a(c(a(a(b(a(b(c(b(c(x1)))))))))))))
→S b(a(a(a(c(a(a(c(c(b(c(b(c(x1)))))))))))))
→R b(a(a(a(c(a(a(c(b(c(a(b(c(x1)))))))))))))
→R b(a(a(c(a(a(a(c(b(c(a(b(c(x1)))))))))))))
→R b(a(c(a(a(a(a(c(b(c(a(b(c(x1)))))))))))))
→R b(c(a(a(a(a(a(c(b(c(a(b(c(x1)))))))))))))
→R b(a(a(b(a(a(a(c(b(c(a(b(c(x1)))))))))))))
→S b(a(a(c(c(a(a(c(b(c(a(b(c(x1)))))))))))))
→R b(a(a(c(a(a(b(c(b(c(a(b(c(x1)))))))))))))
→R b(a(c(a(a(a(b(c(b(c(a(b(c(x1)))))))))))))
→R b(a(a(a(b(a(b(c(b(c(a(b(c(x1)))))))))))))
→S b(a(a(a(b(b(b(c(b(c(a(b(c(x1)))))))))))))
→R b(a(a(a(b(b(a(b(b(c(a(b(c(x1)))))))))))))
→R b(a(a(a(b(b(a(b(a(b(a(b(c(x1)))))))))))))
→S b(a(a(a(b(b(a(b(a(c(c(b(c(x1)))))))))))))
→S b(a(a(a(b(b(a(c(c(c(c(b(c(x1)))))))))))))
→R b(a(a(a(b(b(a(c(c(c(b(c(a(x1)))))))))))))
→R b(a(a(a(b(b(a(c(c(b(c(a(a(x1)))))))))))))
→R b(a(a(a(b(b(a(c(b(c(a(a(a(x1)))))))))))))
→R b(a(a(a(b(b(a(b(c(a(a(a(a(x1)))))))))))))
→S b(a(a(a(b(b(b(b(c(a(a(a(a(x1)))))))))))))
→R b(a(a(a(b(b(b(a(b(a(a(a(a(x1)))))))))))))
→S b(a(a(a(b(b(b(a(c(c(a(a(a(x1)))))))))))))
→R b(a(a(a(b(b(b(a(c(a(a(b(a(x1)))))))))))))
→S b(a(a(a(b(b(b(a(c(a(a(c(c(x1)))))))))))))
→R b(a(a(a(b(b(b(c(a(a(a(c(c(x1)))))))))))))
→R b(a(a(a(b(b(a(b(a(a(a(c(c(x1)))))))))))))
→S b(a(a(a(b(b(a(c(c(a(a(c(c(x1)))))))))))))
→R b(a(a(a(b(b(a(c(a(a(b(c(c(x1)))))))))))))
→R b(a(a(a(b(b(c(a(a(a(b(c(c(x1)))))))))))))
→R b(a(a(a(b(a(b(a(a(a(b(c(c(x1)))))))))))))
→S b(a(a(a(b(a(c(c(a(a(b(c(c(x1)))))))))))))
→R b(a(a(a(b(a(c(a(a(b(b(c(c(x1)))))))))))))
→R b(a(a(a(b(a(c(a(a(b(a(b(c(x1)))))))))))))
→S b(a(a(a(b(a(c(a(a(c(c(b(c(x1)))))))))))))
→R b(a(a(a(b(a(a(a(b(c(c(b(c(x1)))))))))))))
= t38
where t38 = t0σ and σ = {x1/x1}