Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-84)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by matchbox @ termCOMP 2023)

1 Loop

The following loop proves nontermination.

t0 = c(c(b(a(b(a(a(b(x1))))))))
→ c(c(b(a(b(b(a(b(c(c(b(x1)))))))))))
→ c(c(b(a(b(b(a(b(c(a(x1))))))))))
→ c(c(b(a(b(b(a(b(a(x1)))))))))
→ c(a(a(b(b(a(b(a(x1))))))))
→ c(b(a(b(c(c(b(b(a(b(a(x1)))))))))))
→ c(b(a(b(c(a(b(a(b(a(x1))))))))))
→ c(b(a(b(a(b(a(b(a(x1)))))))))
→ a(a(b(a(b(a(b(a(x1))))))))
→ b(a(b(c(c(b(a(b(a(b(a(x1)))))))))))
→ b(a(b(c(a(a(b(a(b(a(x1))))))))))
→ b(b(c(a(a(b(a(b(a(x1)))))))))
→ b(b(c(b(a(b(c(c(b(a(b(a(x1))))))))))))
→ b(b(c(b(a(b(c(a(a(b(a(x1)))))))))))
→ b(b(a(a(b(c(a(a(b(a(x1))))))))))
→ b(b(a(a(b(c(b(a(b(c(c(b(a(x1)))))))))))))
→ b(b(a(a(b(c(b(a(b(c(a(a(x1))))))))))))
→ b(b(a(a(b(c(b(b(c(a(a(x1)))))))))))
→ b(b(a(a(b(a(b(c(a(a(x1))))))))))
→ b(b(a(a(b(a(b(c(b(a(b(c(c(x1)))))))))))))
→ b(b(a(a(b(a(b(c(b(a(b(c(x1))))))))))))
→ b(b(a(a(b(a(b(c(b(a(b(x1)))))))))))
→ b(b(a(a(b(a(b(a(a(b(x1))))))))))
→ b(b(b(a(b(c(c(b(a(b(a(a(b(x1)))))))))))))
= t23
where t23 = C[t0σ] and σ = {x1/x1} and C = b(b(b(a(b(☐)))))