Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-406)

The rewrite relation of the following TRS is considered.

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

1.1 Loop

The following loop proves nontermination.

t0 = c(b(b(b(b(b(b(b(b(b(x1))))))))))
→ a(a(b(c(c(a(b(b(b(b(b(b(x1))))))))))))
→ b(a(b(c(c(a(b(b(b(b(b(b(x1))))))))))))
→ b(b(b(c(c(a(b(b(b(b(b(b(x1))))))))))))
→ b(b(b(c(c(b(b(b(b(b(b(b(x1))))))))))))
→ b(b(b(c(a(a(b(c(c(a(b(b(b(b(x1))))))))))))))
→ b(b(b(c(b(a(b(c(c(a(b(b(b(b(x1))))))))))))))
→ b(b(b(c(b(b(b(c(c(a(b(b(b(b(x1))))))))))))))
→ b(b(b(c(b(b(b(c(c(b(b(b(b(b(x1))))))))))))))
→ b(b(b(c(b(b(b(c(a(a(b(c(c(a(b(b(x1))))))))))))))))
→ b(b(b(c(b(b(b(c(b(a(b(c(c(a(b(b(x1))))))))))))))))
→ b(b(b(c(b(b(b(c(b(b(b(c(c(a(b(b(x1))))))))))))))))
→ b(b(b(c(b(b(b(c(b(b(b(c(c(b(b(b(x1))))))))))))))))
→ b(b(b(c(b(b(b(a(a(b(c(c(a(c(c(b(b(b(x1))))))))))))))))))
→ b(b(b(c(b(b(b(b(a(b(c(c(a(c(c(b(b(b(x1))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(a(c(c(b(b(b(x1))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(c(c(b(b(b(x1))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(c(a(a(b(c(c(a(x1))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(c(b(a(b(c(c(a(x1))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(c(b(b(b(c(c(a(x1))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(c(b(b(b(c(c(b(x1))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(a(a(b(c(c(a(c(c(b(x1))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(b(a(b(c(c(a(c(c(b(x1))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(b(b(b(c(c(a(c(c(b(x1))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(c(b(b(b(b(c(c(b(c(c(b(x1))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(a(a(b(c(c(a(b(c(c(b(c(c(b(x1))))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(b(a(b(c(c(a(b(c(c(b(c(c(b(x1))))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(b(b(b(c(c(a(b(c(c(b(c(c(b(x1))))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(c(b(b(b(c(c(b(b(c(c(b(c(c(b(x1))))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(a(a(b(c(c(a(c(c(b(b(c(c(b(c(c(b(x1))))))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(b(a(b(c(c(a(c(c(b(b(c(c(b(c(c(b(x1))))))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(b(b(b(c(c(a(c(c(b(b(c(c(b(c(c(b(x1))))))))))))))))))))))))))
→ b(b(b(c(b(b(b(b(b(b(b(b(b(c(c(b(c(c(b(b(c(c(b(c(c(b(x1))))))))))))))))))))))))))
= t32
where t32 = C[t0σ] and σ = {x1/c(c(b(c(c(b(b(c(c(b(c(c(b(x1)))))))))))))} and C = b(b(b(☐)))