NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(x0) → x0
a(x0) → b(x0)
b(a(c(x0))) → c(c(a(b(a(x0)))))
c(x0) → x0

Proof

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(x0) → x0
a(x0) → b(x0)
c(a(b(x0))) → a(b(a(c(c(x0)))))
c(x0) → x0

1.1 Loop

The following loop proves nontermination.

t0 = c(a(b(a(b(a(c(c(a(c(c(a(b(a(x0))))))))))))))
→1.1.1.1.1.1.1.1.1.1 c(a(b(a(b(a(c(c(a(c(a(b(a(c(c(a(x0))))))))))))))))
→1.1.1.1.1.1.1.1.1.1.1.1 c(a(b(a(b(a(c(c(a(c(a(b(b(c(c(a(x0))))))))))))))))
→1.1.1.1.1.1.1.1.1 c(a(b(a(b(a(c(c(a(a(b(a(c(c(b(c(c(a(x0))))))))))))))))))
→1.1.1.1.1 c(a(b(a(b(c(c(a(a(b(a(c(c(b(c(c(a(x0)))))))))))))))))
→1.1.1.1.1.1.1.1.1.1.1.1.1.1 c(a(b(a(b(c(c(a(a(b(a(c(c(b(c(a(x0))))))))))))))))
→ε a(b(a(c(c(a(b(c(c(a(a(b(a(c(c(b(c(a(x0))))))))))))))))))
→1.1.1.1.1.1.1.1.1 a(b(a(c(c(a(b(c(c(a(b(a(c(c(b(c(a(x0)))))))))))))))))
→1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 a(b(a(c(c(a(b(c(c(a(b(a(c(c(b(a(x0))))))))))))))))
→1.1.1.1.1.1.1.1.1.1.1.1 a(b(a(c(c(a(b(c(c(a(b(a(c(b(a(x0)))))))))))))))
→1.1.1.1.1.1.1.1.1.1.1.1 a(b(a(c(c(a(b(c(c(a(b(a(b(a(x0))))))))))))))
→1.1.1.1.1.1.1.1 a(b(a(c(c(a(b(c(a(b(a(c(c(a(b(a(x0))))))))))))))))
→1.1.1.1.1.1.1 a(b(a(c(c(a(b(a(b(a(c(c(a(c(c(a(b(a(x0))))))))))))))))))
= t12
where t12 = C[t0σ] and σ = {x0/x0} and C = a(b(a(c(☐))))