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

Proof

1 Loop

The following loop proves nontermination.

t0 = a(c(c(a(c(c(a(c(b(a(c(b(a(c(c(a(b(x0)))))))))))))))))
→1 a(c(a(c(c(a(c(b(a(c(b(a(c(c(a(b(x0))))))))))))))))
→1 a(a(c(c(a(c(b(a(c(b(a(c(c(a(b(x0)))))))))))))))
→1.1 a(a(c(a(c(b(a(c(b(a(c(c(a(b(x0))))))))))))))
→1.1 a(a(a(c(b(a(c(b(a(c(c(a(b(x0)))))))))))))
→1.1.1.1.1.1.1.1 a(a(a(c(b(a(c(b(c(c(a(b(x0))))))))))))
→ε b(c(b(a(c(b(c(c(a(b(x0))))))))))
→1.1.1.1.1 b(c(b(a(c(c(a(c(b(c(a(b(x0))))))))))))
→1.1.1 b(c(b(c(c(a(c(b(c(a(b(x0)))))))))))
→1.1 b(c(c(a(c(b(c(a(c(b(c(a(b(x0)))))))))))))
→1.1.1.1.1.1.1.1.1 b(c(c(a(c(b(c(a(c(c(a(c(b(a(b(x0)))))))))))))))
→1.1.1.1.1.1.1.1.1.1.1 b(c(c(a(c(b(c(a(c(c(a(b(a(b(x0))))))))))))))
→ε c(a(c(b(c(a(c(b(c(a(c(c(a(b(a(b(x0))))))))))))))))
→1.1.1.1.1 c(a(c(b(c(c(b(c(a(c(c(a(b(a(b(x0)))))))))))))))
→1.1.1 c(a(c(c(a(c(b(c(b(c(a(c(c(a(b(a(b(x0)))))))))))))))))
→1.1.1.1.1.1 c(a(c(c(a(c(c(a(c(b(b(c(a(c(c(a(b(a(b(x0)))))))))))))))))))
→1.1.1.1.1.1.1.1.1.1 c(a(c(c(a(c(c(a(c(b(c(a(c(b(a(c(c(a(b(a(b(x0)))))))))))))))))))))
→1.1.1.1.1.1.1.1.1 c(a(c(c(a(c(c(a(c(c(a(c(b(a(c(b(a(c(c(a(b(a(b(x0)))))))))))))))))))))))
= t17
where t17 = C[t0σ] and σ = {x0/a(b(x0))} and C = c(a(c(c(☐))))