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

Proof

1 Loop

The following loop proves nontermination.

t0 = a(a(b(a(c(c(b(b(x0))))))))
→1.1.1.1 a(a(b(a(b(b(x0))))))
→1.1.1 a(a(b(b(a(c(a(b(x0))))))))
→1 a(b(a(c(a(b(a(c(a(b(x0))))))))))
→1.1.1.1.1.1 a(b(a(c(a(b(b(c(a(b(x0))))))))))
→1.1.1.1.1.1.1.1 a(b(a(c(a(b(b(c(b(b(x0))))))))))
→1.1 a(b(b(c(a(b(b(c(b(b(x0))))))))))
→ε b(a(c(a(b(c(a(b(b(c(b(b(x0))))))))))))
→1.1.1.1.1.1 b(a(c(a(b(c(b(a(c(a(b(c(b(b(x0))))))))))))))
→1.1.1 b(a(c(b(a(c(a(c(b(a(c(a(b(c(b(b(x0))))))))))))))))
→1.1.1.1.1.1.1.1.1.1.1 b(a(c(b(a(c(a(c(b(a(c(b(a(c(a(c(b(b(x0))))))))))))))))))
→1.1.1.1.1.1.1.1.1 b(a(c(b(a(c(a(c(b(b(c(b(a(c(a(c(b(b(x0))))))))))))))))))
→1.1.1 b(a(c(a(c(a(c(b(b(c(b(a(c(a(c(b(b(x0)))))))))))))))))
→1.1.1 b(a(c(c(a(c(b(b(c(b(a(c(a(c(b(b(x0))))))))))))))))
→1.1 b(a(a(c(b(b(c(b(a(c(a(c(b(b(x0))))))))))))))
→1.1.1.1 b(a(a(c(b(c(b(a(c(a(c(b(b(x0)))))))))))))
→1.1.1.1 b(a(a(c(c(b(a(c(a(c(b(b(x0))))))))))))
→1.1.1 b(a(a(b(a(c(a(c(b(b(x0))))))))))
→1.1.1.1.1.1 b(a(a(b(a(c(c(b(b(x0)))))))))
= t18
where t18 = C[t0σ] and σ = {x0/x0} and C = b(☐)