MAYBE Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

begin(end(x0)) → rewrite(end(x0))
begin(1(x0)) → rotate(cut(C1(guess(x0))))
begin(q0(x0)) → rotate(cut(Cq0(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(q1(x0)) → rotate(cut(Cq1(guess(x0))))
begin(q2(x0)) → rotate(cut(Cq2(guess(x0))))
guess(1(x0)) → C1(guess(x0))
guess(q0(x0)) → Cq0(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(q1(x0)) → Cq1(guess(x0))
guess(q2(x0)) → Cq2(guess(x0))
guess(1(x0)) → moveleft(B1(wait(x0)))
guess(q0(x0)) → moveleft(Bq0(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(q1(x0)) → moveleft(Bq1(wait(x0)))
guess(q2(x0)) → moveleft(Bq2(wait(x0)))
guess(end(x0)) → finish(end(x0))
C1(moveleft(B1(x0))) → moveleft(B1(A1(x0)))
Cq0(moveleft(B1(x0))) → moveleft(B1(Aq0(x0)))
C0(moveleft(B1(x0))) → moveleft(B1(A0(x0)))
Cq1(moveleft(B1(x0))) → moveleft(B1(Aq1(x0)))
Cq2(moveleft(B1(x0))) → moveleft(B1(Aq2(x0)))
C1(moveleft(Bq0(x0))) → moveleft(Bq0(A1(x0)))
Cq0(moveleft(Bq0(x0))) → moveleft(Bq0(Aq0(x0)))
C0(moveleft(Bq0(x0))) → moveleft(Bq0(A0(x0)))
Cq1(moveleft(Bq0(x0))) → moveleft(Bq0(Aq1(x0)))
Cq2(moveleft(Bq0(x0))) → moveleft(Bq0(Aq2(x0)))
C1(moveleft(B0(x0))) → moveleft(B0(A1(x0)))
Cq0(moveleft(B0(x0))) → moveleft(B0(Aq0(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cq1(moveleft(B0(x0))) → moveleft(B0(Aq1(x0)))
Cq2(moveleft(B0(x0))) → moveleft(B0(Aq2(x0)))
C1(moveleft(Bq1(x0))) → moveleft(Bq1(A1(x0)))
Cq0(moveleft(Bq1(x0))) → moveleft(Bq1(Aq0(x0)))
C0(moveleft(Bq1(x0))) → moveleft(Bq1(A0(x0)))
Cq1(moveleft(Bq1(x0))) → moveleft(Bq1(Aq1(x0)))
Cq2(moveleft(Bq1(x0))) → moveleft(Bq1(Aq2(x0)))
C1(moveleft(Bq2(x0))) → moveleft(Bq2(A1(x0)))
Cq0(moveleft(Bq2(x0))) → moveleft(Bq2(Aq0(x0)))
C0(moveleft(Bq2(x0))) → moveleft(Bq2(A0(x0)))
Cq1(moveleft(Bq2(x0))) → moveleft(Bq2(Aq1(x0)))
Cq2(moveleft(Bq2(x0))) → moveleft(Bq2(Aq2(x0)))
cut(moveleft(B1(x0))) → D1(cut(goright(x0)))
cut(moveleft(Bq0(x0))) → Dq0(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bq1(x0))) → Dq1(cut(goright(x0)))
cut(moveleft(Bq2(x0))) → Dq2(cut(goright(x0)))
goright(A1(x0)) → C1(goright(x0))
goright(Aq0(x0)) → Cq0(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Aq1(x0)) → Cq1(goright(x0))
goright(Aq2(x0)) → Cq2(goright(x0))
goright(wait(1(x0))) → moveleft(B1(wait(x0)))
goright(wait(q0(x0))) → moveleft(Bq0(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(q1(x0))) → moveleft(Bq1(wait(x0)))
goright(wait(q2(x0))) → moveleft(Bq2(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
C1(finish(x0)) → finish(1(x0))
Cq0(finish(x0)) → finish(q0(x0))
C0(finish(x0)) → finish(0(x0))
Cq1(finish(x0)) → finish(q1(x0))
Cq2(finish(x0)) → finish(q2(x0))
cut(finish(x0)) → finish2(x0)
D1(finish2(x0)) → finish2(1(x0))
Dq0(finish2(x0)) → finish2(q0(x0))
D0(finish2(x0)) → finish2(0(x0))
Dq1(finish2(x0)) → finish2(q1(x0))
Dq2(finish2(x0)) → finish2(q2(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(1(q0(1(x0)))) → begin(0(1(q1(x0))))
rewrite(1(q0(0(x0)))) → begin(0(0(q1(x0))))
rewrite(1(q1(1(x0)))) → begin(1(1(q1(x0))))
rewrite(1(q1(0(x0)))) → begin(1(0(q1(x0))))
rewrite(0(q1(x0))) → begin(q2(1(x0)))
rewrite(1(q2(x0))) → begin(q2(1(x0)))
rewrite(0(q2(x0))) → begin(0(q0(x0)))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(1(x0)) → rotate(cut(C1(guess(x0))))
begin(q0(x0)) → rotate(cut(Cq0(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(q1(x0)) → rotate(cut(Cq1(guess(x0))))
begin(q2(x0)) → rotate(cut(Cq2(guess(x0))))
guess(1(x0)) → C1(guess(x0))
guess(q0(x0)) → Cq0(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(q1(x0)) → Cq1(guess(x0))
guess(q2(x0)) → Cq2(guess(x0))
guess(1(x0)) → moveleft(B1(wait(x0)))
guess(q0(x0)) → moveleft(Bq0(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(q1(x0)) → moveleft(Bq1(wait(x0)))
guess(q2(x0)) → moveleft(Bq2(wait(x0)))
guess(end(x0)) → finish(end(x0))
C1(moveleft(B1(x0))) → moveleft(B1(A1(x0)))
Cq0(moveleft(B1(x0))) → moveleft(B1(Aq0(x0)))
C0(moveleft(B1(x0))) → moveleft(B1(A0(x0)))
Cq1(moveleft(B1(x0))) → moveleft(B1(Aq1(x0)))
Cq2(moveleft(B1(x0))) → moveleft(B1(Aq2(x0)))
C1(moveleft(Bq0(x0))) → moveleft(Bq0(A1(x0)))
Cq0(moveleft(Bq0(x0))) → moveleft(Bq0(Aq0(x0)))
C0(moveleft(Bq0(x0))) → moveleft(Bq0(A0(x0)))
Cq1(moveleft(Bq0(x0))) → moveleft(Bq0(Aq1(x0)))
Cq2(moveleft(Bq0(x0))) → moveleft(Bq0(Aq2(x0)))
C1(moveleft(B0(x0))) → moveleft(B0(A1(x0)))
Cq0(moveleft(B0(x0))) → moveleft(B0(Aq0(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cq1(moveleft(B0(x0))) → moveleft(B0(Aq1(x0)))
Cq2(moveleft(B0(x0))) → moveleft(B0(Aq2(x0)))
C1(moveleft(Bq1(x0))) → moveleft(Bq1(A1(x0)))
Cq0(moveleft(Bq1(x0))) → moveleft(Bq1(Aq0(x0)))
C0(moveleft(Bq1(x0))) → moveleft(Bq1(A0(x0)))
Cq1(moveleft(Bq1(x0))) → moveleft(Bq1(Aq1(x0)))
Cq2(moveleft(Bq1(x0))) → moveleft(Bq1(Aq2(x0)))
C1(moveleft(Bq2(x0))) → moveleft(Bq2(A1(x0)))
Cq0(moveleft(Bq2(x0))) → moveleft(Bq2(Aq0(x0)))
C0(moveleft(Bq2(x0))) → moveleft(Bq2(A0(x0)))
Cq1(moveleft(Bq2(x0))) → moveleft(Bq2(Aq1(x0)))
Cq2(moveleft(Bq2(x0))) → moveleft(Bq2(Aq2(x0)))
cut(moveleft(B1(x0))) → D1(cut(goright(x0)))
cut(moveleft(Bq0(x0))) → Dq0(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bq1(x0))) → Dq1(cut(goright(x0)))
cut(moveleft(Bq2(x0))) → Dq2(cut(goright(x0)))
goright(A1(x0)) → C1(goright(x0))
goright(Aq0(x0)) → Cq0(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Aq1(x0)) → Cq1(goright(x0))
goright(Aq2(x0)) → Cq2(goright(x0))
goright(wait(1(x0))) → moveleft(B1(wait(x0)))
goright(wait(q0(x0))) → moveleft(Bq0(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(q1(x0))) → moveleft(Bq1(wait(x0)))
goright(wait(q2(x0))) → moveleft(Bq2(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
C1(finish(x0)) → finish(1(x0))
Cq0(finish(x0)) → finish(q0(x0))
C0(finish(x0)) → finish(0(x0))
Cq1(finish(x0)) → finish(q1(x0))
Cq2(finish(x0)) → finish(q2(x0))
cut(finish(x0)) → finish2(x0)
D1(finish2(x0)) → finish2(1(x0))
Dq0(finish2(x0)) → finish2(q0(x0))
D0(finish2(x0)) → finish2(0(x0))
Dq1(finish2(x0)) → finish2(q1(x0))
Dq2(finish2(x0)) → finish2(q2(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(1(q0(1(x0)))) → begin(0(1(q1(x0))))
rewrite(1(q0(0(x0)))) → begin(0(0(q1(x0))))
rewrite(1(q1(1(x0)))) → begin(1(1(q1(x0))))
rewrite(1(q1(0(x0)))) → begin(1(0(q1(x0))))
rewrite(0(q1(x0))) → begin(q2(1(x0)))
rewrite(1(q2(x0))) → begin(q2(1(x0)))
rewrite(0(q2(x0))) → begin(0(q0(x0)))