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(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(t(x0)) → rotate(cut(Ct(guess(x0))))
begin(u(x0)) → rotate(cut(Cu(guess(x0))))
guess(s(x0)) → Cs(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(t(x0)) → Ct(guess(x0))
guess(u(x0)) → Cu(guess(x0))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(t(x0)) → moveleft(Bt(wait(x0)))
guess(u(x0)) → moveleft(Bu(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cb(moveleft(Bs(x0))) → moveleft(Bs(Ab(x0)))
Ct(moveleft(Bs(x0))) → moveleft(Bs(At(x0)))
Cu(moveleft(Bs(x0))) → moveleft(Bs(Au(x0)))
Cs(moveleft(Bb(x0))) → moveleft(Bb(As(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Ct(moveleft(Bb(x0))) → moveleft(Bb(At(x0)))
Cu(moveleft(Bb(x0))) → moveleft(Bb(Au(x0)))
Cs(moveleft(Bt(x0))) → moveleft(Bt(As(x0)))
Cb(moveleft(Bt(x0))) → moveleft(Bt(Ab(x0)))
Ct(moveleft(Bt(x0))) → moveleft(Bt(At(x0)))
Cu(moveleft(Bt(x0))) → moveleft(Bt(Au(x0)))
Cs(moveleft(Bu(x0))) → moveleft(Bu(As(x0)))
Cb(moveleft(Bu(x0))) → moveleft(Bu(Ab(x0)))
Ct(moveleft(Bu(x0))) → moveleft(Bu(At(x0)))
Cu(moveleft(Bu(x0))) → moveleft(Bu(Au(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Bt(x0))) → Dt(cut(goright(x0)))
cut(moveleft(Bu(x0))) → Du(cut(goright(x0)))
goright(As(x0)) → Cs(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(At(x0)) → Ct(goright(x0))
goright(Au(x0)) → Cu(goright(x0))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(t(x0))) → moveleft(Bt(wait(x0)))
goright(wait(u(x0))) → moveleft(Bu(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cs(finish(x0)) → finish(s(x0))
Cb(finish(x0)) → finish(b(x0))
Ct(finish(x0)) → finish(t(x0))
Cu(finish(x0)) → finish(u(x0))
cut(finish(x0)) → finish2(x0)
Ds(finish2(x0)) → finish2(s(x0))
Db(finish2(x0)) → finish2(b(x0))
Dt(finish2(x0)) → finish2(t(x0))
Du(finish2(x0)) → finish2(u(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(s(b(x0))) → begin(b(s(s(s(x0)))))
rewrite(s(b(s(x0)))) → begin(b(t(x0)))
rewrite(t(b(x0))) → begin(b(s(x0)))
rewrite(t(b(s(x0)))) → begin(u(t(b(x0))))
rewrite(b(u(x0))) → begin(b(s(x0)))
rewrite(t(s(x0))) → begin(t(t(x0)))
rewrite(t(u(x0))) → begin(u(t(x0)))
rewrite(s(u(x0))) → begin(s(s(x0)))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(t(x0)) → rotate(cut(Ct(guess(x0))))
begin(u(x0)) → rotate(cut(Cu(guess(x0))))
guess(s(x0)) → Cs(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(t(x0)) → Ct(guess(x0))
guess(u(x0)) → Cu(guess(x0))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(t(x0)) → moveleft(Bt(wait(x0)))
guess(u(x0)) → moveleft(Bu(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cb(moveleft(Bs(x0))) → moveleft(Bs(Ab(x0)))
Ct(moveleft(Bs(x0))) → moveleft(Bs(At(x0)))
Cu(moveleft(Bs(x0))) → moveleft(Bs(Au(x0)))
Cs(moveleft(Bb(x0))) → moveleft(Bb(As(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Ct(moveleft(Bb(x0))) → moveleft(Bb(At(x0)))
Cu(moveleft(Bb(x0))) → moveleft(Bb(Au(x0)))
Cs(moveleft(Bt(x0))) → moveleft(Bt(As(x0)))
Cb(moveleft(Bt(x0))) → moveleft(Bt(Ab(x0)))
Ct(moveleft(Bt(x0))) → moveleft(Bt(At(x0)))
Cu(moveleft(Bt(x0))) → moveleft(Bt(Au(x0)))
Cs(moveleft(Bu(x0))) → moveleft(Bu(As(x0)))
Cb(moveleft(Bu(x0))) → moveleft(Bu(Ab(x0)))
Ct(moveleft(Bu(x0))) → moveleft(Bu(At(x0)))
Cu(moveleft(Bu(x0))) → moveleft(Bu(Au(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Bt(x0))) → Dt(cut(goright(x0)))
cut(moveleft(Bu(x0))) → Du(cut(goright(x0)))
goright(As(x0)) → Cs(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(At(x0)) → Ct(goright(x0))
goright(Au(x0)) → Cu(goright(x0))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(t(x0))) → moveleft(Bt(wait(x0)))
goright(wait(u(x0))) → moveleft(Bu(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cs(finish(x0)) → finish(s(x0))
Cb(finish(x0)) → finish(b(x0))
Ct(finish(x0)) → finish(t(x0))
Cu(finish(x0)) → finish(u(x0))
cut(finish(x0)) → finish2(x0)
Ds(finish2(x0)) → finish2(s(x0))
Db(finish2(x0)) → finish2(b(x0))
Dt(finish2(x0)) → finish2(t(x0))
Du(finish2(x0)) → finish2(u(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(s(b(x0))) → begin(b(s(s(s(x0)))))
rewrite(s(b(s(x0)))) → begin(b(t(x0)))
rewrite(t(b(x0))) → begin(b(s(x0)))
rewrite(t(b(s(x0)))) → begin(u(t(b(x0))))
rewrite(b(u(x0))) → begin(b(s(x0)))
rewrite(t(s(x0))) → begin(t(t(x0)))
rewrite(t(u(x0))) → begin(u(t(x0)))
rewrite(s(u(x0))) → begin(s(s(x0)))