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(a(x0)) → rotate(cut(Ca(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
begin(d(x0)) → rotate(cut(Cd(guess(x0))))
guess(a(x0)) → Ca(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(d(x0)) → Cd(guess(x0))
guess(a(x0)) → moveleft(Ba(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(d(x0)) → moveleft(Bd(wait(x0)))
guess(end(x0)) → finish(end(x0))
Ca(moveleft(Ba(x0))) → moveleft(Ba(Aa(x0)))
Cb(moveleft(Ba(x0))) → moveleft(Ba(Ab(x0)))
Cc(moveleft(Ba(x0))) → moveleft(Ba(Ac(x0)))
Cd(moveleft(Ba(x0))) → moveleft(Ba(Ad(x0)))
Ca(moveleft(Bb(x0))) → moveleft(Bb(Aa(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Cc(moveleft(Bb(x0))) → moveleft(Bb(Ac(x0)))
Cd(moveleft(Bb(x0))) → moveleft(Bb(Ad(x0)))
Ca(moveleft(Bc(x0))) → moveleft(Bc(Aa(x0)))
Cb(moveleft(Bc(x0))) → moveleft(Bc(Ab(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
Cd(moveleft(Bc(x0))) → moveleft(Bc(Ad(x0)))
Ca(moveleft(Bd(x0))) → moveleft(Bd(Aa(x0)))
Cb(moveleft(Bd(x0))) → moveleft(Bd(Ab(x0)))
Cc(moveleft(Bd(x0))) → moveleft(Bd(Ac(x0)))
Cd(moveleft(Bd(x0))) → moveleft(Bd(Ad(x0)))
cut(moveleft(Ba(x0))) → Da(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
cut(moveleft(Bd(x0))) → Dd(cut(goright(x0)))
goright(Aa(x0)) → Ca(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(Ad(x0)) → Cd(goright(x0))
goright(wait(a(x0))) → moveleft(Ba(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(d(x0))) → moveleft(Bd(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Ca(finish(x0)) → finish(a(x0))
Cb(finish(x0)) → finish(b(x0))
Cc(finish(x0)) → finish(c(x0))
Cd(finish(x0)) → finish(d(x0))
cut(finish(x0)) → finish2(x0)
Da(finish2(x0)) → finish2(a(x0))
Db(finish2(x0)) → finish2(b(x0))
Dc(finish2(x0)) → finish2(c(x0))
Dd(finish2(x0)) → finish2(d(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(a(a(x0))) → begin(a(b(a(b(a(x0))))))
rewrite(c(a(x0))) → begin(a(b(a(a(c(x0))))))
rewrite(b(b(b(x0)))) → begin(a(b(x0)))
rewrite(c(b(x0))) → begin(a(a(c(x0))))
rewrite(c(b(x0))) → begin(b(a(d(x0))))
rewrite(d(d(x0))) → begin(d(b(d(b(d(x0))))))
rewrite(c(c(x0))) → begin(c(d(c(x0))))
rewrite(a(a(a(x0)))) → begin(a(b(b(x0))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(a(x0)) → rotate(cut(Ca(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
begin(d(x0)) → rotate(cut(Cd(guess(x0))))
guess(a(x0)) → Ca(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(d(x0)) → Cd(guess(x0))
guess(a(x0)) → moveleft(Ba(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(d(x0)) → moveleft(Bd(wait(x0)))
guess(end(x0)) → finish(end(x0))
Ca(moveleft(Ba(x0))) → moveleft(Ba(Aa(x0)))
Cb(moveleft(Ba(x0))) → moveleft(Ba(Ab(x0)))
Cc(moveleft(Ba(x0))) → moveleft(Ba(Ac(x0)))
Cd(moveleft(Ba(x0))) → moveleft(Ba(Ad(x0)))
Ca(moveleft(Bb(x0))) → moveleft(Bb(Aa(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Cc(moveleft(Bb(x0))) → moveleft(Bb(Ac(x0)))
Cd(moveleft(Bb(x0))) → moveleft(Bb(Ad(x0)))
Ca(moveleft(Bc(x0))) → moveleft(Bc(Aa(x0)))
Cb(moveleft(Bc(x0))) → moveleft(Bc(Ab(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
Cd(moveleft(Bc(x0))) → moveleft(Bc(Ad(x0)))
Ca(moveleft(Bd(x0))) → moveleft(Bd(Aa(x0)))
Cb(moveleft(Bd(x0))) → moveleft(Bd(Ab(x0)))
Cc(moveleft(Bd(x0))) → moveleft(Bd(Ac(x0)))
Cd(moveleft(Bd(x0))) → moveleft(Bd(Ad(x0)))
cut(moveleft(Ba(x0))) → Da(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
cut(moveleft(Bd(x0))) → Dd(cut(goright(x0)))
goright(Aa(x0)) → Ca(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(Ad(x0)) → Cd(goright(x0))
goright(wait(a(x0))) → moveleft(Ba(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(d(x0))) → moveleft(Bd(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Ca(finish(x0)) → finish(a(x0))
Cb(finish(x0)) → finish(b(x0))
Cc(finish(x0)) → finish(c(x0))
Cd(finish(x0)) → finish(d(x0))
cut(finish(x0)) → finish2(x0)
Da(finish2(x0)) → finish2(a(x0))
Db(finish2(x0)) → finish2(b(x0))
Dc(finish2(x0)) → finish2(c(x0))
Dd(finish2(x0)) → finish2(d(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(a(a(x0))) → begin(a(b(a(b(a(x0))))))
rewrite(c(a(x0))) → begin(a(b(a(a(c(x0))))))
rewrite(b(b(b(x0)))) → begin(a(b(x0)))
rewrite(c(b(x0))) → begin(a(a(c(x0))))
rewrite(c(b(x0))) → begin(b(a(d(x0))))
rewrite(d(d(x0))) → begin(d(b(d(b(d(x0))))))
rewrite(c(c(x0))) → begin(c(d(c(x0))))
rewrite(a(a(a(x0)))) → begin(a(b(b(x0))))