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(g(x0)) → rotate(cut(Cg(guess(x0))))
begin(h(x0)) → rotate(cut(Ch(guess(x0))))
begin(f(x0)) → rotate(cut(Cf(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(a(x0)) → rotate(cut(Ca(guess(x0))))
guess(g(x0)) → Cg(guess(x0))
guess(h(x0)) → Ch(guess(x0))
guess(f(x0)) → Cf(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(a(x0)) → Ca(guess(x0))
guess(g(x0)) → moveleft(Bg(wait(x0)))
guess(h(x0)) → moveleft(Bh(wait(x0)))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(a(x0)) → moveleft(Ba(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cg(moveleft(Bg(x0))) → moveleft(Bg(Ag(x0)))
Ch(moveleft(Bg(x0))) → moveleft(Bg(Ah(x0)))
Cf(moveleft(Bg(x0))) → moveleft(Bg(Af(x0)))
Cs(moveleft(Bg(x0))) → moveleft(Bg(As(x0)))
Cb(moveleft(Bg(x0))) → moveleft(Bg(Ab(x0)))
Ca(moveleft(Bg(x0))) → moveleft(Bg(Aa(x0)))
Cg(moveleft(Bh(x0))) → moveleft(Bh(Ag(x0)))
Ch(moveleft(Bh(x0))) → moveleft(Bh(Ah(x0)))
Cf(moveleft(Bh(x0))) → moveleft(Bh(Af(x0)))
Cs(moveleft(Bh(x0))) → moveleft(Bh(As(x0)))
Cb(moveleft(Bh(x0))) → moveleft(Bh(Ab(x0)))
Ca(moveleft(Bh(x0))) → moveleft(Bh(Aa(x0)))
Cg(moveleft(Bf(x0))) → moveleft(Bf(Ag(x0)))
Ch(moveleft(Bf(x0))) → moveleft(Bf(Ah(x0)))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cb(moveleft(Bf(x0))) → moveleft(Bf(Ab(x0)))
Ca(moveleft(Bf(x0))) → moveleft(Bf(Aa(x0)))
Cg(moveleft(Bs(x0))) → moveleft(Bs(Ag(x0)))
Ch(moveleft(Bs(x0))) → moveleft(Bs(Ah(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cb(moveleft(Bs(x0))) → moveleft(Bs(Ab(x0)))
Ca(moveleft(Bs(x0))) → moveleft(Bs(Aa(x0)))
Cg(moveleft(Bb(x0))) → moveleft(Bb(Ag(x0)))
Ch(moveleft(Bb(x0))) → moveleft(Bb(Ah(x0)))
Cf(moveleft(Bb(x0))) → moveleft(Bb(Af(x0)))
Cs(moveleft(Bb(x0))) → moveleft(Bb(As(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Ca(moveleft(Bb(x0))) → moveleft(Bb(Aa(x0)))
Cg(moveleft(Ba(x0))) → moveleft(Ba(Ag(x0)))
Ch(moveleft(Ba(x0))) → moveleft(Ba(Ah(x0)))
Cf(moveleft(Ba(x0))) → moveleft(Ba(Af(x0)))
Cs(moveleft(Ba(x0))) → moveleft(Ba(As(x0)))
Cb(moveleft(Ba(x0))) → moveleft(Ba(Ab(x0)))
Ca(moveleft(Ba(x0))) → moveleft(Ba(Aa(x0)))
cut(moveleft(Bg(x0))) → Dg(cut(goright(x0)))
cut(moveleft(Bh(x0))) → Dh(cut(goright(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Ba(x0))) → Da(cut(goright(x0)))
goright(Ag(x0)) → Cg(goright(x0))
goright(Ah(x0)) → Ch(goright(x0))
goright(Af(x0)) → Cf(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(Aa(x0)) → Ca(goright(x0))
goright(wait(g(x0))) → moveleft(Bg(wait(x0)))
goright(wait(h(x0))) → moveleft(Bh(wait(x0)))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(a(x0))) → moveleft(Ba(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cg(finish(x0)) → finish(g(x0))
Ch(finish(x0)) → finish(h(x0))
Cf(finish(x0)) → finish(f(x0))
Cs(finish(x0)) → finish(s(x0))
Cb(finish(x0)) → finish(b(x0))
Ca(finish(x0)) → finish(a(x0))
cut(finish(x0)) → finish2(x0)
Dg(finish2(x0)) → finish2(g(x0))
Dh(finish2(x0)) → finish2(h(x0))
Df(finish2(x0)) → finish2(f(x0))
Ds(finish2(x0)) → finish2(s(x0))
Db(finish2(x0)) → finish2(b(x0))
Da(finish2(x0)) → finish2(a(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(g(h(x0))) → begin(g(f(s(x0))))
rewrite(f(s(s(s(x0))))) → begin(h(f(s(h(x0)))))
rewrite(f(h(x0))) → begin(h(f(s(h(x0)))))
rewrite(h(x0)) → begin(x0)
rewrite(f(f(s(s(x0))))) → begin(s(s(s(f(f(x0))))))
rewrite(b(a(x0))) → begin(a(b(x0)))
rewrite(a(a(a(x0)))) → begin(b(a(a(b(x0)))))
rewrite(b(b(b(b(x0))))) → begin(a(x0))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(g(x0)) → rotate(cut(Cg(guess(x0))))
begin(h(x0)) → rotate(cut(Ch(guess(x0))))
begin(f(x0)) → rotate(cut(Cf(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(a(x0)) → rotate(cut(Ca(guess(x0))))
guess(g(x0)) → Cg(guess(x0))
guess(h(x0)) → Ch(guess(x0))
guess(f(x0)) → Cf(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(a(x0)) → Ca(guess(x0))
guess(g(x0)) → moveleft(Bg(wait(x0)))
guess(h(x0)) → moveleft(Bh(wait(x0)))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(a(x0)) → moveleft(Ba(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cg(moveleft(Bg(x0))) → moveleft(Bg(Ag(x0)))
Ch(moveleft(Bg(x0))) → moveleft(Bg(Ah(x0)))
Cf(moveleft(Bg(x0))) → moveleft(Bg(Af(x0)))
Cs(moveleft(Bg(x0))) → moveleft(Bg(As(x0)))
Cb(moveleft(Bg(x0))) → moveleft(Bg(Ab(x0)))
Ca(moveleft(Bg(x0))) → moveleft(Bg(Aa(x0)))
Cg(moveleft(Bh(x0))) → moveleft(Bh(Ag(x0)))
Ch(moveleft(Bh(x0))) → moveleft(Bh(Ah(x0)))
Cf(moveleft(Bh(x0))) → moveleft(Bh(Af(x0)))
Cs(moveleft(Bh(x0))) → moveleft(Bh(As(x0)))
Cb(moveleft(Bh(x0))) → moveleft(Bh(Ab(x0)))
Ca(moveleft(Bh(x0))) → moveleft(Bh(Aa(x0)))
Cg(moveleft(Bf(x0))) → moveleft(Bf(Ag(x0)))
Ch(moveleft(Bf(x0))) → moveleft(Bf(Ah(x0)))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cb(moveleft(Bf(x0))) → moveleft(Bf(Ab(x0)))
Ca(moveleft(Bf(x0))) → moveleft(Bf(Aa(x0)))
Cg(moveleft(Bs(x0))) → moveleft(Bs(Ag(x0)))
Ch(moveleft(Bs(x0))) → moveleft(Bs(Ah(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cb(moveleft(Bs(x0))) → moveleft(Bs(Ab(x0)))
Ca(moveleft(Bs(x0))) → moveleft(Bs(Aa(x0)))
Cg(moveleft(Bb(x0))) → moveleft(Bb(Ag(x0)))
Ch(moveleft(Bb(x0))) → moveleft(Bb(Ah(x0)))
Cf(moveleft(Bb(x0))) → moveleft(Bb(Af(x0)))
Cs(moveleft(Bb(x0))) → moveleft(Bb(As(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Ca(moveleft(Bb(x0))) → moveleft(Bb(Aa(x0)))
Cg(moveleft(Ba(x0))) → moveleft(Ba(Ag(x0)))
Ch(moveleft(Ba(x0))) → moveleft(Ba(Ah(x0)))
Cf(moveleft(Ba(x0))) → moveleft(Ba(Af(x0)))
Cs(moveleft(Ba(x0))) → moveleft(Ba(As(x0)))
Cb(moveleft(Ba(x0))) → moveleft(Ba(Ab(x0)))
Ca(moveleft(Ba(x0))) → moveleft(Ba(Aa(x0)))
cut(moveleft(Bg(x0))) → Dg(cut(goright(x0)))
cut(moveleft(Bh(x0))) → Dh(cut(goright(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Ba(x0))) → Da(cut(goright(x0)))
goright(Ag(x0)) → Cg(goright(x0))
goright(Ah(x0)) → Ch(goright(x0))
goright(Af(x0)) → Cf(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(Aa(x0)) → Ca(goright(x0))
goright(wait(g(x0))) → moveleft(Bg(wait(x0)))
goright(wait(h(x0))) → moveleft(Bh(wait(x0)))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(a(x0))) → moveleft(Ba(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cg(finish(x0)) → finish(g(x0))
Ch(finish(x0)) → finish(h(x0))
Cf(finish(x0)) → finish(f(x0))
Cs(finish(x0)) → finish(s(x0))
Cb(finish(x0)) → finish(b(x0))
Ca(finish(x0)) → finish(a(x0))
cut(finish(x0)) → finish2(x0)
Dg(finish2(x0)) → finish2(g(x0))
Dh(finish2(x0)) → finish2(h(x0))
Df(finish2(x0)) → finish2(f(x0))
Ds(finish2(x0)) → finish2(s(x0))
Db(finish2(x0)) → finish2(b(x0))
Da(finish2(x0)) → finish2(a(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(g(h(x0))) → begin(g(f(s(x0))))
rewrite(f(s(s(s(x0))))) → begin(h(f(s(h(x0)))))
rewrite(f(h(x0))) → begin(h(f(s(h(x0)))))
rewrite(h(x0)) → begin(x0)
rewrite(f(f(s(s(x0))))) → begin(s(s(s(f(f(x0))))))
rewrite(b(a(x0))) → begin(a(b(x0)))
rewrite(a(a(a(x0)))) → begin(b(a(a(b(x0)))))
rewrite(b(b(b(b(x0))))) → begin(a(x0))