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(thrice(x0)) → rotate(cut(Cthrice(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(half(x0)) → rotate(cut(Chalf(guess(x0))))
begin(sixtimes(x0)) → rotate(cut(Csixtimes(guess(x0))))
guess(thrice(x0)) → Cthrice(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(half(x0)) → Chalf(guess(x0))
guess(sixtimes(x0)) → Csixtimes(guess(x0))
guess(thrice(x0)) → moveleft(Bthrice(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(half(x0)) → moveleft(Bhalf(wait(x0)))
guess(sixtimes(x0)) → moveleft(Bsixtimes(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cthrice(moveleft(Bthrice(x0))) → moveleft(Bthrice(Athrice(x0)))
C0(moveleft(Bthrice(x0))) → moveleft(Bthrice(A0(x0)))
Cp(moveleft(Bthrice(x0))) → moveleft(Bthrice(Ap(x0)))
Cs(moveleft(Bthrice(x0))) → moveleft(Bthrice(As(x0)))
Chalf(moveleft(Bthrice(x0))) → moveleft(Bthrice(Ahalf(x0)))
Csixtimes(moveleft(Bthrice(x0))) → moveleft(Bthrice(Asixtimes(x0)))
Cthrice(moveleft(B0(x0))) → moveleft(B0(Athrice(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Chalf(moveleft(B0(x0))) → moveleft(B0(Ahalf(x0)))
Csixtimes(moveleft(B0(x0))) → moveleft(B0(Asixtimes(x0)))
Cthrice(moveleft(Bp(x0))) → moveleft(Bp(Athrice(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Chalf(moveleft(Bp(x0))) → moveleft(Bp(Ahalf(x0)))
Csixtimes(moveleft(Bp(x0))) → moveleft(Bp(Asixtimes(x0)))
Cthrice(moveleft(Bs(x0))) → moveleft(Bs(Athrice(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Chalf(moveleft(Bs(x0))) → moveleft(Bs(Ahalf(x0)))
Csixtimes(moveleft(Bs(x0))) → moveleft(Bs(Asixtimes(x0)))
Cthrice(moveleft(Bhalf(x0))) → moveleft(Bhalf(Athrice(x0)))
C0(moveleft(Bhalf(x0))) → moveleft(Bhalf(A0(x0)))
Cp(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ap(x0)))
Cs(moveleft(Bhalf(x0))) → moveleft(Bhalf(As(x0)))
Chalf(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ahalf(x0)))
Csixtimes(moveleft(Bhalf(x0))) → moveleft(Bhalf(Asixtimes(x0)))
Cthrice(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(Athrice(x0)))
C0(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(A0(x0)))
Cp(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(Ap(x0)))
Cs(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(As(x0)))
Chalf(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(Ahalf(x0)))
Csixtimes(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(Asixtimes(x0)))
cut(moveleft(Bthrice(x0))) → Dthrice(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bhalf(x0))) → Dhalf(cut(goright(x0)))
cut(moveleft(Bsixtimes(x0))) → Dsixtimes(cut(goright(x0)))
goright(Athrice(x0)) → Cthrice(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ahalf(x0)) → Chalf(goright(x0))
goright(Asixtimes(x0)) → Csixtimes(goright(x0))
goright(wait(thrice(x0))) → moveleft(Bthrice(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(half(x0))) → moveleft(Bhalf(wait(x0)))
goright(wait(sixtimes(x0))) → moveleft(Bsixtimes(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cthrice(finish(x0)) → finish(thrice(x0))
C0(finish(x0)) → finish(0(x0))
Cp(finish(x0)) → finish(p(x0))
Cs(finish(x0)) → finish(s(x0))
Chalf(finish(x0)) → finish(half(x0))
Csixtimes(finish(x0)) → finish(sixtimes(x0))
cut(finish(x0)) → finish2(x0)
Dthrice(finish2(x0)) → finish2(thrice(x0))
D0(finish2(x0)) → finish2(0(x0))
Dp(finish2(x0)) → finish2(p(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dhalf(finish2(x0)) → finish2(half(x0))
Dsixtimes(finish2(x0)) → finish2(sixtimes(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(thrice(0(x0))) → begin(p(s(p(p(p(s(s(s(0(p(s(p(s(x0))))))))))))))
rewrite(thrice(s(x0))) → begin(p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x0)))))))))))))))))))
rewrite(half(0(x0))) → begin(p(p(s(s(p(s(0(p(s(s(s(s(x0)))))))))))))
rewrite(half(s(x0))) → begin(p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x0))))))))))))))))))
rewrite(half(s(s(x0)))) → begin(p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x0)))))))))))))))))
rewrite(sixtimes(0(x0))) → begin(p(s(p(s(0(s(s(s(s(s(p(s(p(s(x0)))))))))))))))
rewrite(sixtimes(s(x0))) → begin(p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x0))))))))))))))))))))))))))
rewrite(p(p(s(x0)))) → begin(p(x0))
rewrite(p(s(x0))) → begin(x0)
rewrite(p(0(x0))) → begin(0(s(s(s(s(x0))))))
rewrite(0(x0)) → begin(x0)

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(thrice(x0)) → rotate(cut(Cthrice(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(half(x0)) → rotate(cut(Chalf(guess(x0))))
begin(sixtimes(x0)) → rotate(cut(Csixtimes(guess(x0))))
guess(thrice(x0)) → Cthrice(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(half(x0)) → Chalf(guess(x0))
guess(sixtimes(x0)) → Csixtimes(guess(x0))
guess(thrice(x0)) → moveleft(Bthrice(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(half(x0)) → moveleft(Bhalf(wait(x0)))
guess(sixtimes(x0)) → moveleft(Bsixtimes(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cthrice(moveleft(Bthrice(x0))) → moveleft(Bthrice(Athrice(x0)))
C0(moveleft(Bthrice(x0))) → moveleft(Bthrice(A0(x0)))
Cp(moveleft(Bthrice(x0))) → moveleft(Bthrice(Ap(x0)))
Cs(moveleft(Bthrice(x0))) → moveleft(Bthrice(As(x0)))
Chalf(moveleft(Bthrice(x0))) → moveleft(Bthrice(Ahalf(x0)))
Csixtimes(moveleft(Bthrice(x0))) → moveleft(Bthrice(Asixtimes(x0)))
Cthrice(moveleft(B0(x0))) → moveleft(B0(Athrice(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Chalf(moveleft(B0(x0))) → moveleft(B0(Ahalf(x0)))
Csixtimes(moveleft(B0(x0))) → moveleft(B0(Asixtimes(x0)))
Cthrice(moveleft(Bp(x0))) → moveleft(Bp(Athrice(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Chalf(moveleft(Bp(x0))) → moveleft(Bp(Ahalf(x0)))
Csixtimes(moveleft(Bp(x0))) → moveleft(Bp(Asixtimes(x0)))
Cthrice(moveleft(Bs(x0))) → moveleft(Bs(Athrice(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Chalf(moveleft(Bs(x0))) → moveleft(Bs(Ahalf(x0)))
Csixtimes(moveleft(Bs(x0))) → moveleft(Bs(Asixtimes(x0)))
Cthrice(moveleft(Bhalf(x0))) → moveleft(Bhalf(Athrice(x0)))
C0(moveleft(Bhalf(x0))) → moveleft(Bhalf(A0(x0)))
Cp(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ap(x0)))
Cs(moveleft(Bhalf(x0))) → moveleft(Bhalf(As(x0)))
Chalf(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ahalf(x0)))
Csixtimes(moveleft(Bhalf(x0))) → moveleft(Bhalf(Asixtimes(x0)))
Cthrice(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(Athrice(x0)))
C0(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(A0(x0)))
Cp(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(Ap(x0)))
Cs(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(As(x0)))
Chalf(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(Ahalf(x0)))
Csixtimes(moveleft(Bsixtimes(x0))) → moveleft(Bsixtimes(Asixtimes(x0)))
cut(moveleft(Bthrice(x0))) → Dthrice(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bhalf(x0))) → Dhalf(cut(goright(x0)))
cut(moveleft(Bsixtimes(x0))) → Dsixtimes(cut(goright(x0)))
goright(Athrice(x0)) → Cthrice(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ahalf(x0)) → Chalf(goright(x0))
goright(Asixtimes(x0)) → Csixtimes(goright(x0))
goright(wait(thrice(x0))) → moveleft(Bthrice(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(half(x0))) → moveleft(Bhalf(wait(x0)))
goright(wait(sixtimes(x0))) → moveleft(Bsixtimes(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cthrice(finish(x0)) → finish(thrice(x0))
C0(finish(x0)) → finish(0(x0))
Cp(finish(x0)) → finish(p(x0))
Cs(finish(x0)) → finish(s(x0))
Chalf(finish(x0)) → finish(half(x0))
Csixtimes(finish(x0)) → finish(sixtimes(x0))
cut(finish(x0)) → finish2(x0)
Dthrice(finish2(x0)) → finish2(thrice(x0))
D0(finish2(x0)) → finish2(0(x0))
Dp(finish2(x0)) → finish2(p(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dhalf(finish2(x0)) → finish2(half(x0))
Dsixtimes(finish2(x0)) → finish2(sixtimes(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(thrice(0(x0))) → begin(p(s(p(p(p(s(s(s(0(p(s(p(s(x0))))))))))))))
rewrite(thrice(s(x0))) → begin(p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x0)))))))))))))))))))
rewrite(half(0(x0))) → begin(p(p(s(s(p(s(0(p(s(s(s(s(x0)))))))))))))
rewrite(half(s(x0))) → begin(p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x0))))))))))))))))))
rewrite(half(s(s(x0)))) → begin(p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x0)))))))))))))))))
rewrite(sixtimes(0(x0))) → begin(p(s(p(s(0(s(s(s(s(s(p(s(p(s(x0)))))))))))))))
rewrite(sixtimes(s(x0))) → begin(p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x0))))))))))))))))))))))))))
rewrite(p(p(s(x0)))) → begin(p(x0))
rewrite(p(s(x0))) → begin(x0)
rewrite(p(0(x0))) → begin(0(s(s(s(s(x0))))))
rewrite(0(x0)) → begin(x0)