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(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(f(x0)) → rotate(cut(Cf(guess(x0))))
begin(g(x0)) → rotate(cut(Cg(guess(x0))))
begin(j(x0)) → rotate(cut(Cj(guess(x0))))
begin(half(x0)) → rotate(cut(Chalf(guess(x0))))
begin(rd(x0)) → rotate(cut(Crd(guess(x0))))
guess(p(x0)) → Cp(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(f(x0)) → Cf(guess(x0))
guess(g(x0)) → Cg(guess(x0))
guess(j(x0)) → Cj(guess(x0))
guess(half(x0)) → Chalf(guess(x0))
guess(rd(x0)) → Crd(guess(x0))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(g(x0)) → moveleft(Bg(wait(x0)))
guess(j(x0)) → moveleft(Bj(wait(x0)))
guess(half(x0)) → moveleft(Bhalf(wait(x0)))
guess(rd(x0)) → moveleft(Brd(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cf(moveleft(Bp(x0))) → moveleft(Bp(Af(x0)))
Cg(moveleft(Bp(x0))) → moveleft(Bp(Ag(x0)))
Cj(moveleft(Bp(x0))) → moveleft(Bp(Aj(x0)))
Chalf(moveleft(Bp(x0))) → moveleft(Bp(Ahalf(x0)))
Crd(moveleft(Bp(x0))) → moveleft(Bp(Ard(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cf(moveleft(B0(x0))) → moveleft(B0(Af(x0)))
Cg(moveleft(B0(x0))) → moveleft(B0(Ag(x0)))
Cj(moveleft(B0(x0))) → moveleft(B0(Aj(x0)))
Chalf(moveleft(B0(x0))) → moveleft(B0(Ahalf(x0)))
Crd(moveleft(B0(x0))) → moveleft(B0(Ard(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
Cg(moveleft(Bs(x0))) → moveleft(Bs(Ag(x0)))
Cj(moveleft(Bs(x0))) → moveleft(Bs(Aj(x0)))
Chalf(moveleft(Bs(x0))) → moveleft(Bs(Ahalf(x0)))
Crd(moveleft(Bs(x0))) → moveleft(Bs(Ard(x0)))
Cp(moveleft(Bf(x0))) → moveleft(Bf(Ap(x0)))
C0(moveleft(Bf(x0))) → moveleft(Bf(A0(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
Cg(moveleft(Bf(x0))) → moveleft(Bf(Ag(x0)))
Cj(moveleft(Bf(x0))) → moveleft(Bf(Aj(x0)))
Chalf(moveleft(Bf(x0))) → moveleft(Bf(Ahalf(x0)))
Crd(moveleft(Bf(x0))) → moveleft(Bf(Ard(x0)))
Cp(moveleft(Bg(x0))) → moveleft(Bg(Ap(x0)))
C0(moveleft(Bg(x0))) → moveleft(Bg(A0(x0)))
Cs(moveleft(Bg(x0))) → moveleft(Bg(As(x0)))
Cf(moveleft(Bg(x0))) → moveleft(Bg(Af(x0)))
Cg(moveleft(Bg(x0))) → moveleft(Bg(Ag(x0)))
Cj(moveleft(Bg(x0))) → moveleft(Bg(Aj(x0)))
Chalf(moveleft(Bg(x0))) → moveleft(Bg(Ahalf(x0)))
Crd(moveleft(Bg(x0))) → moveleft(Bg(Ard(x0)))
Cp(moveleft(Bj(x0))) → moveleft(Bj(Ap(x0)))
C0(moveleft(Bj(x0))) → moveleft(Bj(A0(x0)))
Cs(moveleft(Bj(x0))) → moveleft(Bj(As(x0)))
Cf(moveleft(Bj(x0))) → moveleft(Bj(Af(x0)))
Cg(moveleft(Bj(x0))) → moveleft(Bj(Ag(x0)))
Cj(moveleft(Bj(x0))) → moveleft(Bj(Aj(x0)))
Chalf(moveleft(Bj(x0))) → moveleft(Bj(Ahalf(x0)))
Crd(moveleft(Bj(x0))) → moveleft(Bj(Ard(x0)))
Cp(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ap(x0)))
C0(moveleft(Bhalf(x0))) → moveleft(Bhalf(A0(x0)))
Cs(moveleft(Bhalf(x0))) → moveleft(Bhalf(As(x0)))
Cf(moveleft(Bhalf(x0))) → moveleft(Bhalf(Af(x0)))
Cg(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ag(x0)))
Cj(moveleft(Bhalf(x0))) → moveleft(Bhalf(Aj(x0)))
Chalf(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ahalf(x0)))
Crd(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ard(x0)))
Cp(moveleft(Brd(x0))) → moveleft(Brd(Ap(x0)))
C0(moveleft(Brd(x0))) → moveleft(Brd(A0(x0)))
Cs(moveleft(Brd(x0))) → moveleft(Brd(As(x0)))
Cf(moveleft(Brd(x0))) → moveleft(Brd(Af(x0)))
Cg(moveleft(Brd(x0))) → moveleft(Brd(Ag(x0)))
Cj(moveleft(Brd(x0))) → moveleft(Brd(Aj(x0)))
Chalf(moveleft(Brd(x0))) → moveleft(Brd(Ahalf(x0)))
Crd(moveleft(Brd(x0))) → moveleft(Brd(Ard(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(Bg(x0))) → Dg(cut(goright(x0)))
cut(moveleft(Bj(x0))) → Dj(cut(goright(x0)))
cut(moveleft(Bhalf(x0))) → Dhalf(cut(goright(x0)))
cut(moveleft(Brd(x0))) → Drd(cut(goright(x0)))
goright(Ap(x0)) → Cp(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Af(x0)) → Cf(goright(x0))
goright(Ag(x0)) → Cg(goright(x0))
goright(Aj(x0)) → Cj(goright(x0))
goright(Ahalf(x0)) → Chalf(goright(x0))
goright(Ard(x0)) → Crd(goright(x0))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(g(x0))) → moveleft(Bg(wait(x0)))
goright(wait(j(x0))) → moveleft(Bj(wait(x0)))
goright(wait(half(x0))) → moveleft(Bhalf(wait(x0)))
goright(wait(rd(x0))) → moveleft(Brd(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cp(finish(x0)) → finish(p(x0))
C0(finish(x0)) → finish(0(x0))
Cs(finish(x0)) → finish(s(x0))
Cf(finish(x0)) → finish(f(x0))
Cg(finish(x0)) → finish(g(x0))
Cj(finish(x0)) → finish(j(x0))
Chalf(finish(x0)) → finish(half(x0))
Crd(finish(x0)) → finish(rd(x0))
cut(finish(x0)) → finish2(x0)
Dp(finish2(x0)) → finish2(p(x0))
D0(finish2(x0)) → finish2(0(x0))
Ds(finish2(x0)) → finish2(s(x0))
Df(finish2(x0)) → finish2(f(x0))
Dg(finish2(x0)) → finish2(g(x0))
Dj(finish2(x0)) → finish2(j(x0))
Dhalf(finish2(x0)) → finish2(half(x0))
Drd(finish2(x0)) → finish2(rd(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(p(0(x0))) → begin(0(s(s(p(x0)))))
rewrite(p(s(x0))) → begin(x0)
rewrite(p(p(s(x0)))) → begin(p(x0))
rewrite(f(s(x0))) → begin(p(s(g(p(s(s(x0)))))))
rewrite(g(s(x0))) → begin(p(p(s(s(s(j(s(p(s(p(s(x0))))))))))))
rewrite(j(s(x0))) → begin(p(s(s(p(s(f(p(s(p(p(s(x0))))))))))))
rewrite(half(0(x0))) → begin(0(s(s(half(p(s(p(s(x0)))))))))
rewrite(half(s(s(x0)))) → begin(s(half(p(p(s(s(x0)))))))
rewrite(rd(0(x0))) → begin(0(s(0(0(0(0(s(0(rd(x0))))))))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(f(x0)) → rotate(cut(Cf(guess(x0))))
begin(g(x0)) → rotate(cut(Cg(guess(x0))))
begin(j(x0)) → rotate(cut(Cj(guess(x0))))
begin(half(x0)) → rotate(cut(Chalf(guess(x0))))
begin(rd(x0)) → rotate(cut(Crd(guess(x0))))
guess(p(x0)) → Cp(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(f(x0)) → Cf(guess(x0))
guess(g(x0)) → Cg(guess(x0))
guess(j(x0)) → Cj(guess(x0))
guess(half(x0)) → Chalf(guess(x0))
guess(rd(x0)) → Crd(guess(x0))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(g(x0)) → moveleft(Bg(wait(x0)))
guess(j(x0)) → moveleft(Bj(wait(x0)))
guess(half(x0)) → moveleft(Bhalf(wait(x0)))
guess(rd(x0)) → moveleft(Brd(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cf(moveleft(Bp(x0))) → moveleft(Bp(Af(x0)))
Cg(moveleft(Bp(x0))) → moveleft(Bp(Ag(x0)))
Cj(moveleft(Bp(x0))) → moveleft(Bp(Aj(x0)))
Chalf(moveleft(Bp(x0))) → moveleft(Bp(Ahalf(x0)))
Crd(moveleft(Bp(x0))) → moveleft(Bp(Ard(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cf(moveleft(B0(x0))) → moveleft(B0(Af(x0)))
Cg(moveleft(B0(x0))) → moveleft(B0(Ag(x0)))
Cj(moveleft(B0(x0))) → moveleft(B0(Aj(x0)))
Chalf(moveleft(B0(x0))) → moveleft(B0(Ahalf(x0)))
Crd(moveleft(B0(x0))) → moveleft(B0(Ard(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
Cg(moveleft(Bs(x0))) → moveleft(Bs(Ag(x0)))
Cj(moveleft(Bs(x0))) → moveleft(Bs(Aj(x0)))
Chalf(moveleft(Bs(x0))) → moveleft(Bs(Ahalf(x0)))
Crd(moveleft(Bs(x0))) → moveleft(Bs(Ard(x0)))
Cp(moveleft(Bf(x0))) → moveleft(Bf(Ap(x0)))
C0(moveleft(Bf(x0))) → moveleft(Bf(A0(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
Cg(moveleft(Bf(x0))) → moveleft(Bf(Ag(x0)))
Cj(moveleft(Bf(x0))) → moveleft(Bf(Aj(x0)))
Chalf(moveleft(Bf(x0))) → moveleft(Bf(Ahalf(x0)))
Crd(moveleft(Bf(x0))) → moveleft(Bf(Ard(x0)))
Cp(moveleft(Bg(x0))) → moveleft(Bg(Ap(x0)))
C0(moveleft(Bg(x0))) → moveleft(Bg(A0(x0)))
Cs(moveleft(Bg(x0))) → moveleft(Bg(As(x0)))
Cf(moveleft(Bg(x0))) → moveleft(Bg(Af(x0)))
Cg(moveleft(Bg(x0))) → moveleft(Bg(Ag(x0)))
Cj(moveleft(Bg(x0))) → moveleft(Bg(Aj(x0)))
Chalf(moveleft(Bg(x0))) → moveleft(Bg(Ahalf(x0)))
Crd(moveleft(Bg(x0))) → moveleft(Bg(Ard(x0)))
Cp(moveleft(Bj(x0))) → moveleft(Bj(Ap(x0)))
C0(moveleft(Bj(x0))) → moveleft(Bj(A0(x0)))
Cs(moveleft(Bj(x0))) → moveleft(Bj(As(x0)))
Cf(moveleft(Bj(x0))) → moveleft(Bj(Af(x0)))
Cg(moveleft(Bj(x0))) → moveleft(Bj(Ag(x0)))
Cj(moveleft(Bj(x0))) → moveleft(Bj(Aj(x0)))
Chalf(moveleft(Bj(x0))) → moveleft(Bj(Ahalf(x0)))
Crd(moveleft(Bj(x0))) → moveleft(Bj(Ard(x0)))
Cp(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ap(x0)))
C0(moveleft(Bhalf(x0))) → moveleft(Bhalf(A0(x0)))
Cs(moveleft(Bhalf(x0))) → moveleft(Bhalf(As(x0)))
Cf(moveleft(Bhalf(x0))) → moveleft(Bhalf(Af(x0)))
Cg(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ag(x0)))
Cj(moveleft(Bhalf(x0))) → moveleft(Bhalf(Aj(x0)))
Chalf(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ahalf(x0)))
Crd(moveleft(Bhalf(x0))) → moveleft(Bhalf(Ard(x0)))
Cp(moveleft(Brd(x0))) → moveleft(Brd(Ap(x0)))
C0(moveleft(Brd(x0))) → moveleft(Brd(A0(x0)))
Cs(moveleft(Brd(x0))) → moveleft(Brd(As(x0)))
Cf(moveleft(Brd(x0))) → moveleft(Brd(Af(x0)))
Cg(moveleft(Brd(x0))) → moveleft(Brd(Ag(x0)))
Cj(moveleft(Brd(x0))) → moveleft(Brd(Aj(x0)))
Chalf(moveleft(Brd(x0))) → moveleft(Brd(Ahalf(x0)))
Crd(moveleft(Brd(x0))) → moveleft(Brd(Ard(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(Bg(x0))) → Dg(cut(goright(x0)))
cut(moveleft(Bj(x0))) → Dj(cut(goright(x0)))
cut(moveleft(Bhalf(x0))) → Dhalf(cut(goright(x0)))
cut(moveleft(Brd(x0))) → Drd(cut(goright(x0)))
goright(Ap(x0)) → Cp(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Af(x0)) → Cf(goright(x0))
goright(Ag(x0)) → Cg(goright(x0))
goright(Aj(x0)) → Cj(goright(x0))
goright(Ahalf(x0)) → Chalf(goright(x0))
goright(Ard(x0)) → Crd(goright(x0))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(g(x0))) → moveleft(Bg(wait(x0)))
goright(wait(j(x0))) → moveleft(Bj(wait(x0)))
goright(wait(half(x0))) → moveleft(Bhalf(wait(x0)))
goright(wait(rd(x0))) → moveleft(Brd(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cp(finish(x0)) → finish(p(x0))
C0(finish(x0)) → finish(0(x0))
Cs(finish(x0)) → finish(s(x0))
Cf(finish(x0)) → finish(f(x0))
Cg(finish(x0)) → finish(g(x0))
Cj(finish(x0)) → finish(j(x0))
Chalf(finish(x0)) → finish(half(x0))
Crd(finish(x0)) → finish(rd(x0))
cut(finish(x0)) → finish2(x0)
Dp(finish2(x0)) → finish2(p(x0))
D0(finish2(x0)) → finish2(0(x0))
Ds(finish2(x0)) → finish2(s(x0))
Df(finish2(x0)) → finish2(f(x0))
Dg(finish2(x0)) → finish2(g(x0))
Dj(finish2(x0)) → finish2(j(x0))
Dhalf(finish2(x0)) → finish2(half(x0))
Drd(finish2(x0)) → finish2(rd(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(p(0(x0))) → begin(0(s(s(p(x0)))))
rewrite(p(s(x0))) → begin(x0)
rewrite(p(p(s(x0)))) → begin(p(x0))
rewrite(f(s(x0))) → begin(p(s(g(p(s(s(x0)))))))
rewrite(g(s(x0))) → begin(p(p(s(s(s(j(s(p(s(p(s(x0))))))))))))
rewrite(j(s(x0))) → begin(p(s(s(p(s(f(p(s(p(p(s(x0))))))))))))
rewrite(half(0(x0))) → begin(0(s(s(half(p(s(p(s(x0)))))))))
rewrite(half(s(s(x0)))) → begin(s(half(p(p(s(s(x0)))))))
rewrite(rd(0(x0))) → begin(0(s(0(0(0(0(s(0(rd(x0))))))))))