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(R(x0)) → rotate(cut(CR(guess(x0))))
begin(2(x0)) → rotate(cut(C2(guess(x0))))
begin(3(x0)) → rotate(cut(C3(guess(x0))))
begin(1(x0)) → rotate(cut(C1(guess(x0))))
begin(L(x0)) → rotate(cut(CL(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
guess(R(x0)) → CR(guess(x0))
guess(2(x0)) → C2(guess(x0))
guess(3(x0)) → C3(guess(x0))
guess(1(x0)) → C1(guess(x0))
guess(L(x0)) → CL(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(R(x0)) → moveleft(BR(wait(x0)))
guess(2(x0)) → moveleft(B2(wait(x0)))
guess(3(x0)) → moveleft(B3(wait(x0)))
guess(1(x0)) → moveleft(B1(wait(x0)))
guess(L(x0)) → moveleft(BL(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(end(x0)) → finish(end(x0))
CR(moveleft(BR(x0))) → moveleft(BR(AR(x0)))
C2(moveleft(BR(x0))) → moveleft(BR(A2(x0)))
C3(moveleft(BR(x0))) → moveleft(BR(A3(x0)))
C1(moveleft(BR(x0))) → moveleft(BR(A1(x0)))
CL(moveleft(BR(x0))) → moveleft(BR(AL(x0)))
C0(moveleft(BR(x0))) → moveleft(BR(A0(x0)))
Cb(moveleft(BR(x0))) → moveleft(BR(Ab(x0)))
Cc(moveleft(BR(x0))) → moveleft(BR(Ac(x0)))
CR(moveleft(B2(x0))) → moveleft(B2(AR(x0)))
C2(moveleft(B2(x0))) → moveleft(B2(A2(x0)))
C3(moveleft(B2(x0))) → moveleft(B2(A3(x0)))
C1(moveleft(B2(x0))) → moveleft(B2(A1(x0)))
CL(moveleft(B2(x0))) → moveleft(B2(AL(x0)))
C0(moveleft(B2(x0))) → moveleft(B2(A0(x0)))
Cb(moveleft(B2(x0))) → moveleft(B2(Ab(x0)))
Cc(moveleft(B2(x0))) → moveleft(B2(Ac(x0)))
CR(moveleft(B3(x0))) → moveleft(B3(AR(x0)))
C2(moveleft(B3(x0))) → moveleft(B3(A2(x0)))
C3(moveleft(B3(x0))) → moveleft(B3(A3(x0)))
C1(moveleft(B3(x0))) → moveleft(B3(A1(x0)))
CL(moveleft(B3(x0))) → moveleft(B3(AL(x0)))
C0(moveleft(B3(x0))) → moveleft(B3(A0(x0)))
Cb(moveleft(B3(x0))) → moveleft(B3(Ab(x0)))
Cc(moveleft(B3(x0))) → moveleft(B3(Ac(x0)))
CR(moveleft(B1(x0))) → moveleft(B1(AR(x0)))
C2(moveleft(B1(x0))) → moveleft(B1(A2(x0)))
C3(moveleft(B1(x0))) → moveleft(B1(A3(x0)))
C1(moveleft(B1(x0))) → moveleft(B1(A1(x0)))
CL(moveleft(B1(x0))) → moveleft(B1(AL(x0)))
C0(moveleft(B1(x0))) → moveleft(B1(A0(x0)))
Cb(moveleft(B1(x0))) → moveleft(B1(Ab(x0)))
Cc(moveleft(B1(x0))) → moveleft(B1(Ac(x0)))
CR(moveleft(BL(x0))) → moveleft(BL(AR(x0)))
C2(moveleft(BL(x0))) → moveleft(BL(A2(x0)))
C3(moveleft(BL(x0))) → moveleft(BL(A3(x0)))
C1(moveleft(BL(x0))) → moveleft(BL(A1(x0)))
CL(moveleft(BL(x0))) → moveleft(BL(AL(x0)))
C0(moveleft(BL(x0))) → moveleft(BL(A0(x0)))
Cb(moveleft(BL(x0))) → moveleft(BL(Ab(x0)))
Cc(moveleft(BL(x0))) → moveleft(BL(Ac(x0)))
CR(moveleft(B0(x0))) → moveleft(B0(AR(x0)))
C2(moveleft(B0(x0))) → moveleft(B0(A2(x0)))
C3(moveleft(B0(x0))) → moveleft(B0(A3(x0)))
C1(moveleft(B0(x0))) → moveleft(B0(A1(x0)))
CL(moveleft(B0(x0))) → moveleft(B0(AL(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cb(moveleft(B0(x0))) → moveleft(B0(Ab(x0)))
Cc(moveleft(B0(x0))) → moveleft(B0(Ac(x0)))
CR(moveleft(Bb(x0))) → moveleft(Bb(AR(x0)))
C2(moveleft(Bb(x0))) → moveleft(Bb(A2(x0)))
C3(moveleft(Bb(x0))) → moveleft(Bb(A3(x0)))
C1(moveleft(Bb(x0))) → moveleft(Bb(A1(x0)))
CL(moveleft(Bb(x0))) → moveleft(Bb(AL(x0)))
C0(moveleft(Bb(x0))) → moveleft(Bb(A0(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Cc(moveleft(Bb(x0))) → moveleft(Bb(Ac(x0)))
CR(moveleft(Bc(x0))) → moveleft(Bc(AR(x0)))
C2(moveleft(Bc(x0))) → moveleft(Bc(A2(x0)))
C3(moveleft(Bc(x0))) → moveleft(Bc(A3(x0)))
C1(moveleft(Bc(x0))) → moveleft(Bc(A1(x0)))
CL(moveleft(Bc(x0))) → moveleft(Bc(AL(x0)))
C0(moveleft(Bc(x0))) → moveleft(Bc(A0(x0)))
Cb(moveleft(Bc(x0))) → moveleft(Bc(Ab(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
cut(moveleft(BR(x0))) → DR(cut(goright(x0)))
cut(moveleft(B2(x0))) → D2(cut(goright(x0)))
cut(moveleft(B3(x0))) → D3(cut(goright(x0)))
cut(moveleft(B1(x0))) → D1(cut(goright(x0)))
cut(moveleft(BL(x0))) → DL(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
goright(AR(x0)) → CR(goright(x0))
goright(A2(x0)) → C2(goright(x0))
goright(A3(x0)) → C3(goright(x0))
goright(A1(x0)) → C1(goright(x0))
goright(AL(x0)) → CL(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(wait(R(x0))) → moveleft(BR(wait(x0)))
goright(wait(2(x0))) → moveleft(B2(wait(x0)))
goright(wait(3(x0))) → moveleft(B3(wait(x0)))
goright(wait(1(x0))) → moveleft(B1(wait(x0)))
goright(wait(L(x0))) → moveleft(BL(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
CR(finish(x0)) → finish(R(x0))
C2(finish(x0)) → finish(2(x0))
C3(finish(x0)) → finish(3(x0))
C1(finish(x0)) → finish(1(x0))
CL(finish(x0)) → finish(L(x0))
C0(finish(x0)) → finish(0(x0))
Cb(finish(x0)) → finish(b(x0))
Cc(finish(x0)) → finish(c(x0))
cut(finish(x0)) → finish2(x0)
DR(finish2(x0)) → finish2(R(x0))
D2(finish2(x0)) → finish2(2(x0))
D3(finish2(x0)) → finish2(3(x0))
D1(finish2(x0)) → finish2(1(x0))
DL(finish2(x0)) → finish2(L(x0))
D0(finish2(x0)) → finish2(0(x0))
Db(finish2(x0)) → finish2(b(x0))
Dc(finish2(x0)) → finish2(c(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(R(2(x0))) → begin(2(R(x0)))
rewrite(R(3(x0))) → begin(3(R(x0)))
rewrite(R(1(x0))) → begin(L(3(x0)))
rewrite(3(L(x0))) → begin(L(3(x0)))
rewrite(2(L(x0))) → begin(L(2(x0)))
rewrite(0(L(x0))) → begin(2(R(x0)))
rewrite(R(b(x0))) → begin(c(1(b(x0))))
rewrite(3(c(x0))) → begin(c(1(x0)))
rewrite(2(c(1(x0)))) → begin(c(0(R(1(x0)))))
rewrite(2(c(0(x0)))) → begin(c(0(0(x0))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(R(x0)) → rotate(cut(CR(guess(x0))))
begin(2(x0)) → rotate(cut(C2(guess(x0))))
begin(3(x0)) → rotate(cut(C3(guess(x0))))
begin(1(x0)) → rotate(cut(C1(guess(x0))))
begin(L(x0)) → rotate(cut(CL(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
guess(R(x0)) → CR(guess(x0))
guess(2(x0)) → C2(guess(x0))
guess(3(x0)) → C3(guess(x0))
guess(1(x0)) → C1(guess(x0))
guess(L(x0)) → CL(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(R(x0)) → moveleft(BR(wait(x0)))
guess(2(x0)) → moveleft(B2(wait(x0)))
guess(3(x0)) → moveleft(B3(wait(x0)))
guess(1(x0)) → moveleft(B1(wait(x0)))
guess(L(x0)) → moveleft(BL(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(end(x0)) → finish(end(x0))
CR(moveleft(BR(x0))) → moveleft(BR(AR(x0)))
C2(moveleft(BR(x0))) → moveleft(BR(A2(x0)))
C3(moveleft(BR(x0))) → moveleft(BR(A3(x0)))
C1(moveleft(BR(x0))) → moveleft(BR(A1(x0)))
CL(moveleft(BR(x0))) → moveleft(BR(AL(x0)))
C0(moveleft(BR(x0))) → moveleft(BR(A0(x0)))
Cb(moveleft(BR(x0))) → moveleft(BR(Ab(x0)))
Cc(moveleft(BR(x0))) → moveleft(BR(Ac(x0)))
CR(moveleft(B2(x0))) → moveleft(B2(AR(x0)))
C2(moveleft(B2(x0))) → moveleft(B2(A2(x0)))
C3(moveleft(B2(x0))) → moveleft(B2(A3(x0)))
C1(moveleft(B2(x0))) → moveleft(B2(A1(x0)))
CL(moveleft(B2(x0))) → moveleft(B2(AL(x0)))
C0(moveleft(B2(x0))) → moveleft(B2(A0(x0)))
Cb(moveleft(B2(x0))) → moveleft(B2(Ab(x0)))
Cc(moveleft(B2(x0))) → moveleft(B2(Ac(x0)))
CR(moveleft(B3(x0))) → moveleft(B3(AR(x0)))
C2(moveleft(B3(x0))) → moveleft(B3(A2(x0)))
C3(moveleft(B3(x0))) → moveleft(B3(A3(x0)))
C1(moveleft(B3(x0))) → moveleft(B3(A1(x0)))
CL(moveleft(B3(x0))) → moveleft(B3(AL(x0)))
C0(moveleft(B3(x0))) → moveleft(B3(A0(x0)))
Cb(moveleft(B3(x0))) → moveleft(B3(Ab(x0)))
Cc(moveleft(B3(x0))) → moveleft(B3(Ac(x0)))
CR(moveleft(B1(x0))) → moveleft(B1(AR(x0)))
C2(moveleft(B1(x0))) → moveleft(B1(A2(x0)))
C3(moveleft(B1(x0))) → moveleft(B1(A3(x0)))
C1(moveleft(B1(x0))) → moveleft(B1(A1(x0)))
CL(moveleft(B1(x0))) → moveleft(B1(AL(x0)))
C0(moveleft(B1(x0))) → moveleft(B1(A0(x0)))
Cb(moveleft(B1(x0))) → moveleft(B1(Ab(x0)))
Cc(moveleft(B1(x0))) → moveleft(B1(Ac(x0)))
CR(moveleft(BL(x0))) → moveleft(BL(AR(x0)))
C2(moveleft(BL(x0))) → moveleft(BL(A2(x0)))
C3(moveleft(BL(x0))) → moveleft(BL(A3(x0)))
C1(moveleft(BL(x0))) → moveleft(BL(A1(x0)))
CL(moveleft(BL(x0))) → moveleft(BL(AL(x0)))
C0(moveleft(BL(x0))) → moveleft(BL(A0(x0)))
Cb(moveleft(BL(x0))) → moveleft(BL(Ab(x0)))
Cc(moveleft(BL(x0))) → moveleft(BL(Ac(x0)))
CR(moveleft(B0(x0))) → moveleft(B0(AR(x0)))
C2(moveleft(B0(x0))) → moveleft(B0(A2(x0)))
C3(moveleft(B0(x0))) → moveleft(B0(A3(x0)))
C1(moveleft(B0(x0))) → moveleft(B0(A1(x0)))
CL(moveleft(B0(x0))) → moveleft(B0(AL(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cb(moveleft(B0(x0))) → moveleft(B0(Ab(x0)))
Cc(moveleft(B0(x0))) → moveleft(B0(Ac(x0)))
CR(moveleft(Bb(x0))) → moveleft(Bb(AR(x0)))
C2(moveleft(Bb(x0))) → moveleft(Bb(A2(x0)))
C3(moveleft(Bb(x0))) → moveleft(Bb(A3(x0)))
C1(moveleft(Bb(x0))) → moveleft(Bb(A1(x0)))
CL(moveleft(Bb(x0))) → moveleft(Bb(AL(x0)))
C0(moveleft(Bb(x0))) → moveleft(Bb(A0(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Cc(moveleft(Bb(x0))) → moveleft(Bb(Ac(x0)))
CR(moveleft(Bc(x0))) → moveleft(Bc(AR(x0)))
C2(moveleft(Bc(x0))) → moveleft(Bc(A2(x0)))
C3(moveleft(Bc(x0))) → moveleft(Bc(A3(x0)))
C1(moveleft(Bc(x0))) → moveleft(Bc(A1(x0)))
CL(moveleft(Bc(x0))) → moveleft(Bc(AL(x0)))
C0(moveleft(Bc(x0))) → moveleft(Bc(A0(x0)))
Cb(moveleft(Bc(x0))) → moveleft(Bc(Ab(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
cut(moveleft(BR(x0))) → DR(cut(goright(x0)))
cut(moveleft(B2(x0))) → D2(cut(goright(x0)))
cut(moveleft(B3(x0))) → D3(cut(goright(x0)))
cut(moveleft(B1(x0))) → D1(cut(goright(x0)))
cut(moveleft(BL(x0))) → DL(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
goright(AR(x0)) → CR(goright(x0))
goright(A2(x0)) → C2(goright(x0))
goright(A3(x0)) → C3(goright(x0))
goright(A1(x0)) → C1(goright(x0))
goright(AL(x0)) → CL(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(wait(R(x0))) → moveleft(BR(wait(x0)))
goright(wait(2(x0))) → moveleft(B2(wait(x0)))
goright(wait(3(x0))) → moveleft(B3(wait(x0)))
goright(wait(1(x0))) → moveleft(B1(wait(x0)))
goright(wait(L(x0))) → moveleft(BL(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
CR(finish(x0)) → finish(R(x0))
C2(finish(x0)) → finish(2(x0))
C3(finish(x0)) → finish(3(x0))
C1(finish(x0)) → finish(1(x0))
CL(finish(x0)) → finish(L(x0))
C0(finish(x0)) → finish(0(x0))
Cb(finish(x0)) → finish(b(x0))
Cc(finish(x0)) → finish(c(x0))
cut(finish(x0)) → finish2(x0)
DR(finish2(x0)) → finish2(R(x0))
D2(finish2(x0)) → finish2(2(x0))
D3(finish2(x0)) → finish2(3(x0))
D1(finish2(x0)) → finish2(1(x0))
DL(finish2(x0)) → finish2(L(x0))
D0(finish2(x0)) → finish2(0(x0))
Db(finish2(x0)) → finish2(b(x0))
Dc(finish2(x0)) → finish2(c(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(R(2(x0))) → begin(2(R(x0)))
rewrite(R(3(x0))) → begin(3(R(x0)))
rewrite(R(1(x0))) → begin(L(3(x0)))
rewrite(3(L(x0))) → begin(L(3(x0)))
rewrite(2(L(x0))) → begin(L(2(x0)))
rewrite(0(L(x0))) → begin(2(R(x0)))
rewrite(R(b(x0))) → begin(c(1(b(x0))))
rewrite(3(c(x0))) → begin(c(1(x0)))
rewrite(2(c(1(x0)))) → begin(c(0(R(1(x0)))))
rewrite(2(c(0(x0)))) → begin(c(0(0(x0))))