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(i(x0)) → rotate(cut(Ci(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(j(x0)) → rotate(cut(Cj(guess(x0))))
guess(i(x0)) → Ci(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(j(x0)) → Cj(guess(x0))
guess(i(x0)) → moveleft(Bi(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(j(x0)) → moveleft(Bj(wait(x0)))
guess(end(x0)) → finish(end(x0))
Ci(moveleft(Bi(x0))) → moveleft(Bi(Ai(x0)))
C0(moveleft(Bi(x0))) → moveleft(Bi(A0(x0)))
Cp(moveleft(Bi(x0))) → moveleft(Bi(Ap(x0)))
Cs(moveleft(Bi(x0))) → moveleft(Bi(As(x0)))
Cj(moveleft(Bi(x0))) → moveleft(Bi(Aj(x0)))
Ci(moveleft(B0(x0))) → moveleft(B0(Ai(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)))
Cj(moveleft(B0(x0))) → moveleft(B0(Aj(x0)))
Ci(moveleft(Bp(x0))) → moveleft(Bp(Ai(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)))
Cj(moveleft(Bp(x0))) → moveleft(Bp(Aj(x0)))
Ci(moveleft(Bs(x0))) → moveleft(Bs(Ai(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)))
Cj(moveleft(Bs(x0))) → moveleft(Bs(Aj(x0)))
Ci(moveleft(Bj(x0))) → moveleft(Bj(Ai(x0)))
C0(moveleft(Bj(x0))) → moveleft(Bj(A0(x0)))
Cp(moveleft(Bj(x0))) → moveleft(Bj(Ap(x0)))
Cs(moveleft(Bj(x0))) → moveleft(Bj(As(x0)))
Cj(moveleft(Bj(x0))) → moveleft(Bj(Aj(x0)))
cut(moveleft(Bi(x0))) → Di(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(Bj(x0))) → Dj(cut(goright(x0)))
goright(Ai(x0)) → Ci(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Aj(x0)) → Cj(goright(x0))
goright(wait(i(x0))) → moveleft(Bi(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(j(x0))) → moveleft(Bj(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Ci(finish(x0)) → finish(i(x0))
C0(finish(x0)) → finish(0(x0))
Cp(finish(x0)) → finish(p(x0))
Cs(finish(x0)) → finish(s(x0))
Cj(finish(x0)) → finish(j(x0))
cut(finish(x0)) → finish2(x0)
Di(finish2(x0)) → finish2(i(x0))
D0(finish2(x0)) → finish2(0(x0))
Dp(finish2(x0)) → finish2(p(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dj(finish2(x0)) → finish2(j(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(i(0(x0))) → begin(p(s(p(s(0(p(s(p(s(x0))))))))))
rewrite(i(s(x0))) → begin(p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x0)))))))))))))))))))
rewrite(j(0(x0))) → begin(p(s(p(p(s(s(0(p(s(p(s(x0))))))))))))
rewrite(j(s(x0))) → begin(s(s(s(s(p(p(s(s(i(p(s(p(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(s(s(s(s(x0))))))))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(i(x0)) → rotate(cut(Ci(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(j(x0)) → rotate(cut(Cj(guess(x0))))
guess(i(x0)) → Ci(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(j(x0)) → Cj(guess(x0))
guess(i(x0)) → moveleft(Bi(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(j(x0)) → moveleft(Bj(wait(x0)))
guess(end(x0)) → finish(end(x0))
Ci(moveleft(Bi(x0))) → moveleft(Bi(Ai(x0)))
C0(moveleft(Bi(x0))) → moveleft(Bi(A0(x0)))
Cp(moveleft(Bi(x0))) → moveleft(Bi(Ap(x0)))
Cs(moveleft(Bi(x0))) → moveleft(Bi(As(x0)))
Cj(moveleft(Bi(x0))) → moveleft(Bi(Aj(x0)))
Ci(moveleft(B0(x0))) → moveleft(B0(Ai(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)))
Cj(moveleft(B0(x0))) → moveleft(B0(Aj(x0)))
Ci(moveleft(Bp(x0))) → moveleft(Bp(Ai(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)))
Cj(moveleft(Bp(x0))) → moveleft(Bp(Aj(x0)))
Ci(moveleft(Bs(x0))) → moveleft(Bs(Ai(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)))
Cj(moveleft(Bs(x0))) → moveleft(Bs(Aj(x0)))
Ci(moveleft(Bj(x0))) → moveleft(Bj(Ai(x0)))
C0(moveleft(Bj(x0))) → moveleft(Bj(A0(x0)))
Cp(moveleft(Bj(x0))) → moveleft(Bj(Ap(x0)))
Cs(moveleft(Bj(x0))) → moveleft(Bj(As(x0)))
Cj(moveleft(Bj(x0))) → moveleft(Bj(Aj(x0)))
cut(moveleft(Bi(x0))) → Di(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(Bj(x0))) → Dj(cut(goright(x0)))
goright(Ai(x0)) → Ci(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Aj(x0)) → Cj(goright(x0))
goright(wait(i(x0))) → moveleft(Bi(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(j(x0))) → moveleft(Bj(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Ci(finish(x0)) → finish(i(x0))
C0(finish(x0)) → finish(0(x0))
Cp(finish(x0)) → finish(p(x0))
Cs(finish(x0)) → finish(s(x0))
Cj(finish(x0)) → finish(j(x0))
cut(finish(x0)) → finish2(x0)
Di(finish2(x0)) → finish2(i(x0))
D0(finish2(x0)) → finish2(0(x0))
Dp(finish2(x0)) → finish2(p(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dj(finish2(x0)) → finish2(j(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(i(0(x0))) → begin(p(s(p(s(0(p(s(p(s(x0))))))))))
rewrite(i(s(x0))) → begin(p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x0)))))))))))))))))))
rewrite(j(0(x0))) → begin(p(s(p(p(s(s(0(p(s(p(s(x0))))))))))))
rewrite(j(s(x0))) → begin(s(s(s(s(p(p(s(s(i(p(s(p(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(s(s(s(s(x0))))))))))