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(v(x0)) → rotate(cut(Cv(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(w(x0)) → rotate(cut(Cw(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
guess(v(x0)) → Cv(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(w(x0)) → Cw(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(v(x0)) → moveleft(Bv(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(w(x0)) → moveleft(Bw(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cv(moveleft(Bv(x0))) → moveleft(Bv(Av(x0)))
Cs(moveleft(Bv(x0))) → moveleft(Bv(As(x0)))
Cp(moveleft(Bv(x0))) → moveleft(Bv(Ap(x0)))
Cw(moveleft(Bv(x0))) → moveleft(Bv(Aw(x0)))
C0(moveleft(Bv(x0))) → moveleft(Bv(A0(x0)))
Cv(moveleft(Bs(x0))) → moveleft(Bs(Av(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
Cw(moveleft(Bs(x0))) → moveleft(Bs(Aw(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cv(moveleft(Bp(x0))) → moveleft(Bp(Av(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
Cw(moveleft(Bp(x0))) → moveleft(Bp(Aw(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cv(moveleft(Bw(x0))) → moveleft(Bw(Av(x0)))
Cs(moveleft(Bw(x0))) → moveleft(Bw(As(x0)))
Cp(moveleft(Bw(x0))) → moveleft(Bw(Ap(x0)))
Cw(moveleft(Bw(x0))) → moveleft(Bw(Aw(x0)))
C0(moveleft(Bw(x0))) → moveleft(Bw(A0(x0)))
Cv(moveleft(B0(x0))) → moveleft(B0(Av(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
Cw(moveleft(B0(x0))) → moveleft(B0(Aw(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
cut(moveleft(Bv(x0))) → Dv(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(Bw(x0))) → Dw(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
goright(Av(x0)) → Cv(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(Aw(x0)) → Cw(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(wait(v(x0))) → moveleft(Bv(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(w(x0))) → moveleft(Bw(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cv(finish(x0)) → finish(v(x0))
Cs(finish(x0)) → finish(s(x0))
Cp(finish(x0)) → finish(p(x0))
Cw(finish(x0)) → finish(w(x0))
C0(finish(x0)) → finish(0(x0))
cut(finish(x0)) → finish2(x0)
Dv(finish2(x0)) → finish2(v(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dp(finish2(x0)) → finish2(p(x0))
Dw(finish2(x0)) → finish2(w(x0))
D0(finish2(x0)) → finish2(0(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(v(s(x0))) → begin(s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x0)))))))))))))))))))))
rewrite(v(0(x0))) → begin(p(p(s(s(0(p(p(s(s(s(s(s(x0)))))))))))))
rewrite(w(s(x0))) → begin(s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x0)))))))))))))))))))))
rewrite(w(0(x0))) → begin(p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(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(s(s(s(p(s(x0)))))))))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(v(x0)) → rotate(cut(Cv(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(w(x0)) → rotate(cut(Cw(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
guess(v(x0)) → Cv(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(w(x0)) → Cw(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(v(x0)) → moveleft(Bv(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(w(x0)) → moveleft(Bw(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cv(moveleft(Bv(x0))) → moveleft(Bv(Av(x0)))
Cs(moveleft(Bv(x0))) → moveleft(Bv(As(x0)))
Cp(moveleft(Bv(x0))) → moveleft(Bv(Ap(x0)))
Cw(moveleft(Bv(x0))) → moveleft(Bv(Aw(x0)))
C0(moveleft(Bv(x0))) → moveleft(Bv(A0(x0)))
Cv(moveleft(Bs(x0))) → moveleft(Bs(Av(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
Cw(moveleft(Bs(x0))) → moveleft(Bs(Aw(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cv(moveleft(Bp(x0))) → moveleft(Bp(Av(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
Cw(moveleft(Bp(x0))) → moveleft(Bp(Aw(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cv(moveleft(Bw(x0))) → moveleft(Bw(Av(x0)))
Cs(moveleft(Bw(x0))) → moveleft(Bw(As(x0)))
Cp(moveleft(Bw(x0))) → moveleft(Bw(Ap(x0)))
Cw(moveleft(Bw(x0))) → moveleft(Bw(Aw(x0)))
C0(moveleft(Bw(x0))) → moveleft(Bw(A0(x0)))
Cv(moveleft(B0(x0))) → moveleft(B0(Av(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
Cw(moveleft(B0(x0))) → moveleft(B0(Aw(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
cut(moveleft(Bv(x0))) → Dv(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(Bw(x0))) → Dw(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
goright(Av(x0)) → Cv(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(Aw(x0)) → Cw(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(wait(v(x0))) → moveleft(Bv(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(w(x0))) → moveleft(Bw(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cv(finish(x0)) → finish(v(x0))
Cs(finish(x0)) → finish(s(x0))
Cp(finish(x0)) → finish(p(x0))
Cw(finish(x0)) → finish(w(x0))
C0(finish(x0)) → finish(0(x0))
cut(finish(x0)) → finish2(x0)
Dv(finish2(x0)) → finish2(v(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dp(finish2(x0)) → finish2(p(x0))
Dw(finish2(x0)) → finish2(w(x0))
D0(finish2(x0)) → finish2(0(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(v(s(x0))) → begin(s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x0)))))))))))))))))))))
rewrite(v(0(x0))) → begin(p(p(s(s(0(p(p(s(s(s(s(s(x0)))))))))))))
rewrite(w(s(x0))) → begin(s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x0)))))))))))))))))))))
rewrite(w(0(x0))) → begin(p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(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(s(s(s(p(s(x0)))))))))))