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(q(x0)) → rotate(cut(Cq(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(r(x0)) → rotate(cut(Cr(guess(x0))))
guess(q(x0)) → Cq(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(r(x0)) → Cr(guess(x0))
guess(q(x0)) → moveleft(Bq(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(r(x0)) → moveleft(Br(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cq(moveleft(Bq(x0))) → moveleft(Bq(Aq(x0)))
C0(moveleft(Bq(x0))) → moveleft(Bq(A0(x0)))
Cp(moveleft(Bq(x0))) → moveleft(Bq(Ap(x0)))
Cs(moveleft(Bq(x0))) → moveleft(Bq(As(x0)))
Cr(moveleft(Bq(x0))) → moveleft(Bq(Ar(x0)))
Cq(moveleft(B0(x0))) → moveleft(B0(Aq(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)))
Cr(moveleft(B0(x0))) → moveleft(B0(Ar(x0)))
Cq(moveleft(Bp(x0))) → moveleft(Bp(Aq(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)))
Cr(moveleft(Bp(x0))) → moveleft(Bp(Ar(x0)))
Cq(moveleft(Bs(x0))) → moveleft(Bs(Aq(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)))
Cr(moveleft(Bs(x0))) → moveleft(Bs(Ar(x0)))
Cq(moveleft(Br(x0))) → moveleft(Br(Aq(x0)))
C0(moveleft(Br(x0))) → moveleft(Br(A0(x0)))
Cp(moveleft(Br(x0))) → moveleft(Br(Ap(x0)))
Cs(moveleft(Br(x0))) → moveleft(Br(As(x0)))
Cr(moveleft(Br(x0))) → moveleft(Br(Ar(x0)))
cut(moveleft(Bq(x0))) → Dq(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(Br(x0))) → Dr(cut(goright(x0)))
goright(Aq(x0)) → Cq(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ar(x0)) → Cr(goright(x0))
goright(wait(q(x0))) → moveleft(Bq(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(r(x0))) → moveleft(Br(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cq(finish(x0)) → finish(q(x0))
C0(finish(x0)) → finish(0(x0))
Cp(finish(x0)) → finish(p(x0))
Cs(finish(x0)) → finish(s(x0))
Cr(finish(x0)) → finish(r(x0))
cut(finish(x0)) → finish2(x0)
Dq(finish2(x0)) → finish2(q(x0))
D0(finish2(x0)) → finish2(0(x0))
Dp(finish2(x0)) → finish2(p(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dr(finish2(x0)) → finish2(r(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(q(0(x0))) → begin(p(p(s(s(0(s(s(s(s(x0))))))))))
rewrite(q(s(x0))) → begin(p(p(s(s(s(s(s(s(r(p(p(s(s(x0))))))))))))))
rewrite(r(0(x0))) → begin(p(s(p(s(0(p(p(p(s(s(s(x0))))))))))))
rewrite(r(s(x0))) → begin(p(s(p(s(s(q(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(x0)))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(q(x0)) → rotate(cut(Cq(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(r(x0)) → rotate(cut(Cr(guess(x0))))
guess(q(x0)) → Cq(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(r(x0)) → Cr(guess(x0))
guess(q(x0)) → moveleft(Bq(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(r(x0)) → moveleft(Br(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cq(moveleft(Bq(x0))) → moveleft(Bq(Aq(x0)))
C0(moveleft(Bq(x0))) → moveleft(Bq(A0(x0)))
Cp(moveleft(Bq(x0))) → moveleft(Bq(Ap(x0)))
Cs(moveleft(Bq(x0))) → moveleft(Bq(As(x0)))
Cr(moveleft(Bq(x0))) → moveleft(Bq(Ar(x0)))
Cq(moveleft(B0(x0))) → moveleft(B0(Aq(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)))
Cr(moveleft(B0(x0))) → moveleft(B0(Ar(x0)))
Cq(moveleft(Bp(x0))) → moveleft(Bp(Aq(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)))
Cr(moveleft(Bp(x0))) → moveleft(Bp(Ar(x0)))
Cq(moveleft(Bs(x0))) → moveleft(Bs(Aq(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)))
Cr(moveleft(Bs(x0))) → moveleft(Bs(Ar(x0)))
Cq(moveleft(Br(x0))) → moveleft(Br(Aq(x0)))
C0(moveleft(Br(x0))) → moveleft(Br(A0(x0)))
Cp(moveleft(Br(x0))) → moveleft(Br(Ap(x0)))
Cs(moveleft(Br(x0))) → moveleft(Br(As(x0)))
Cr(moveleft(Br(x0))) → moveleft(Br(Ar(x0)))
cut(moveleft(Bq(x0))) → Dq(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(Br(x0))) → Dr(cut(goright(x0)))
goright(Aq(x0)) → Cq(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ar(x0)) → Cr(goright(x0))
goright(wait(q(x0))) → moveleft(Bq(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(r(x0))) → moveleft(Br(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cq(finish(x0)) → finish(q(x0))
C0(finish(x0)) → finish(0(x0))
Cp(finish(x0)) → finish(p(x0))
Cs(finish(x0)) → finish(s(x0))
Cr(finish(x0)) → finish(r(x0))
cut(finish(x0)) → finish2(x0)
Dq(finish2(x0)) → finish2(q(x0))
D0(finish2(x0)) → finish2(0(x0))
Dp(finish2(x0)) → finish2(p(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dr(finish2(x0)) → finish2(r(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(q(0(x0))) → begin(p(p(s(s(0(s(s(s(s(x0))))))))))
rewrite(q(s(x0))) → begin(p(p(s(s(s(s(s(s(r(p(p(s(s(x0))))))))))))))
rewrite(r(0(x0))) → begin(p(s(p(s(0(p(p(p(s(s(s(x0))))))))))))
rewrite(r(s(x0))) → begin(p(s(p(s(s(q(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(x0)))))