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(sq(x0)) → rotate(cut(Csq(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(twice(x0)) → rotate(cut(Ctwice(guess(x0))))
guess(sq(x0)) → Csq(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(twice(x0)) → Ctwice(guess(x0))
guess(sq(x0)) → moveleft(Bsq(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(twice(x0)) → moveleft(Btwice(wait(x0)))
guess(end(x0)) → finish(end(x0))
Csq(moveleft(Bsq(x0))) → moveleft(Bsq(Asq(x0)))
C0(moveleft(Bsq(x0))) → moveleft(Bsq(A0(x0)))
Cp(moveleft(Bsq(x0))) → moveleft(Bsq(Ap(x0)))
Cs(moveleft(Bsq(x0))) → moveleft(Bsq(As(x0)))
Ctwice(moveleft(Bsq(x0))) → moveleft(Bsq(Atwice(x0)))
Csq(moveleft(B0(x0))) → moveleft(B0(Asq(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)))
Ctwice(moveleft(B0(x0))) → moveleft(B0(Atwice(x0)))
Csq(moveleft(Bp(x0))) → moveleft(Bp(Asq(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)))
Ctwice(moveleft(Bp(x0))) → moveleft(Bp(Atwice(x0)))
Csq(moveleft(Bs(x0))) → moveleft(Bs(Asq(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)))
Ctwice(moveleft(Bs(x0))) → moveleft(Bs(Atwice(x0)))
Csq(moveleft(Btwice(x0))) → moveleft(Btwice(Asq(x0)))
C0(moveleft(Btwice(x0))) → moveleft(Btwice(A0(x0)))
Cp(moveleft(Btwice(x0))) → moveleft(Btwice(Ap(x0)))
Cs(moveleft(Btwice(x0))) → moveleft(Btwice(As(x0)))
Ctwice(moveleft(Btwice(x0))) → moveleft(Btwice(Atwice(x0)))
cut(moveleft(Bsq(x0))) → Dsq(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(Btwice(x0))) → Dtwice(cut(goright(x0)))
goright(Asq(x0)) → Csq(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Atwice(x0)) → Ctwice(goright(x0))
goright(wait(sq(x0))) → moveleft(Bsq(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(twice(x0))) → moveleft(Btwice(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Csq(finish(x0)) → finish(sq(x0))
C0(finish(x0)) → finish(0(x0))
Cp(finish(x0)) → finish(p(x0))
Cs(finish(x0)) → finish(s(x0))
Ctwice(finish(x0)) → finish(twice(x0))
cut(finish(x0)) → finish2(x0)
Dsq(finish2(x0)) → finish2(sq(x0))
D0(finish2(x0)) → finish2(0(x0))
Dp(finish2(x0)) → finish2(p(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dtwice(finish2(x0)) → finish2(twice(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(sq(0(x0))) → begin(p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x0))))))))))))))))))
rewrite(sq(s(x0))) → begin(s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x0))))))))))))))))))))))))))))))))))
rewrite(twice(0(x0))) → begin(p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x0))))))))))))))))))))))))))))
rewrite(twice(s(x0))) → begin(p(p(s(s(s(p(p(s(s(s(twice(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(s(s(s(x0)))))))))))))
rewrite(0(x0)) → begin(x0)

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(sq(x0)) → rotate(cut(Csq(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(twice(x0)) → rotate(cut(Ctwice(guess(x0))))
guess(sq(x0)) → Csq(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(twice(x0)) → Ctwice(guess(x0))
guess(sq(x0)) → moveleft(Bsq(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(twice(x0)) → moveleft(Btwice(wait(x0)))
guess(end(x0)) → finish(end(x0))
Csq(moveleft(Bsq(x0))) → moveleft(Bsq(Asq(x0)))
C0(moveleft(Bsq(x0))) → moveleft(Bsq(A0(x0)))
Cp(moveleft(Bsq(x0))) → moveleft(Bsq(Ap(x0)))
Cs(moveleft(Bsq(x0))) → moveleft(Bsq(As(x0)))
Ctwice(moveleft(Bsq(x0))) → moveleft(Bsq(Atwice(x0)))
Csq(moveleft(B0(x0))) → moveleft(B0(Asq(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)))
Ctwice(moveleft(B0(x0))) → moveleft(B0(Atwice(x0)))
Csq(moveleft(Bp(x0))) → moveleft(Bp(Asq(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)))
Ctwice(moveleft(Bp(x0))) → moveleft(Bp(Atwice(x0)))
Csq(moveleft(Bs(x0))) → moveleft(Bs(Asq(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)))
Ctwice(moveleft(Bs(x0))) → moveleft(Bs(Atwice(x0)))
Csq(moveleft(Btwice(x0))) → moveleft(Btwice(Asq(x0)))
C0(moveleft(Btwice(x0))) → moveleft(Btwice(A0(x0)))
Cp(moveleft(Btwice(x0))) → moveleft(Btwice(Ap(x0)))
Cs(moveleft(Btwice(x0))) → moveleft(Btwice(As(x0)))
Ctwice(moveleft(Btwice(x0))) → moveleft(Btwice(Atwice(x0)))
cut(moveleft(Bsq(x0))) → Dsq(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(Btwice(x0))) → Dtwice(cut(goright(x0)))
goright(Asq(x0)) → Csq(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Atwice(x0)) → Ctwice(goright(x0))
goright(wait(sq(x0))) → moveleft(Bsq(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(twice(x0))) → moveleft(Btwice(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Csq(finish(x0)) → finish(sq(x0))
C0(finish(x0)) → finish(0(x0))
Cp(finish(x0)) → finish(p(x0))
Cs(finish(x0)) → finish(s(x0))
Ctwice(finish(x0)) → finish(twice(x0))
cut(finish(x0)) → finish2(x0)
Dsq(finish2(x0)) → finish2(sq(x0))
D0(finish2(x0)) → finish2(0(x0))
Dp(finish2(x0)) → finish2(p(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dtwice(finish2(x0)) → finish2(twice(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(sq(0(x0))) → begin(p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x0))))))))))))))))))
rewrite(sq(s(x0))) → begin(s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x0))))))))))))))))))))))))))))))))))
rewrite(twice(0(x0))) → begin(p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x0))))))))))))))))))))))))))))
rewrite(twice(s(x0))) → begin(p(p(s(s(s(p(p(s(s(s(twice(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(s(s(s(x0)))))))))))))
rewrite(0(x0)) → begin(x0)