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(a(x0)) → rotate(cut(Ca(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(C(x0)) → rotate(cut(CC(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
begin(A(x0)) → rotate(cut(CA(guess(x0))))
begin(B(x0)) → rotate(cut(CB(guess(x0))))
guess(a(x0)) → Ca(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(C(x0)) → CC(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(A(x0)) → CA(guess(x0))
guess(B(x0)) → CB(guess(x0))
guess(a(x0)) → moveleft(Ba(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(C(x0)) → moveleft(BC(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(A(x0)) → moveleft(BA(wait(x0)))
guess(B(x0)) → moveleft(BB(wait(x0)))
guess(end(x0)) → finish(end(x0))
Ca(moveleft(Ba(x0))) → moveleft(Ba(Aa(x0)))
Cb(moveleft(Ba(x0))) → moveleft(Ba(Ab(x0)))
CC(moveleft(Ba(x0))) → moveleft(Ba(AC(x0)))
Cc(moveleft(Ba(x0))) → moveleft(Ba(Ac(x0)))
CA(moveleft(Ba(x0))) → moveleft(Ba(AA(x0)))
CB(moveleft(Ba(x0))) → moveleft(Ba(AB(x0)))
Ca(moveleft(Bb(x0))) → moveleft(Bb(Aa(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
CC(moveleft(Bb(x0))) → moveleft(Bb(AC(x0)))
Cc(moveleft(Bb(x0))) → moveleft(Bb(Ac(x0)))
CA(moveleft(Bb(x0))) → moveleft(Bb(AA(x0)))
CB(moveleft(Bb(x0))) → moveleft(Bb(AB(x0)))
Ca(moveleft(BC(x0))) → moveleft(BC(Aa(x0)))
Cb(moveleft(BC(x0))) → moveleft(BC(Ab(x0)))
CC(moveleft(BC(x0))) → moveleft(BC(AC(x0)))
Cc(moveleft(BC(x0))) → moveleft(BC(Ac(x0)))
CA(moveleft(BC(x0))) → moveleft(BC(AA(x0)))
CB(moveleft(BC(x0))) → moveleft(BC(AB(x0)))
Ca(moveleft(Bc(x0))) → moveleft(Bc(Aa(x0)))
Cb(moveleft(Bc(x0))) → moveleft(Bc(Ab(x0)))
CC(moveleft(Bc(x0))) → moveleft(Bc(AC(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
CA(moveleft(Bc(x0))) → moveleft(Bc(AA(x0)))
CB(moveleft(Bc(x0))) → moveleft(Bc(AB(x0)))
Ca(moveleft(BA(x0))) → moveleft(BA(Aa(x0)))
Cb(moveleft(BA(x0))) → moveleft(BA(Ab(x0)))
CC(moveleft(BA(x0))) → moveleft(BA(AC(x0)))
Cc(moveleft(BA(x0))) → moveleft(BA(Ac(x0)))
CA(moveleft(BA(x0))) → moveleft(BA(AA(x0)))
CB(moveleft(BA(x0))) → moveleft(BA(AB(x0)))
Ca(moveleft(BB(x0))) → moveleft(BB(Aa(x0)))
Cb(moveleft(BB(x0))) → moveleft(BB(Ab(x0)))
CC(moveleft(BB(x0))) → moveleft(BB(AC(x0)))
Cc(moveleft(BB(x0))) → moveleft(BB(Ac(x0)))
CA(moveleft(BB(x0))) → moveleft(BB(AA(x0)))
CB(moveleft(BB(x0))) → moveleft(BB(AB(x0)))
cut(moveleft(Ba(x0))) → Da(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(BC(x0))) → DC(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
cut(moveleft(BA(x0))) → DA(cut(goright(x0)))
cut(moveleft(BB(x0))) → DB(cut(goright(x0)))
goright(Aa(x0)) → Ca(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(AC(x0)) → CC(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(AA(x0)) → CA(goright(x0))
goright(AB(x0)) → CB(goright(x0))
goright(wait(a(x0))) → moveleft(Ba(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(C(x0))) → moveleft(BC(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(A(x0))) → moveleft(BA(wait(x0)))
goright(wait(B(x0))) → moveleft(BB(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Ca(finish(x0)) → finish(a(x0))
Cb(finish(x0)) → finish(b(x0))
CC(finish(x0)) → finish(C(x0))
Cc(finish(x0)) → finish(c(x0))
CA(finish(x0)) → finish(A(x0))
CB(finish(x0)) → finish(B(x0))
cut(finish(x0)) → finish2(x0)
Da(finish2(x0)) → finish2(a(x0))
Db(finish2(x0)) → finish2(b(x0))
DC(finish2(x0)) → finish2(C(x0))
Dc(finish2(x0)) → finish2(c(x0))
DA(finish2(x0)) → finish2(A(x0))
DB(finish2(x0)) → finish2(B(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(a(a(b(b(x0))))) → begin(C(C(x0)))
rewrite(b(b(c(c(x0))))) → begin(A(A(x0)))
rewrite(c(c(a(a(x0))))) → begin(B(B(x0)))
rewrite(A(A(C(C(x0))))) → begin(b(b(x0)))
rewrite(C(C(B(B(x0))))) → begin(a(a(x0)))
rewrite(B(B(A(A(x0))))) → begin(c(c(x0)))
rewrite(a(a(a(a(a(a(a(a(a(a(x0))))))))))) → begin(A(A(A(A(A(A(x0)))))))
rewrite(A(A(A(A(A(A(A(A(x0))))))))) → begin(a(a(a(a(a(a(a(a(x0)))))))))
rewrite(b(b(b(b(b(b(b(b(b(b(x0))))))))))) → begin(B(B(B(B(B(B(x0)))))))
rewrite(B(B(B(B(B(B(B(B(x0))))))))) → begin(b(b(b(b(b(b(b(b(x0)))))))))
rewrite(c(c(c(c(c(c(c(c(c(c(x0))))))))))) → begin(C(C(C(C(C(C(x0)))))))
rewrite(C(C(C(C(C(C(C(C(x0))))))))) → begin(c(c(c(c(c(c(c(c(x0)))))))))
rewrite(B(B(a(a(a(a(a(a(a(a(x0))))))))))) → begin(c(c(A(A(A(A(A(A(x0)))))))))
rewrite(A(A(A(A(A(A(b(b(x0))))))))) → begin(a(a(a(a(a(a(a(a(C(C(x0)))))))))))
rewrite(C(C(b(b(b(b(b(b(b(b(x0))))))))))) → begin(a(a(B(B(B(B(B(B(x0)))))))))
rewrite(B(B(B(B(B(B(c(c(x0))))))))) → begin(b(b(b(b(b(b(b(b(A(A(x0)))))))))))
rewrite(A(A(c(c(c(c(c(c(c(c(x0))))))))))) → begin(b(b(C(C(C(C(C(C(x0)))))))))
rewrite(C(C(C(C(C(C(a(a(x0))))))))) → begin(c(c(c(c(c(c(c(c(B(B(x0)))))))))))
rewrite(a(a(A(A(x0))))) → begin(x0)
rewrite(A(A(a(a(x0))))) → begin(x0)
rewrite(b(b(B(B(x0))))) → begin(x0)
rewrite(B(B(b(b(x0))))) → begin(x0)
rewrite(c(c(C(C(x0))))) → begin(x0)
rewrite(C(C(c(c(x0))))) → begin(x0)

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(a(x0)) → rotate(cut(Ca(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(C(x0)) → rotate(cut(CC(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
begin(A(x0)) → rotate(cut(CA(guess(x0))))
begin(B(x0)) → rotate(cut(CB(guess(x0))))
guess(a(x0)) → Ca(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(C(x0)) → CC(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(A(x0)) → CA(guess(x0))
guess(B(x0)) → CB(guess(x0))
guess(a(x0)) → moveleft(Ba(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(C(x0)) → moveleft(BC(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(A(x0)) → moveleft(BA(wait(x0)))
guess(B(x0)) → moveleft(BB(wait(x0)))
guess(end(x0)) → finish(end(x0))
Ca(moveleft(Ba(x0))) → moveleft(Ba(Aa(x0)))
Cb(moveleft(Ba(x0))) → moveleft(Ba(Ab(x0)))
CC(moveleft(Ba(x0))) → moveleft(Ba(AC(x0)))
Cc(moveleft(Ba(x0))) → moveleft(Ba(Ac(x0)))
CA(moveleft(Ba(x0))) → moveleft(Ba(AA(x0)))
CB(moveleft(Ba(x0))) → moveleft(Ba(AB(x0)))
Ca(moveleft(Bb(x0))) → moveleft(Bb(Aa(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
CC(moveleft(Bb(x0))) → moveleft(Bb(AC(x0)))
Cc(moveleft(Bb(x0))) → moveleft(Bb(Ac(x0)))
CA(moveleft(Bb(x0))) → moveleft(Bb(AA(x0)))
CB(moveleft(Bb(x0))) → moveleft(Bb(AB(x0)))
Ca(moveleft(BC(x0))) → moveleft(BC(Aa(x0)))
Cb(moveleft(BC(x0))) → moveleft(BC(Ab(x0)))
CC(moveleft(BC(x0))) → moveleft(BC(AC(x0)))
Cc(moveleft(BC(x0))) → moveleft(BC(Ac(x0)))
CA(moveleft(BC(x0))) → moveleft(BC(AA(x0)))
CB(moveleft(BC(x0))) → moveleft(BC(AB(x0)))
Ca(moveleft(Bc(x0))) → moveleft(Bc(Aa(x0)))
Cb(moveleft(Bc(x0))) → moveleft(Bc(Ab(x0)))
CC(moveleft(Bc(x0))) → moveleft(Bc(AC(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
CA(moveleft(Bc(x0))) → moveleft(Bc(AA(x0)))
CB(moveleft(Bc(x0))) → moveleft(Bc(AB(x0)))
Ca(moveleft(BA(x0))) → moveleft(BA(Aa(x0)))
Cb(moveleft(BA(x0))) → moveleft(BA(Ab(x0)))
CC(moveleft(BA(x0))) → moveleft(BA(AC(x0)))
Cc(moveleft(BA(x0))) → moveleft(BA(Ac(x0)))
CA(moveleft(BA(x0))) → moveleft(BA(AA(x0)))
CB(moveleft(BA(x0))) → moveleft(BA(AB(x0)))
Ca(moveleft(BB(x0))) → moveleft(BB(Aa(x0)))
Cb(moveleft(BB(x0))) → moveleft(BB(Ab(x0)))
CC(moveleft(BB(x0))) → moveleft(BB(AC(x0)))
Cc(moveleft(BB(x0))) → moveleft(BB(Ac(x0)))
CA(moveleft(BB(x0))) → moveleft(BB(AA(x0)))
CB(moveleft(BB(x0))) → moveleft(BB(AB(x0)))
cut(moveleft(Ba(x0))) → Da(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(BC(x0))) → DC(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
cut(moveleft(BA(x0))) → DA(cut(goright(x0)))
cut(moveleft(BB(x0))) → DB(cut(goright(x0)))
goright(Aa(x0)) → Ca(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(AC(x0)) → CC(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(AA(x0)) → CA(goright(x0))
goright(AB(x0)) → CB(goright(x0))
goright(wait(a(x0))) → moveleft(Ba(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(C(x0))) → moveleft(BC(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(A(x0))) → moveleft(BA(wait(x0)))
goright(wait(B(x0))) → moveleft(BB(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Ca(finish(x0)) → finish(a(x0))
Cb(finish(x0)) → finish(b(x0))
CC(finish(x0)) → finish(C(x0))
Cc(finish(x0)) → finish(c(x0))
CA(finish(x0)) → finish(A(x0))
CB(finish(x0)) → finish(B(x0))
cut(finish(x0)) → finish2(x0)
Da(finish2(x0)) → finish2(a(x0))
Db(finish2(x0)) → finish2(b(x0))
DC(finish2(x0)) → finish2(C(x0))
Dc(finish2(x0)) → finish2(c(x0))
DA(finish2(x0)) → finish2(A(x0))
DB(finish2(x0)) → finish2(B(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(a(a(b(b(x0))))) → begin(C(C(x0)))
rewrite(b(b(c(c(x0))))) → begin(A(A(x0)))
rewrite(c(c(a(a(x0))))) → begin(B(B(x0)))
rewrite(A(A(C(C(x0))))) → begin(b(b(x0)))
rewrite(C(C(B(B(x0))))) → begin(a(a(x0)))
rewrite(B(B(A(A(x0))))) → begin(c(c(x0)))
rewrite(a(a(a(a(a(a(a(a(a(a(x0))))))))))) → begin(A(A(A(A(A(A(x0)))))))
rewrite(A(A(A(A(A(A(A(A(x0))))))))) → begin(a(a(a(a(a(a(a(a(x0)))))))))
rewrite(b(b(b(b(b(b(b(b(b(b(x0))))))))))) → begin(B(B(B(B(B(B(x0)))))))
rewrite(B(B(B(B(B(B(B(B(x0))))))))) → begin(b(b(b(b(b(b(b(b(x0)))))))))
rewrite(c(c(c(c(c(c(c(c(c(c(x0))))))))))) → begin(C(C(C(C(C(C(x0)))))))
rewrite(C(C(C(C(C(C(C(C(x0))))))))) → begin(c(c(c(c(c(c(c(c(x0)))))))))
rewrite(B(B(a(a(a(a(a(a(a(a(x0))))))))))) → begin(c(c(A(A(A(A(A(A(x0)))))))))
rewrite(A(A(A(A(A(A(b(b(x0))))))))) → begin(a(a(a(a(a(a(a(a(C(C(x0)))))))))))
rewrite(C(C(b(b(b(b(b(b(b(b(x0))))))))))) → begin(a(a(B(B(B(B(B(B(x0)))))))))
rewrite(B(B(B(B(B(B(c(c(x0))))))))) → begin(b(b(b(b(b(b(b(b(A(A(x0)))))))))))
rewrite(A(A(c(c(c(c(c(c(c(c(x0))))))))))) → begin(b(b(C(C(C(C(C(C(x0)))))))))
rewrite(C(C(C(C(C(C(a(a(x0))))))))) → begin(c(c(c(c(c(c(c(c(B(B(x0)))))))))))
rewrite(a(a(A(A(x0))))) → begin(x0)
rewrite(A(A(a(a(x0))))) → begin(x0)
rewrite(b(b(B(B(x0))))) → begin(x0)
rewrite(B(B(b(b(x0))))) → begin(x0)
rewrite(c(c(C(C(x0))))) → begin(x0)
rewrite(C(C(c(c(x0))))) → begin(x0)