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(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(f(x0)) → rotate(cut(Cf(guess(x0))))
begin(g(x0)) → rotate(cut(Cg(guess(x0))))
begin(q(x0)) → rotate(cut(Cq(guess(x0))))
begin(i(x0)) → rotate(cut(Ci(guess(x0))))
guess(p(x0)) → Cp(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(f(x0)) → Cf(guess(x0))
guess(g(x0)) → Cg(guess(x0))
guess(q(x0)) → Cq(guess(x0))
guess(i(x0)) → Ci(guess(x0))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(g(x0)) → moveleft(Bg(wait(x0)))
guess(q(x0)) → moveleft(Bq(wait(x0)))
guess(i(x0)) → moveleft(Bi(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cf(moveleft(Bp(x0))) → moveleft(Bp(Af(x0)))
Cg(moveleft(Bp(x0))) → moveleft(Bp(Ag(x0)))
Cq(moveleft(Bp(x0))) → moveleft(Bp(Aq(x0)))
Ci(moveleft(Bp(x0))) → moveleft(Bp(Ai(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cf(moveleft(B0(x0))) → moveleft(B0(Af(x0)))
Cg(moveleft(B0(x0))) → moveleft(B0(Ag(x0)))
Cq(moveleft(B0(x0))) → moveleft(B0(Aq(x0)))
Ci(moveleft(B0(x0))) → moveleft(B0(Ai(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
Cg(moveleft(Bs(x0))) → moveleft(Bs(Ag(x0)))
Cq(moveleft(Bs(x0))) → moveleft(Bs(Aq(x0)))
Ci(moveleft(Bs(x0))) → moveleft(Bs(Ai(x0)))
Cp(moveleft(Bf(x0))) → moveleft(Bf(Ap(x0)))
C0(moveleft(Bf(x0))) → moveleft(Bf(A0(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
Cg(moveleft(Bf(x0))) → moveleft(Bf(Ag(x0)))
Cq(moveleft(Bf(x0))) → moveleft(Bf(Aq(x0)))
Ci(moveleft(Bf(x0))) → moveleft(Bf(Ai(x0)))
Cp(moveleft(Bg(x0))) → moveleft(Bg(Ap(x0)))
C0(moveleft(Bg(x0))) → moveleft(Bg(A0(x0)))
Cs(moveleft(Bg(x0))) → moveleft(Bg(As(x0)))
Cf(moveleft(Bg(x0))) → moveleft(Bg(Af(x0)))
Cg(moveleft(Bg(x0))) → moveleft(Bg(Ag(x0)))
Cq(moveleft(Bg(x0))) → moveleft(Bg(Aq(x0)))
Ci(moveleft(Bg(x0))) → moveleft(Bg(Ai(x0)))
Cp(moveleft(Bq(x0))) → moveleft(Bq(Ap(x0)))
C0(moveleft(Bq(x0))) → moveleft(Bq(A0(x0)))
Cs(moveleft(Bq(x0))) → moveleft(Bq(As(x0)))
Cf(moveleft(Bq(x0))) → moveleft(Bq(Af(x0)))
Cg(moveleft(Bq(x0))) → moveleft(Bq(Ag(x0)))
Cq(moveleft(Bq(x0))) → moveleft(Bq(Aq(x0)))
Ci(moveleft(Bq(x0))) → moveleft(Bq(Ai(x0)))
Cp(moveleft(Bi(x0))) → moveleft(Bi(Ap(x0)))
C0(moveleft(Bi(x0))) → moveleft(Bi(A0(x0)))
Cs(moveleft(Bi(x0))) → moveleft(Bi(As(x0)))
Cf(moveleft(Bi(x0))) → moveleft(Bi(Af(x0)))
Cg(moveleft(Bi(x0))) → moveleft(Bi(Ag(x0)))
Cq(moveleft(Bi(x0))) → moveleft(Bi(Aq(x0)))
Ci(moveleft(Bi(x0))) → moveleft(Bi(Ai(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(Bg(x0))) → Dg(cut(goright(x0)))
cut(moveleft(Bq(x0))) → Dq(cut(goright(x0)))
cut(moveleft(Bi(x0))) → Di(cut(goright(x0)))
goright(Ap(x0)) → Cp(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Af(x0)) → Cf(goright(x0))
goright(Ag(x0)) → Cg(goright(x0))
goright(Aq(x0)) → Cq(goright(x0))
goright(Ai(x0)) → Ci(goright(x0))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(g(x0))) → moveleft(Bg(wait(x0)))
goright(wait(q(x0))) → moveleft(Bq(wait(x0)))
goright(wait(i(x0))) → moveleft(Bi(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cp(finish(x0)) → finish(p(x0))
C0(finish(x0)) → finish(0(x0))
Cs(finish(x0)) → finish(s(x0))
Cf(finish(x0)) → finish(f(x0))
Cg(finish(x0)) → finish(g(x0))
Cq(finish(x0)) → finish(q(x0))
Ci(finish(x0)) → finish(i(x0))
cut(finish(x0)) → finish2(x0)
Dp(finish2(x0)) → finish2(p(x0))
D0(finish2(x0)) → finish2(0(x0))
Ds(finish2(x0)) → finish2(s(x0))
Df(finish2(x0)) → finish2(f(x0))
Dg(finish2(x0)) → finish2(g(x0))
Dq(finish2(x0)) → finish2(q(x0))
Di(finish2(x0)) → finish2(i(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(p(0(x0))) → begin(s(s(0(s(s(p(x0)))))))
rewrite(p(s(0(x0)))) → begin(0(x0))
rewrite(p(s(s(x0)))) → begin(s(p(s(x0))))
rewrite(f(s(x0))) → begin(g(q(i(x0))))
rewrite(g(x0)) → begin(f(p(p(x0))))
rewrite(q(i(x0))) → begin(q(s(x0)))
rewrite(q(s(x0))) → begin(s(s(x0)))
rewrite(i(x0)) → begin(s(x0))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(f(x0)) → rotate(cut(Cf(guess(x0))))
begin(g(x0)) → rotate(cut(Cg(guess(x0))))
begin(q(x0)) → rotate(cut(Cq(guess(x0))))
begin(i(x0)) → rotate(cut(Ci(guess(x0))))
guess(p(x0)) → Cp(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(f(x0)) → Cf(guess(x0))
guess(g(x0)) → Cg(guess(x0))
guess(q(x0)) → Cq(guess(x0))
guess(i(x0)) → Ci(guess(x0))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(g(x0)) → moveleft(Bg(wait(x0)))
guess(q(x0)) → moveleft(Bq(wait(x0)))
guess(i(x0)) → moveleft(Bi(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cf(moveleft(Bp(x0))) → moveleft(Bp(Af(x0)))
Cg(moveleft(Bp(x0))) → moveleft(Bp(Ag(x0)))
Cq(moveleft(Bp(x0))) → moveleft(Bp(Aq(x0)))
Ci(moveleft(Bp(x0))) → moveleft(Bp(Ai(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cf(moveleft(B0(x0))) → moveleft(B0(Af(x0)))
Cg(moveleft(B0(x0))) → moveleft(B0(Ag(x0)))
Cq(moveleft(B0(x0))) → moveleft(B0(Aq(x0)))
Ci(moveleft(B0(x0))) → moveleft(B0(Ai(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
Cg(moveleft(Bs(x0))) → moveleft(Bs(Ag(x0)))
Cq(moveleft(Bs(x0))) → moveleft(Bs(Aq(x0)))
Ci(moveleft(Bs(x0))) → moveleft(Bs(Ai(x0)))
Cp(moveleft(Bf(x0))) → moveleft(Bf(Ap(x0)))
C0(moveleft(Bf(x0))) → moveleft(Bf(A0(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
Cg(moveleft(Bf(x0))) → moveleft(Bf(Ag(x0)))
Cq(moveleft(Bf(x0))) → moveleft(Bf(Aq(x0)))
Ci(moveleft(Bf(x0))) → moveleft(Bf(Ai(x0)))
Cp(moveleft(Bg(x0))) → moveleft(Bg(Ap(x0)))
C0(moveleft(Bg(x0))) → moveleft(Bg(A0(x0)))
Cs(moveleft(Bg(x0))) → moveleft(Bg(As(x0)))
Cf(moveleft(Bg(x0))) → moveleft(Bg(Af(x0)))
Cg(moveleft(Bg(x0))) → moveleft(Bg(Ag(x0)))
Cq(moveleft(Bg(x0))) → moveleft(Bg(Aq(x0)))
Ci(moveleft(Bg(x0))) → moveleft(Bg(Ai(x0)))
Cp(moveleft(Bq(x0))) → moveleft(Bq(Ap(x0)))
C0(moveleft(Bq(x0))) → moveleft(Bq(A0(x0)))
Cs(moveleft(Bq(x0))) → moveleft(Bq(As(x0)))
Cf(moveleft(Bq(x0))) → moveleft(Bq(Af(x0)))
Cg(moveleft(Bq(x0))) → moveleft(Bq(Ag(x0)))
Cq(moveleft(Bq(x0))) → moveleft(Bq(Aq(x0)))
Ci(moveleft(Bq(x0))) → moveleft(Bq(Ai(x0)))
Cp(moveleft(Bi(x0))) → moveleft(Bi(Ap(x0)))
C0(moveleft(Bi(x0))) → moveleft(Bi(A0(x0)))
Cs(moveleft(Bi(x0))) → moveleft(Bi(As(x0)))
Cf(moveleft(Bi(x0))) → moveleft(Bi(Af(x0)))
Cg(moveleft(Bi(x0))) → moveleft(Bi(Ag(x0)))
Cq(moveleft(Bi(x0))) → moveleft(Bi(Aq(x0)))
Ci(moveleft(Bi(x0))) → moveleft(Bi(Ai(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(Bg(x0))) → Dg(cut(goright(x0)))
cut(moveleft(Bq(x0))) → Dq(cut(goright(x0)))
cut(moveleft(Bi(x0))) → Di(cut(goright(x0)))
goright(Ap(x0)) → Cp(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Af(x0)) → Cf(goright(x0))
goright(Ag(x0)) → Cg(goright(x0))
goright(Aq(x0)) → Cq(goright(x0))
goright(Ai(x0)) → Ci(goright(x0))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(g(x0))) → moveleft(Bg(wait(x0)))
goright(wait(q(x0))) → moveleft(Bq(wait(x0)))
goright(wait(i(x0))) → moveleft(Bi(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cp(finish(x0)) → finish(p(x0))
C0(finish(x0)) → finish(0(x0))
Cs(finish(x0)) → finish(s(x0))
Cf(finish(x0)) → finish(f(x0))
Cg(finish(x0)) → finish(g(x0))
Cq(finish(x0)) → finish(q(x0))
Ci(finish(x0)) → finish(i(x0))
cut(finish(x0)) → finish2(x0)
Dp(finish2(x0)) → finish2(p(x0))
D0(finish2(x0)) → finish2(0(x0))
Ds(finish2(x0)) → finish2(s(x0))
Df(finish2(x0)) → finish2(f(x0))
Dg(finish2(x0)) → finish2(g(x0))
Dq(finish2(x0)) → finish2(q(x0))
Di(finish2(x0)) → finish2(i(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(p(0(x0))) → begin(s(s(0(s(s(p(x0)))))))
rewrite(p(s(0(x0)))) → begin(0(x0))
rewrite(p(s(s(x0)))) → begin(s(p(s(x0))))
rewrite(f(s(x0))) → begin(g(q(i(x0))))
rewrite(g(x0)) → begin(f(p(p(x0))))
rewrite(q(i(x0))) → begin(q(s(x0)))
rewrite(q(s(x0))) → begin(s(s(x0)))
rewrite(i(x0)) → begin(s(x0))