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(f(x0)) → rotate(cut(Cf(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(d(x0)) → rotate(cut(Cd(guess(x0))))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
guess(f(x0)) → Cf(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(d(x0)) → Cd(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(d(x0)) → moveleft(Bd(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
C0(moveleft(Bf(x0))) → moveleft(Bf(A0(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cd(moveleft(Bf(x0))) → moveleft(Bf(Ad(x0)))
Cp(moveleft(Bf(x0))) → moveleft(Bf(Ap(x0)))
Cf(moveleft(B0(x0))) → moveleft(B0(Af(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cd(moveleft(B0(x0))) → moveleft(B0(Ad(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cd(moveleft(Bs(x0))) → moveleft(Bs(Ad(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
Cf(moveleft(Bd(x0))) → moveleft(Bd(Af(x0)))
C0(moveleft(Bd(x0))) → moveleft(Bd(A0(x0)))
Cs(moveleft(Bd(x0))) → moveleft(Bd(As(x0)))
Cd(moveleft(Bd(x0))) → moveleft(Bd(Ad(x0)))
Cp(moveleft(Bd(x0))) → moveleft(Bd(Ap(x0)))
Cf(moveleft(Bp(x0))) → moveleft(Bp(Af(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cd(moveleft(Bp(x0))) → moveleft(Bp(Ad(x0)))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bd(x0))) → Dd(cut(goright(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
goright(Af(x0)) → Cf(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ad(x0)) → Cd(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(d(x0))) → moveleft(Bd(wait(x0)))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cf(finish(x0)) → finish(f(x0))
C0(finish(x0)) → finish(0(x0))
Cs(finish(x0)) → finish(s(x0))
Cd(finish(x0)) → finish(d(x0))
Cp(finish(x0)) → finish(p(x0))
cut(finish(x0)) → finish2(x0)
Df(finish2(x0)) → finish2(f(x0))
D0(finish2(x0)) → finish2(0(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dd(finish2(x0)) → finish2(d(x0))
Dp(finish2(x0)) → finish2(p(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(f(0(x0))) → begin(s(0(x0)))
rewrite(d(0(x0))) → begin(0(x0))
rewrite(d(s(x0))) → begin(s(s(d(p(s(x0))))))
rewrite(f(s(x0))) → begin(d(f(p(s(x0)))))
rewrite(p(s(x0))) → begin(x0)

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(f(x0)) → rotate(cut(Cf(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(d(x0)) → rotate(cut(Cd(guess(x0))))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
guess(f(x0)) → Cf(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(d(x0)) → Cd(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(d(x0)) → moveleft(Bd(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
C0(moveleft(Bf(x0))) → moveleft(Bf(A0(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cd(moveleft(Bf(x0))) → moveleft(Bf(Ad(x0)))
Cp(moveleft(Bf(x0))) → moveleft(Bf(Ap(x0)))
Cf(moveleft(B0(x0))) → moveleft(B0(Af(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cd(moveleft(B0(x0))) → moveleft(B0(Ad(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cd(moveleft(Bs(x0))) → moveleft(Bs(Ad(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
Cf(moveleft(Bd(x0))) → moveleft(Bd(Af(x0)))
C0(moveleft(Bd(x0))) → moveleft(Bd(A0(x0)))
Cs(moveleft(Bd(x0))) → moveleft(Bd(As(x0)))
Cd(moveleft(Bd(x0))) → moveleft(Bd(Ad(x0)))
Cp(moveleft(Bd(x0))) → moveleft(Bd(Ap(x0)))
Cf(moveleft(Bp(x0))) → moveleft(Bp(Af(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cd(moveleft(Bp(x0))) → moveleft(Bp(Ad(x0)))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bd(x0))) → Dd(cut(goright(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
goright(Af(x0)) → Cf(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ad(x0)) → Cd(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(d(x0))) → moveleft(Bd(wait(x0)))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cf(finish(x0)) → finish(f(x0))
C0(finish(x0)) → finish(0(x0))
Cs(finish(x0)) → finish(s(x0))
Cd(finish(x0)) → finish(d(x0))
Cp(finish(x0)) → finish(p(x0))
cut(finish(x0)) → finish2(x0)
Df(finish2(x0)) → finish2(f(x0))
D0(finish2(x0)) → finish2(0(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dd(finish2(x0)) → finish2(d(x0))
Dp(finish2(x0)) → finish2(p(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(f(0(x0))) → begin(s(0(x0)))
rewrite(d(0(x0))) → begin(0(x0))
rewrite(d(s(x0))) → begin(s(s(d(p(s(x0))))))
rewrite(f(s(x0))) → begin(d(f(p(s(x0)))))
rewrite(p(s(x0))) → begin(x0)