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(n(x0)) → rotate(cut(Cn(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
begin(a(x0)) → rotate(cut(Ca(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
guess(f(x0)) → Cf(guess(x0))
guess(n(x0)) → Cn(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(a(x0)) → Ca(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(n(x0)) → moveleft(Bn(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(a(x0)) → moveleft(Ba(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
Cn(moveleft(Bf(x0))) → moveleft(Bf(An(x0)))
Cc(moveleft(Bf(x0))) → moveleft(Bf(Ac(x0)))
Ca(moveleft(Bf(x0))) → moveleft(Bf(Aa(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cf(moveleft(Bn(x0))) → moveleft(Bn(Af(x0)))
Cn(moveleft(Bn(x0))) → moveleft(Bn(An(x0)))
Cc(moveleft(Bn(x0))) → moveleft(Bn(Ac(x0)))
Ca(moveleft(Bn(x0))) → moveleft(Bn(Aa(x0)))
Cs(moveleft(Bn(x0))) → moveleft(Bn(As(x0)))
Cf(moveleft(Bc(x0))) → moveleft(Bc(Af(x0)))
Cn(moveleft(Bc(x0))) → moveleft(Bc(An(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
Ca(moveleft(Bc(x0))) → moveleft(Bc(Aa(x0)))
Cs(moveleft(Bc(x0))) → moveleft(Bc(As(x0)))
Cf(moveleft(Ba(x0))) → moveleft(Ba(Af(x0)))
Cn(moveleft(Ba(x0))) → moveleft(Ba(An(x0)))
Cc(moveleft(Ba(x0))) → moveleft(Ba(Ac(x0)))
Ca(moveleft(Ba(x0))) → moveleft(Ba(Aa(x0)))
Cs(moveleft(Ba(x0))) → moveleft(Ba(As(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
Cn(moveleft(Bs(x0))) → moveleft(Bs(An(x0)))
Cc(moveleft(Bs(x0))) → moveleft(Bs(Ac(x0)))
Ca(moveleft(Bs(x0))) → moveleft(Bs(Aa(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(Bn(x0))) → Dn(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
cut(moveleft(Ba(x0))) → Da(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
goright(Af(x0)) → Cf(goright(x0))
goright(An(x0)) → Cn(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(Aa(x0)) → Ca(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(n(x0))) → moveleft(Bn(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(a(x0))) → moveleft(Ba(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cf(finish(x0)) → finish(f(x0))
Cn(finish(x0)) → finish(n(x0))
Cc(finish(x0)) → finish(c(x0))
Ca(finish(x0)) → finish(a(x0))
Cs(finish(x0)) → finish(s(x0))
cut(finish(x0)) → finish2(x0)
Df(finish2(x0)) → finish2(f(x0))
Dn(finish2(x0)) → finish2(n(x0))
Dc(finish2(x0)) → finish2(c(x0))
Da(finish2(x0)) → finish2(a(x0))
Ds(finish2(x0)) → finish2(s(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(f(x0)) → begin(n(c(n(a(x0)))))
rewrite(c(f(x0))) → begin(f(n(a(c(x0)))))
rewrite(n(a(x0))) → begin(c(x0))
rewrite(c(c(x0))) → begin(c(x0))
rewrite(n(s(x0))) → begin(f(s(s(x0))))
rewrite(n(f(x0))) → begin(f(n(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(n(x0)) → rotate(cut(Cn(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
begin(a(x0)) → rotate(cut(Ca(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
guess(f(x0)) → Cf(guess(x0))
guess(n(x0)) → Cn(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(a(x0)) → Ca(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(f(x0)) → moveleft(Bf(wait(x0)))
guess(n(x0)) → moveleft(Bn(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(a(x0)) → moveleft(Ba(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cf(moveleft(Bf(x0))) → moveleft(Bf(Af(x0)))
Cn(moveleft(Bf(x0))) → moveleft(Bf(An(x0)))
Cc(moveleft(Bf(x0))) → moveleft(Bf(Ac(x0)))
Ca(moveleft(Bf(x0))) → moveleft(Bf(Aa(x0)))
Cs(moveleft(Bf(x0))) → moveleft(Bf(As(x0)))
Cf(moveleft(Bn(x0))) → moveleft(Bn(Af(x0)))
Cn(moveleft(Bn(x0))) → moveleft(Bn(An(x0)))
Cc(moveleft(Bn(x0))) → moveleft(Bn(Ac(x0)))
Ca(moveleft(Bn(x0))) → moveleft(Bn(Aa(x0)))
Cs(moveleft(Bn(x0))) → moveleft(Bn(As(x0)))
Cf(moveleft(Bc(x0))) → moveleft(Bc(Af(x0)))
Cn(moveleft(Bc(x0))) → moveleft(Bc(An(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
Ca(moveleft(Bc(x0))) → moveleft(Bc(Aa(x0)))
Cs(moveleft(Bc(x0))) → moveleft(Bc(As(x0)))
Cf(moveleft(Ba(x0))) → moveleft(Ba(Af(x0)))
Cn(moveleft(Ba(x0))) → moveleft(Ba(An(x0)))
Cc(moveleft(Ba(x0))) → moveleft(Ba(Ac(x0)))
Ca(moveleft(Ba(x0))) → moveleft(Ba(Aa(x0)))
Cs(moveleft(Ba(x0))) → moveleft(Ba(As(x0)))
Cf(moveleft(Bs(x0))) → moveleft(Bs(Af(x0)))
Cn(moveleft(Bs(x0))) → moveleft(Bs(An(x0)))
Cc(moveleft(Bs(x0))) → moveleft(Bs(Ac(x0)))
Ca(moveleft(Bs(x0))) → moveleft(Bs(Aa(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
cut(moveleft(Bf(x0))) → Df(cut(goright(x0)))
cut(moveleft(Bn(x0))) → Dn(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
cut(moveleft(Ba(x0))) → Da(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
goright(Af(x0)) → Cf(goright(x0))
goright(An(x0)) → Cn(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(Aa(x0)) → Ca(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(wait(f(x0))) → moveleft(Bf(wait(x0)))
goright(wait(n(x0))) → moveleft(Bn(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(a(x0))) → moveleft(Ba(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cf(finish(x0)) → finish(f(x0))
Cn(finish(x0)) → finish(n(x0))
Cc(finish(x0)) → finish(c(x0))
Ca(finish(x0)) → finish(a(x0))
Cs(finish(x0)) → finish(s(x0))
cut(finish(x0)) → finish2(x0)
Df(finish2(x0)) → finish2(f(x0))
Dn(finish2(x0)) → finish2(n(x0))
Dc(finish2(x0)) → finish2(c(x0))
Da(finish2(x0)) → finish2(a(x0))
Ds(finish2(x0)) → finish2(s(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(f(x0)) → begin(n(c(n(a(x0)))))
rewrite(c(f(x0))) → begin(f(n(a(c(x0)))))
rewrite(n(a(x0))) → begin(c(x0))
rewrite(c(c(x0))) → begin(c(x0))
rewrite(n(s(x0))) → begin(f(s(s(x0))))
rewrite(n(f(x0))) → begin(f(n(x0)))