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(r(x0)) → rotate(cut(Cr(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(n(x0)) → rotate(cut(Cn(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(u(x0)) → rotate(cut(Cu(guess(x0))))
begin(t(x0)) → rotate(cut(Ct(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
guess(r(x0)) → Cr(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(n(x0)) → Cn(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(u(x0)) → Cu(guess(x0))
guess(t(x0)) → Ct(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(r(x0)) → moveleft(Br(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(n(x0)) → moveleft(Bn(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(u(x0)) → moveleft(Bu(wait(x0)))
guess(t(x0)) → moveleft(Bt(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cr(moveleft(Br(x0))) → moveleft(Br(Ar(x0)))
Cs(moveleft(Br(x0))) → moveleft(Br(As(x0)))
Cn(moveleft(Br(x0))) → moveleft(Br(An(x0)))
Cb(moveleft(Br(x0))) → moveleft(Br(Ab(x0)))
Cu(moveleft(Br(x0))) → moveleft(Br(Au(x0)))
Ct(moveleft(Br(x0))) → moveleft(Br(At(x0)))
Cc(moveleft(Br(x0))) → moveleft(Br(Ac(x0)))
Cr(moveleft(Bs(x0))) → moveleft(Bs(Ar(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cn(moveleft(Bs(x0))) → moveleft(Bs(An(x0)))
Cb(moveleft(Bs(x0))) → moveleft(Bs(Ab(x0)))
Cu(moveleft(Bs(x0))) → moveleft(Bs(Au(x0)))
Ct(moveleft(Bs(x0))) → moveleft(Bs(At(x0)))
Cc(moveleft(Bs(x0))) → moveleft(Bs(Ac(x0)))
Cr(moveleft(Bn(x0))) → moveleft(Bn(Ar(x0)))
Cs(moveleft(Bn(x0))) → moveleft(Bn(As(x0)))
Cn(moveleft(Bn(x0))) → moveleft(Bn(An(x0)))
Cb(moveleft(Bn(x0))) → moveleft(Bn(Ab(x0)))
Cu(moveleft(Bn(x0))) → moveleft(Bn(Au(x0)))
Ct(moveleft(Bn(x0))) → moveleft(Bn(At(x0)))
Cc(moveleft(Bn(x0))) → moveleft(Bn(Ac(x0)))
Cr(moveleft(Bb(x0))) → moveleft(Bb(Ar(x0)))
Cs(moveleft(Bb(x0))) → moveleft(Bb(As(x0)))
Cn(moveleft(Bb(x0))) → moveleft(Bb(An(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Cu(moveleft(Bb(x0))) → moveleft(Bb(Au(x0)))
Ct(moveleft(Bb(x0))) → moveleft(Bb(At(x0)))
Cc(moveleft(Bb(x0))) → moveleft(Bb(Ac(x0)))
Cr(moveleft(Bu(x0))) → moveleft(Bu(Ar(x0)))
Cs(moveleft(Bu(x0))) → moveleft(Bu(As(x0)))
Cn(moveleft(Bu(x0))) → moveleft(Bu(An(x0)))
Cb(moveleft(Bu(x0))) → moveleft(Bu(Ab(x0)))
Cu(moveleft(Bu(x0))) → moveleft(Bu(Au(x0)))
Ct(moveleft(Bu(x0))) → moveleft(Bu(At(x0)))
Cc(moveleft(Bu(x0))) → moveleft(Bu(Ac(x0)))
Cr(moveleft(Bt(x0))) → moveleft(Bt(Ar(x0)))
Cs(moveleft(Bt(x0))) → moveleft(Bt(As(x0)))
Cn(moveleft(Bt(x0))) → moveleft(Bt(An(x0)))
Cb(moveleft(Bt(x0))) → moveleft(Bt(Ab(x0)))
Cu(moveleft(Bt(x0))) → moveleft(Bt(Au(x0)))
Ct(moveleft(Bt(x0))) → moveleft(Bt(At(x0)))
Cc(moveleft(Bt(x0))) → moveleft(Bt(Ac(x0)))
Cr(moveleft(Bc(x0))) → moveleft(Bc(Ar(x0)))
Cs(moveleft(Bc(x0))) → moveleft(Bc(As(x0)))
Cn(moveleft(Bc(x0))) → moveleft(Bc(An(x0)))
Cb(moveleft(Bc(x0))) → moveleft(Bc(Ab(x0)))
Cu(moveleft(Bc(x0))) → moveleft(Bc(Au(x0)))
Ct(moveleft(Bc(x0))) → moveleft(Bc(At(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
cut(moveleft(Br(x0))) → Dr(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bn(x0))) → Dn(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Bu(x0))) → Du(cut(goright(x0)))
cut(moveleft(Bt(x0))) → Dt(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
goright(Ar(x0)) → Cr(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(An(x0)) → Cn(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(Au(x0)) → Cu(goright(x0))
goright(At(x0)) → Ct(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(wait(r(x0))) → moveleft(Br(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(n(x0))) → moveleft(Bn(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(u(x0))) → moveleft(Bu(wait(x0)))
goright(wait(t(x0))) → moveleft(Bt(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cr(finish(x0)) → finish(r(x0))
Cs(finish(x0)) → finish(s(x0))
Cn(finish(x0)) → finish(n(x0))
Cb(finish(x0)) → finish(b(x0))
Cu(finish(x0)) → finish(u(x0))
Ct(finish(x0)) → finish(t(x0))
Cc(finish(x0)) → finish(c(x0))
cut(finish(x0)) → finish2(x0)
Dr(finish2(x0)) → finish2(r(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dn(finish2(x0)) → finish2(n(x0))
Db(finish2(x0)) → finish2(b(x0))
Du(finish2(x0)) → finish2(u(x0))
Dt(finish2(x0)) → finish2(t(x0))
Dc(finish2(x0)) → finish2(c(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(r(r(x0))) → begin(s(r(x0)))
rewrite(r(s(x0))) → begin(s(r(x0)))
rewrite(r(n(x0))) → begin(s(r(x0)))
rewrite(r(b(x0))) → begin(u(s(b(x0))))
rewrite(r(u(x0))) → begin(u(r(x0)))
rewrite(s(u(x0))) → begin(u(s(x0)))
rewrite(n(u(x0))) → begin(u(n(x0)))
rewrite(t(r(u(x0)))) → begin(t(c(r(x0))))
rewrite(t(s(u(x0)))) → begin(t(c(r(x0))))
rewrite(t(n(u(x0)))) → begin(t(c(r(x0))))
rewrite(c(u(x0))) → begin(u(c(x0)))
rewrite(c(s(x0))) → begin(s(c(x0)))
rewrite(c(r(x0))) → begin(r(c(x0)))
rewrite(c(n(x0))) → begin(n(c(x0)))
rewrite(c(n(x0))) → begin(n(x0))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(r(x0)) → rotate(cut(Cr(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(n(x0)) → rotate(cut(Cn(guess(x0))))
begin(b(x0)) → rotate(cut(Cb(guess(x0))))
begin(u(x0)) → rotate(cut(Cu(guess(x0))))
begin(t(x0)) → rotate(cut(Ct(guess(x0))))
begin(c(x0)) → rotate(cut(Cc(guess(x0))))
guess(r(x0)) → Cr(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(n(x0)) → Cn(guess(x0))
guess(b(x0)) → Cb(guess(x0))
guess(u(x0)) → Cu(guess(x0))
guess(t(x0)) → Ct(guess(x0))
guess(c(x0)) → Cc(guess(x0))
guess(r(x0)) → moveleft(Br(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(n(x0)) → moveleft(Bn(wait(x0)))
guess(b(x0)) → moveleft(Bb(wait(x0)))
guess(u(x0)) → moveleft(Bu(wait(x0)))
guess(t(x0)) → moveleft(Bt(wait(x0)))
guess(c(x0)) → moveleft(Bc(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cr(moveleft(Br(x0))) → moveleft(Br(Ar(x0)))
Cs(moveleft(Br(x0))) → moveleft(Br(As(x0)))
Cn(moveleft(Br(x0))) → moveleft(Br(An(x0)))
Cb(moveleft(Br(x0))) → moveleft(Br(Ab(x0)))
Cu(moveleft(Br(x0))) → moveleft(Br(Au(x0)))
Ct(moveleft(Br(x0))) → moveleft(Br(At(x0)))
Cc(moveleft(Br(x0))) → moveleft(Br(Ac(x0)))
Cr(moveleft(Bs(x0))) → moveleft(Bs(Ar(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cn(moveleft(Bs(x0))) → moveleft(Bs(An(x0)))
Cb(moveleft(Bs(x0))) → moveleft(Bs(Ab(x0)))
Cu(moveleft(Bs(x0))) → moveleft(Bs(Au(x0)))
Ct(moveleft(Bs(x0))) → moveleft(Bs(At(x0)))
Cc(moveleft(Bs(x0))) → moveleft(Bs(Ac(x0)))
Cr(moveleft(Bn(x0))) → moveleft(Bn(Ar(x0)))
Cs(moveleft(Bn(x0))) → moveleft(Bn(As(x0)))
Cn(moveleft(Bn(x0))) → moveleft(Bn(An(x0)))
Cb(moveleft(Bn(x0))) → moveleft(Bn(Ab(x0)))
Cu(moveleft(Bn(x0))) → moveleft(Bn(Au(x0)))
Ct(moveleft(Bn(x0))) → moveleft(Bn(At(x0)))
Cc(moveleft(Bn(x0))) → moveleft(Bn(Ac(x0)))
Cr(moveleft(Bb(x0))) → moveleft(Bb(Ar(x0)))
Cs(moveleft(Bb(x0))) → moveleft(Bb(As(x0)))
Cn(moveleft(Bb(x0))) → moveleft(Bb(An(x0)))
Cb(moveleft(Bb(x0))) → moveleft(Bb(Ab(x0)))
Cu(moveleft(Bb(x0))) → moveleft(Bb(Au(x0)))
Ct(moveleft(Bb(x0))) → moveleft(Bb(At(x0)))
Cc(moveleft(Bb(x0))) → moveleft(Bb(Ac(x0)))
Cr(moveleft(Bu(x0))) → moveleft(Bu(Ar(x0)))
Cs(moveleft(Bu(x0))) → moveleft(Bu(As(x0)))
Cn(moveleft(Bu(x0))) → moveleft(Bu(An(x0)))
Cb(moveleft(Bu(x0))) → moveleft(Bu(Ab(x0)))
Cu(moveleft(Bu(x0))) → moveleft(Bu(Au(x0)))
Ct(moveleft(Bu(x0))) → moveleft(Bu(At(x0)))
Cc(moveleft(Bu(x0))) → moveleft(Bu(Ac(x0)))
Cr(moveleft(Bt(x0))) → moveleft(Bt(Ar(x0)))
Cs(moveleft(Bt(x0))) → moveleft(Bt(As(x0)))
Cn(moveleft(Bt(x0))) → moveleft(Bt(An(x0)))
Cb(moveleft(Bt(x0))) → moveleft(Bt(Ab(x0)))
Cu(moveleft(Bt(x0))) → moveleft(Bt(Au(x0)))
Ct(moveleft(Bt(x0))) → moveleft(Bt(At(x0)))
Cc(moveleft(Bt(x0))) → moveleft(Bt(Ac(x0)))
Cr(moveleft(Bc(x0))) → moveleft(Bc(Ar(x0)))
Cs(moveleft(Bc(x0))) → moveleft(Bc(As(x0)))
Cn(moveleft(Bc(x0))) → moveleft(Bc(An(x0)))
Cb(moveleft(Bc(x0))) → moveleft(Bc(Ab(x0)))
Cu(moveleft(Bc(x0))) → moveleft(Bc(Au(x0)))
Ct(moveleft(Bc(x0))) → moveleft(Bc(At(x0)))
Cc(moveleft(Bc(x0))) → moveleft(Bc(Ac(x0)))
cut(moveleft(Br(x0))) → Dr(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bn(x0))) → Dn(cut(goright(x0)))
cut(moveleft(Bb(x0))) → Db(cut(goright(x0)))
cut(moveleft(Bu(x0))) → Du(cut(goright(x0)))
cut(moveleft(Bt(x0))) → Dt(cut(goright(x0)))
cut(moveleft(Bc(x0))) → Dc(cut(goright(x0)))
goright(Ar(x0)) → Cr(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(An(x0)) → Cn(goright(x0))
goright(Ab(x0)) → Cb(goright(x0))
goright(Au(x0)) → Cu(goright(x0))
goright(At(x0)) → Ct(goright(x0))
goright(Ac(x0)) → Cc(goright(x0))
goright(wait(r(x0))) → moveleft(Br(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(n(x0))) → moveleft(Bn(wait(x0)))
goright(wait(b(x0))) → moveleft(Bb(wait(x0)))
goright(wait(u(x0))) → moveleft(Bu(wait(x0)))
goright(wait(t(x0))) → moveleft(Bt(wait(x0)))
goright(wait(c(x0))) → moveleft(Bc(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cr(finish(x0)) → finish(r(x0))
Cs(finish(x0)) → finish(s(x0))
Cn(finish(x0)) → finish(n(x0))
Cb(finish(x0)) → finish(b(x0))
Cu(finish(x0)) → finish(u(x0))
Ct(finish(x0)) → finish(t(x0))
Cc(finish(x0)) → finish(c(x0))
cut(finish(x0)) → finish2(x0)
Dr(finish2(x0)) → finish2(r(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dn(finish2(x0)) → finish2(n(x0))
Db(finish2(x0)) → finish2(b(x0))
Du(finish2(x0)) → finish2(u(x0))
Dt(finish2(x0)) → finish2(t(x0))
Dc(finish2(x0)) → finish2(c(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(r(r(x0))) → begin(s(r(x0)))
rewrite(r(s(x0))) → begin(s(r(x0)))
rewrite(r(n(x0))) → begin(s(r(x0)))
rewrite(r(b(x0))) → begin(u(s(b(x0))))
rewrite(r(u(x0))) → begin(u(r(x0)))
rewrite(s(u(x0))) → begin(u(s(x0)))
rewrite(n(u(x0))) → begin(u(n(x0)))
rewrite(t(r(u(x0)))) → begin(t(c(r(x0))))
rewrite(t(s(u(x0)))) → begin(t(c(r(x0))))
rewrite(t(n(u(x0)))) → begin(t(c(r(x0))))
rewrite(c(u(x0))) → begin(u(c(x0)))
rewrite(c(s(x0))) → begin(s(c(x0)))
rewrite(c(r(x0))) → begin(r(c(x0)))
rewrite(c(n(x0))) → begin(n(c(x0)))
rewrite(c(n(x0))) → begin(n(x0))