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(foo(x0)) → rotate(cut(Cfoo(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(bar(x0)) → rotate(cut(Cbar(guess(x0))))
guess(foo(x0)) → Cfoo(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(bar(x0)) → Cbar(guess(x0))
guess(foo(x0)) → moveleft(Bfoo(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(bar(x0)) → moveleft(Bbar(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cfoo(moveleft(Bfoo(x0))) → moveleft(Bfoo(Afoo(x0)))
C0(moveleft(Bfoo(x0))) → moveleft(Bfoo(A0(x0)))
Cs(moveleft(Bfoo(x0))) → moveleft(Bfoo(As(x0)))
Cp(moveleft(Bfoo(x0))) → moveleft(Bfoo(Ap(x0)))
Cbar(moveleft(Bfoo(x0))) → moveleft(Bfoo(Abar(x0)))
Cfoo(moveleft(B0(x0))) → moveleft(B0(Afoo(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
Cbar(moveleft(B0(x0))) → moveleft(B0(Abar(x0)))
Cfoo(moveleft(Bs(x0))) → moveleft(Bs(Afoo(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
Cbar(moveleft(Bs(x0))) → moveleft(Bs(Abar(x0)))
Cfoo(moveleft(Bp(x0))) → moveleft(Bp(Afoo(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
Cbar(moveleft(Bp(x0))) → moveleft(Bp(Abar(x0)))
Cfoo(moveleft(Bbar(x0))) → moveleft(Bbar(Afoo(x0)))
C0(moveleft(Bbar(x0))) → moveleft(Bbar(A0(x0)))
Cs(moveleft(Bbar(x0))) → moveleft(Bbar(As(x0)))
Cp(moveleft(Bbar(x0))) → moveleft(Bbar(Ap(x0)))
Cbar(moveleft(Bbar(x0))) → moveleft(Bbar(Abar(x0)))
cut(moveleft(Bfoo(x0))) → Dfoo(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(Bbar(x0))) → Dbar(cut(goright(x0)))
goright(Afoo(x0)) → Cfoo(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(Abar(x0)) → Cbar(goright(x0))
goright(wait(foo(x0))) → moveleft(Bfoo(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(bar(x0))) → moveleft(Bbar(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cfoo(finish(x0)) → finish(foo(x0))
C0(finish(x0)) → finish(0(x0))
Cs(finish(x0)) → finish(s(x0))
Cp(finish(x0)) → finish(p(x0))
Cbar(finish(x0)) → finish(bar(x0))
cut(finish(x0)) → finish2(x0)
Dfoo(finish2(x0)) → finish2(foo(x0))
D0(finish2(x0)) → finish2(0(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dp(finish2(x0)) → finish2(p(x0))
Dbar(finish2(x0)) → finish2(bar(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(foo(0(x0))) → begin(0(s(p(p(p(s(s(s(p(s(x0)))))))))))
rewrite(foo(s(x0))) → begin(p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x0)))))))))))))))))))))))))))
rewrite(bar(0(x0))) → begin(0(p(s(s(s(x0))))))
rewrite(bar(s(x0))) → begin(p(s(p(p(s(s(foo(s(p(p(s(s(x0)))))))))))))
rewrite(p(p(s(x0)))) → begin(p(x0))
rewrite(p(s(x0))) → begin(x0)
rewrite(p(0(x0))) → begin(0(s(s(s(s(x0))))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(foo(x0)) → rotate(cut(Cfoo(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(s(x0)) → rotate(cut(Cs(guess(x0))))
begin(p(x0)) → rotate(cut(Cp(guess(x0))))
begin(bar(x0)) → rotate(cut(Cbar(guess(x0))))
guess(foo(x0)) → Cfoo(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(s(x0)) → Cs(guess(x0))
guess(p(x0)) → Cp(guess(x0))
guess(bar(x0)) → Cbar(guess(x0))
guess(foo(x0)) → moveleft(Bfoo(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(s(x0)) → moveleft(Bs(wait(x0)))
guess(p(x0)) → moveleft(Bp(wait(x0)))
guess(bar(x0)) → moveleft(Bbar(wait(x0)))
guess(end(x0)) → finish(end(x0))
Cfoo(moveleft(Bfoo(x0))) → moveleft(Bfoo(Afoo(x0)))
C0(moveleft(Bfoo(x0))) → moveleft(Bfoo(A0(x0)))
Cs(moveleft(Bfoo(x0))) → moveleft(Bfoo(As(x0)))
Cp(moveleft(Bfoo(x0))) → moveleft(Bfoo(Ap(x0)))
Cbar(moveleft(Bfoo(x0))) → moveleft(Bfoo(Abar(x0)))
Cfoo(moveleft(B0(x0))) → moveleft(B0(Afoo(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
Cs(moveleft(B0(x0))) → moveleft(B0(As(x0)))
Cp(moveleft(B0(x0))) → moveleft(B0(Ap(x0)))
Cbar(moveleft(B0(x0))) → moveleft(B0(Abar(x0)))
Cfoo(moveleft(Bs(x0))) → moveleft(Bs(Afoo(x0)))
C0(moveleft(Bs(x0))) → moveleft(Bs(A0(x0)))
Cs(moveleft(Bs(x0))) → moveleft(Bs(As(x0)))
Cp(moveleft(Bs(x0))) → moveleft(Bs(Ap(x0)))
Cbar(moveleft(Bs(x0))) → moveleft(Bs(Abar(x0)))
Cfoo(moveleft(Bp(x0))) → moveleft(Bp(Afoo(x0)))
C0(moveleft(Bp(x0))) → moveleft(Bp(A0(x0)))
Cs(moveleft(Bp(x0))) → moveleft(Bp(As(x0)))
Cp(moveleft(Bp(x0))) → moveleft(Bp(Ap(x0)))
Cbar(moveleft(Bp(x0))) → moveleft(Bp(Abar(x0)))
Cfoo(moveleft(Bbar(x0))) → moveleft(Bbar(Afoo(x0)))
C0(moveleft(Bbar(x0))) → moveleft(Bbar(A0(x0)))
Cs(moveleft(Bbar(x0))) → moveleft(Bbar(As(x0)))
Cp(moveleft(Bbar(x0))) → moveleft(Bbar(Ap(x0)))
Cbar(moveleft(Bbar(x0))) → moveleft(Bbar(Abar(x0)))
cut(moveleft(Bfoo(x0))) → Dfoo(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(Bs(x0))) → Ds(cut(goright(x0)))
cut(moveleft(Bp(x0))) → Dp(cut(goright(x0)))
cut(moveleft(Bbar(x0))) → Dbar(cut(goright(x0)))
goright(Afoo(x0)) → Cfoo(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(As(x0)) → Cs(goright(x0))
goright(Ap(x0)) → Cp(goright(x0))
goright(Abar(x0)) → Cbar(goright(x0))
goright(wait(foo(x0))) → moveleft(Bfoo(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(s(x0))) → moveleft(Bs(wait(x0)))
goright(wait(p(x0))) → moveleft(Bp(wait(x0)))
goright(wait(bar(x0))) → moveleft(Bbar(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
Cfoo(finish(x0)) → finish(foo(x0))
C0(finish(x0)) → finish(0(x0))
Cs(finish(x0)) → finish(s(x0))
Cp(finish(x0)) → finish(p(x0))
Cbar(finish(x0)) → finish(bar(x0))
cut(finish(x0)) → finish2(x0)
Dfoo(finish2(x0)) → finish2(foo(x0))
D0(finish2(x0)) → finish2(0(x0))
Ds(finish2(x0)) → finish2(s(x0))
Dp(finish2(x0)) → finish2(p(x0))
Dbar(finish2(x0)) → finish2(bar(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(foo(0(x0))) → begin(0(s(p(p(p(s(s(s(p(s(x0)))))))))))
rewrite(foo(s(x0))) → begin(p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x0)))))))))))))))))))))))))))
rewrite(bar(0(x0))) → begin(0(p(s(s(s(x0))))))
rewrite(bar(s(x0))) → begin(p(s(p(p(s(s(foo(s(p(p(s(s(x0)))))))))))))
rewrite(p(p(s(x0)))) → begin(p(x0))
rewrite(p(s(x0))) → begin(x0)
rewrite(p(0(x0))) → begin(0(s(s(s(s(x0))))))