  We propose a general framework for computer-assisted verification of the
presence of symmetry breaking, period-tupling and touch-and-go bifurcations of
symmetric periodic orbits for reversible maps. The framework is then adopted to
Poincar\'e maps in reversible autonomous Hamiltonian systems.
  In order to justify the applicability of the method, we study bifurcations of
halo orbits in the Circular Restricted Three Body Problem. We give a
computer-assisted proof of the existence of wide branches of halo orbits
bifurcating from $L_{1,2,3}$-Lyapunov families and for wide range of mass
parameter. For two physically relevant mass parameters we prove, that halo
orbits undergo multiple period doubling, quadrupling and third-order
touch-and-go bifurcations.
