End cs_preserve.
Ltac make_cs_preserve cs_from cs_to :=
let coeq0 := eval cbv in (cs_preserve_env cs_from cs_to) in
let Hcoeq0 := constr: (eq_refl: coeq0 = cs_preserve_env cs_from cs_to) in
let coeq0_consistent := constr: (cs_preserve_env_consisent cs_from cs_to coeq0 Hcoeq0) in
let coeq0_complete := constr: (cs_preserve_completeness cs_from cs_to coeq0 Hcoeq0) in
refine (  {| coeq := coeq0 ;
coeq_consistent := coeq0_consistent;
coeq_complete := coeq0_complete |}).

Lemma complete_legal_cosu_type_change_composite {cs_from cs_to} {CCE: change_composite_env cs_from cs_to}: forall (t: type),
cs_preserve_type cs_from cs_to (coeq _ _) t = true ->
@complete_legal_cosu_type (@cenv_cs cs_from) t = @complete_legal_cosu_type (@cenv_cs cs_to) t.