+nlemma eq_to_eqasttype : ∀t1,t2.t1 = t2 → eq_asttype t1 t2 = true.
+ #t1;
+ napply (ast_type_index … t1);
+ ##[ ##1: #b1; #t2; ncases t2;
+ ##[ ##1: #b2; #H; nrewrite > (asttype_destruct_base_base … H);
+ nchange with ((eq_astbasetype b2 b2) = true);
+ nrewrite > (eq_to_eqastbasetype b2 b2 (refl_eq …));
+ napply refl_eq
+ ##| ##2: #st2; #n2; #H; ndestruct (*napply (asttype_destruct … H)*)
+ ##| ##3: #nl2; #H; ndestruct (*napply (asttype_destruct … H)*)
+ ##]
+ ##| ##2: #st1; #n1; #H; #t2; ncases t2;
+ ##[ ##2: #st2; #n2; #H1; nchange with (((eq_asttype st1 st2)⊗(eq_nat n1 n2)) = true);
+ nrewrite > (H st2 (asttype_destruct_array_array_1 … H1));
+ nrewrite > (eq_to_eqnat n1 n2 (asttype_destruct_array_array_2 … H1));
+ nnormalize;
+ napply refl_eq
+ ##| ##1: #b2; #H1; ndestruct (*napply (asttype_destruct … H1)*)
+ ##| ##3: #nl2; #H1; ndestruct (*napply (asttype_destruct … H1)*)
+ ##]
+ ##| ##3: #hh1; #H; #t2; ncases t2;
+ ##[ ##3: #nl2; ncases nl2;
+ ##[ ##1: #hh2; #H1; nchange with ((eq_asttype hh1 hh2) = true);
+ nrewrite > (H hh2 (nelist_destruct_nil_nil ? hh1 hh2 (asttype_destruct_struct_struct … H1)));
+ napply refl_eq
+ ##| ##2: #hh2; #ll2; #H1;
+ (* !!! ndestruct non va *)
+ nelim (nelist_destruct_nil_cons ? hh1 hh2 ll2 (asttype_destruct_struct_struct … H1))
+ ##]
+ ##| ##1: #b2; #H1; ndestruct (*napply (asttype_destruct … H1)*)
+ ##| ##2: #st2; #n2; #H1; ndestruct (*napply (asttype_destruct … H1)*)
+ ##]
+ ##| ##4: #hh1; #ll1; #H; #H1; #t2; ncases t2;
+ ##[ ##3: #nl2; ncases nl2;
+ ##[ ##1: #hh2; #H2;
+ (* !!! ndestruct non va *)
+ nelim (nelist_destruct_cons_nil ? hh1 hh2 ll1 (asttype_destruct_struct_struct … H2))
+ ##| ##2: #hh2; #ll2; #H2; nchange with (((eq_asttype hh1 hh2)⊗(bfold_right_neList2 ? (λx,y.eq_asttype x y) ll1 ll2)) = true);
+ nrewrite > (H hh2 (nelist_destruct_cons_cons_1 … (asttype_destruct_struct_struct … H2)));
+ nrewrite > (eq_to_eqasttype_aux1 ll1 ll2 (H1 (AST_TYPE_STRUCT ll2) ?));
+ ##[ ##1: nnormalize; napply refl_eq
+ ##| ##2: nrewrite > (nelist_destruct_cons_cons_2 … (asttype_destruct_struct_struct … H2));
+ napply refl_eq
+ ##]
+ ##]
+ ##| ##1: #b2; #H2; ndestruct (*napply (asttype_destruct … H2)*)
+ ##| ##2: #st2; #n2; #H2; ndestruct (*napply (asttype_destruct … H2)*)
+ ##]
+ ##]
+nqed.
+
+nlemma neqasttype_to_neq : ∀n1,n2.eq_asttype n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_asttype n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqasttype n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
+
+nlemma eqasttype_to_eq : ∀t1,t2.eq_asttype t1 t2 = true → t1 = t2.
+ #t1;
+ napply (ast_type_index … t1);