(* *)
(**************************************************************************)
-include "basic_2/notation/functions/voidstar_2.ma".
+include "basic_2/notation/relations/rvoidstar_4.ma".
include "basic_2/syntax/lenv.ma".
-(* EXTENSION OF A LOCAL ENVIRONMENT WITH EXCLUSION BINDERS ******************)
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
-rec definition voids (L:lenv) (n:nat) on n: lenv ≝ match n with
-[ O ⇒ L | S m ⇒ (voids L m).ⓧ ].
+inductive voids: bi_relation nat lenv ≝
+| voids_atom : voids 0 (⋆) 0 (⋆)
+| voids_pair_sn: ∀I1,I2,K1,K2,V1,n. voids n K1 n K2 →
+ voids 0 (K1.ⓑ{I1}V1) 0 (K2.ⓘ{I2})
+| voids_pair_dx: ∀I1,I2,K1,K2,V2,n. voids n K1 n K2 →
+ voids 0 (K1.ⓘ{I1}) 0 (K2.ⓑ{I2}V2)
+| voids_void_sn: ∀K1,K2,n1,n2. voids n1 K1 n2 K2 →
+ voids (⫯n1) (K1.ⓧ) n2 K2
+| voids_void_dx: ∀K1,K2,n1,n2. voids n1 K1 n2 K2 →
+ voids n1 K1 (⫯n2) (K2.ⓧ)
+.
-interpretation "extension with exclusion binders (local environment)"
- 'VoidStar n L = (voids L n).
+interpretation "equivalence up to exclusion binders (local environment)"
+ 'RVoidStar n1 L1 n2 L2 = (voids n1 L1 n2 L2).
(* Basic properties *********************************************************)
+lemma voids_refl: ∀L. ∃n. ⓧ*[n]L ≋ ⓧ*[n]L.
+#L elim L -L /2 width=2 by ex_intro, voids_atom/
+#L #I * #n #IH cases I -I /3 width=2 by ex_intro, voids_pair_dx/
+* /4 width=2 by ex_intro, voids_void_sn, voids_void_dx/
+qed-.
+
+lemma voids_sym: bi_symmetric … voids.
+#n1 #n2 #L1 #L2 #H elim H -L1 -L2 -n1 -n2
+/2 width=2 by voids_atom, voids_pair_sn, voids_pair_dx, voids_void_sn, voids_void_dx/
+qed-.
+
+(*
+
+
+
+
lemma voids_zero: ∀L. L = ⓧ*[0]L.
// qed.
elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)
/2 width=1 by/
qed-.
+*)
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