- work in progress proceeds for the new definition of voids ...

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / syntax / voids.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma
index 766eefd5c2ccfa8c1027b62cfb4b7ded9bffc94e..ca5c4fc58a691c8537e4c7d55a9780147e41743a 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma
@@ -12,37 +12,52 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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_zero: ∀L. L = ⓧ*[0]L.
-// qed.
-
-lemma voids_succ: ∀L,n. (ⓧ*[n]L).ⓧ = ⓧ*[⫯n]L.
-// qed.
-
-(* Advanced properties ******************************************************)
-
-lemma voids_next: ∀n,L. ⓧ*[n](L.ⓧ) = ⓧ*[⫯n]L.
-#n elim n -n //
-qed.
+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-.
 
-(* Main inversion properties ************************************************)
+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-.
 
-theorem voids_inj: ∀n. injective … (λL. ⓧ*[n]L).
-#n elim n -n //
-#n #IH #L1 #L2
-<voids_succ <voids_succ #H
-elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)
-/2 width=1 by/
+(* Advanced Inversion lemmas ************************************************)
+
+fact voids_inv_void_dx_aux: ∀L1,L2,n1,n2. ⓧ*[n1]L1 ≋ ⓧ*[n2]L2 →
+                            ∀K2,m2. n2 = ⫯m2 → L2 = K2.ⓧ → ⓧ*[n1]L1 ≋ ⓧ*[m2]K2.
+#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2
+[ #K2 #m2 #H destruct
+| #I1 #I2 #L1 #L2 #V #n #_ #_ #K2 #m2 #H destruct
+| #I1 #I2 #L1 #L2 #V #n #_ #_ #K2 #m2 #H destruct
+| #L1 #L2 #n1 #n2 #_ #IH #K2 #m2 #H1 #H2 destruct
+  /3 width=1 by voids_void_sn/
+| #L1 #L2 #n1 #n2 #HL12 #_ #K2 #m2 #H1 #H2 destruct //
+]
 qed-.
+
+lemma voids_inv_void_dx: ∀L1,L2,n1,n2. ⓧ*[n1]L1 ≋ ⓧ*[⫯n2]L2.ⓧ → ⓧ*[n1]L1 ≋ ⓧ*[n2]L2.
+/2 width=5 by voids_inv_void_dx_aux/ qed-.