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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/notation/functions/voidstar_2.ma".
16 include "basic_2/syntax/lenv.ma".
18 (* EXTENSION OF A LOCAL ENVIRONMENT WITH EXCLUSION BINDERS ******************)
20 rec definition voids (L:lenv) (n:nat) on n: lenv ≝ match n with
21 [ O ⇒ L | S m ⇒ (voids L m).ⓧ ].
23 interpretation "extension with exclusion binders (local environment)"
24 'VoidStar n L = (voids L n).
26 (* Basic properties *********************************************************)
28 lemma voids_zero: ∀L. L = ⓧ*[0]L.
31 lemma voids_succ: ∀L,n. (ⓧ*[n]L).ⓧ = ⓧ*[⫯n]L.
34 (* Advanced properties ******************************************************)
36 lemma voids_next: ∀n,L. ⓧ*[n](L.ⓧ) = ⓧ*[⫯n]L.
40 (* Basic inversion lemmas ***************************************************)
42 lemma voids_atom_inv: ∀K,n. ⓧ*[n]K = ⋆ → ∧∧ ⋆ = K & 0 = n.
43 #K * /2 width=1 by conj/
44 #n <voids_succ #H destruct
47 lemma voids_pair_inv: ∀I,K1,K2,V,n. ⓧ*[n]K1 = K2.ⓑ{I}V →
48 ∧∧ K2.ⓑ{I}V = K1 & 0 = n.
49 #I #K1 #K2 #V * /2 width=1 by conj/
50 #n <voids_succ #H destruct
53 (* Advanced inversion lemmas ************************************************)
55 lemma voids_inv_atom_sn: ∀n1,K2,n2. ⓧ*[n1]⋆ = ⓧ*[n2]K2 →
56 ∧∧ ⓧ*[n1-n2]⋆ = K2 & n2 ≤ n1.
58 [ #K2 <voids_zero * /2 width=1 by conj/
59 #n1 <voids_succ #H destruct
61 [ <voids_zero #H destruct /2 width=1 by conj/
62 | #n2 <voids_succ <voids_succ >minus_S_S #H
63 elim (destruct_lbind_lbind_aux … H) -H #HK #_ (**) (* destruct lemma needed *)
64 elim (IH … HK) -IH -HK #H #Hn destruct /3 width=1 by conj, le_S_S/
69 lemma voids_inv_pair_sn: ∀I,V,n1,K1,K2,n2. ⓧ*[n1]K1.ⓑ{I}V = ⓧ*[n2]K2 →
70 ∧∧ ⓧ*[n1-n2]K1.ⓑ{I}V = K2 & n2 ≤ n1.
72 [ #K1 #K2 <voids_zero * /2 width=1 by conj/
73 #n1 <voids_succ #H destruct
75 [ <voids_zero #H destruct /2 width=1 by conj/
76 | #n2 <voids_succ <voids_succ >minus_S_S #H
77 elim (destruct_lbind_lbind_aux … H) -H #HK #_ (**) (* destruct lemma needed *)
78 elim (IH … HK) -IH -HK #H #Hn destruct /3 width=1 by conj, le_S_S/
83 (* Main inversion properties ************************************************)
85 theorem voids_inj: ∀n. injective … (λL. ⓧ*[n]L).
88 <voids_succ <voids_succ #H
89 elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)