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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/relocation/lifts_tdeq.ma".
16 include "basic_2/static/lfxs_drops.ma".
17 include "basic_2/static/lfdeq_fqup.ma".
19 (* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
21 (* Properties with generic slicing for local environments *******************)
23 (* Basic_2A1: includes: lleq_lift_le lleq_lift_ge *)
24 lemma lfdeq_lifts_sn: ∀h,o. dedropable_sn (cdeq h o).
25 /3 width=5 by lfxs_liftable_dedropable_sn, tdeq_lifts_sn/ qed-.
27 (* Inversion lemmas with generic slicing for local environments *************)
29 (* Basic_2A1: restricts: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
30 lemma lfdeq_inv_lifts_sn: ∀h,o. dropable_sn (cdeq h o).
31 /2 width=5 by lfxs_dropable_sn/ qed-.
33 (* Note: missing in basic_2A1 *)
34 lemma lfdeq_inv_lifts_dx: ∀h,o. dropable_dx (cdeq h o).
35 /2 width=5 by lfxs_dropable_dx/ qed-.
37 (* Note: missing in basic_2A1 *)
38 lemma lfdeq_inv_lifts_bi: ∀h,o,L1,L2,U. L1 ≡[h, o, U] L2 →
39 ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 →
40 ∀T. ⬆*[i] T ≡ U → K1 ≡[h, o, T] K2.
41 /2 width=8 by lfxs_inv_lifts_bi/ qed-.
43 lemma lfdeq_inv_lref_sn: ∀h,o,L1,L2,i. L1 ≡[h, o, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 →
44 ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & K1 ≡[h, o, V1] K2 & V1 ≡[h, o] V2.
45 /2 width=3 by lfxs_inv_lref_sn/ qed-.
47 lemma lfdeq_inv_lref_dx: ∀h,o,L1,L2,i. L1 ≡[h, o, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 →
48 ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & K1 ≡[h, o, V1] K2 & V1 ≡[h, o] V2.
49 /2 width=3 by lfxs_inv_lref_dx/ qed-.