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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 "static_2/relocation/lifts_tdeq.ma".
16 include "static_2/static/rex_drops.ma".
17 include "static_2/static/rdeq.ma".
19 (* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
21 (* Properties with generic slicing for local environments *******************)
23 lemma rdeq_lifts_sn: f_dedropable_sn cdeq.
24 /3 width=5 by rex_liftable_dedropable_sn, tdeq_lifts_sn/ qed-.
26 (* Inversion lemmas with generic slicing for local environments *************)
28 lemma rdeq_inv_lifts_sn: f_dropable_sn cdeq.
29 /2 width=5 by rex_dropable_sn/ qed-.
31 lemma rdeq_inv_lifts_dx: f_dropable_dx cdeq.
32 /2 width=5 by rex_dropable_dx/ qed-.
34 lemma rdeq_inv_lifts_bi: ∀L1,L2,U. L1 ≛[U] L2 → ∀b,f. 𝐔⦃f⦄ →
35 ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
36 ∀T. ⬆*[f] T ≘ U → K1 ≛[T] K2.
37 /2 width=10 by rex_inv_lifts_bi/ qed-.
39 lemma rdeq_inv_lref_pair_sn: ∀L1,L2,i. L1 ≛[#i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
40 ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ≛[V1] K2 & V1 ≛ V2.
41 /2 width=3 by rex_inv_lref_pair_sn/ qed-.
43 lemma rdeq_inv_lref_pair_dx: ∀L1,L2,i. L1 ≛[#i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
44 ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ≛[V1] K2 & V1 ≛ V2.
45 /2 width=3 by rex_inv_lref_pair_dx/ qed-.
47 lemma rdeq_inv_lref_pair_bi (L1) (L2) (i):
49 ∀I1,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I1}V1 →
50 ∀I2,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I2}V2 →
51 ∧∧ K1 ≛[V1] K2 & V1 ≛ V2 & I1 = I2.
52 /2 width=6 by rex_inv_lref_pair_bi/ qed-.