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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 "ground/xoa/ex_4_3.ma".
16 include "basic_2A/static/lsubr.ma".
18 (* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
20 (* Auxiliary inversion lemmas ***********************************************)
22 fact lsubr_inv_pair1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
24 | ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓑ{I}X
25 | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
28 [ #L #J #K1 #X #H destruct /2 width=1 by or3_intro0/
29 | #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by or3_intro1, ex2_intro/
30 | #L1 #L2 #V #W #HL12 #J #K1 #X #H destruct /3 width=6 by or3_intro2, ex4_3_intro/
34 lemma lsubr_inv_pair1: ∀I,K1,L2,X. K1.ⓑ{I}X ⫃ L2 →
36 | ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓑ{I}X
37 | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
39 /2 width=3 by lsubr_inv_pair1_aux/ qed-.
41 (* Main properties **********************************************************)
43 theorem lsubr_trans: Transitive … lsubr.
44 #L1 #L #H elim H -L1 -L
46 lapply (lsubr_inv_atom1 … H) -H //
47 | #I #L1 #L #V #_ #IHL1 #X #H
48 elim (lsubr_inv_pair1 … H) -H // *
49 #L2 [2: #V2 #W2 ] #HL2 #H1 [ #H2 #H3 ] destruct /3 width=1 by lsubr_pair, lsubr_beta/
50 | #L1 #L #V1 #W #_ #IHL1 #X #H
51 elim (lsubr_inv_abst1 … H) -H // *
52 #L2 #HL2 #H destruct /3 width=1 by lsubr_beta/