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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 *)
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11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/substitution/lsubr.ma".
17 (* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
19 fact lsubr_inv_abbr1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W. L1 = K1.ⓓW →
21 | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓓW
22 | ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
24 [ #L #K1 #W #H destruct /2 width=1/
25 | #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3/
26 | #I #L1 #L2 #V1 #V2 #HL12 #K1 #W #H destruct /3 width=4/
30 lemma lsubr_inv_abbr1: ∀K1,L2,W. K1.ⓓW ⊑ L2 →
32 | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓓW
33 | ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
34 /2 width=3 by lsubr_inv_abbr1_aux/ qed-.
36 fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W1. L1 = K1.ⓛW1 →
38 ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
40 [ #L #K1 #W1 #H destruct /2 width=1/
41 | #L1 #L2 #V #_ #K1 #W1 #H destruct
42 | #I #L1 #L2 #V1 #V2 #HL12 #K1 #W1 #H destruct /3 width=4/
46 lemma lsubr_inv_abst1: ∀K1,L2,W1. K1.ⓛW1 ⊑ L2 →
48 ∃∃K2,W2. K1 ⊑ K2 & L2 = K2.ⓛW2.
49 /2 width=4 by lsubr_inv_abst1_aux/ qed-.
51 (* Main properties **********************************************************)
53 theorem lsubr_trans: Transitive … lsubr.
54 #L1 #L #H elim H -L1 -L
56 lapply (lsubr_inv_atom1 … H) -H //
57 | #L1 #L #V #_ #IHL1 #X #H
58 elim (lsubr_inv_abbr1 … H) -H // *
59 #L2 [2: #V2 ] #HL2 #H destruct /3 width=1/
60 | #I #L1 #L #V1 #V #_ #IHL1 #X #H
61 elim (lsubr_inv_abst1 … H) -H // *
62 #L2 #V2 #HL2 #H destruct /3 width=1/