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15 include "basic_2/notation/relations/extlrsubeq_2.ma".
16 include "basic_2/relocation/ldrop.ma".
18 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
20 inductive lsuby: relation lenv ≝
21 | lsuby_atom: ∀L. lsuby L (⋆)
22 | lsuby_pair: ∀I1,I2,L1,L2,V. lsuby L1 L2 → lsuby (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
26 "local environment refinement (extended substitution)"
27 'ExtLRSubEq L1 L2 = (lsuby L1 L2).
29 (* Basic properties *********************************************************)
31 lemma lsuby_refl: ∀L. L ⊑× L.
32 #L elim L -L // /2 width=1/
35 (* Basic inversion lemmas ***************************************************)
37 fact lsuby_inv_atom1_aux: ∀L1,L2. L1 ⊑× L2 → L1 = ⋆ → L2 = ⋆.
39 #I1 #I2 #L1 #L2 #V #_ #H destruct
42 lemma lsuby_inv_atom1: ∀L2. ⋆ ⊑× L2 → L2 = ⋆.
43 /2 width=3 by lsuby_inv_atom1_aux/ qed-.
45 fact lsuby_inv_pair1_aux: ∀L1,L2. L1 ⊑× L2 → ∀J1,K1,W. L1 = K1.ⓑ{J1}W →
46 L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊑× K2 & L2 = K2.ⓑ{I2}W.
48 [ #L #J1 #K1 #W #H destruct /2 width=1 by or_introl/
49 | #I1 #I2 #L1 #L2 #V #HL12 #J1 #K1 #W #H destruct /3 width=4 by ex2_2_intro, or_intror/
53 lemma lsuby_inv_pair1: ∀I1,K1,L2,W. K1.ⓑ{I1}W ⊑× L2 →
54 L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊑× K2 & L2 = K2.ⓑ{I2}W.
55 /2 width=4 by lsuby_inv_pair1_aux/ qed-.
57 (* Basic forward lemmas *****************************************************)
59 lemma lsuby_fwd_length: ∀L1,L2. L1 ⊑× L2 → |L2| ≤ |L1|.
60 #L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
63 lemma lsuby_fwd_ldrop2_pair: ∀L1,L2. L1 ⊑× L2 →
64 ∀I2,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I2}W →
65 ∃∃I1,K1. K1 ⊑× K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
66 #L1 #L2 #H elim H -L1 -L2
68 elim (ldrop_inv_atom1 … H) -H #H destruct
69 | #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #i #H
70 elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
71 [ /3 width=4 by ldrop_pair, ex2_2_intro/
72 | elim (IHL12 … HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/