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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/notation/relations/lrsubeq_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 'LRSubEq L1 L2 = (lsuby L1 L2).
29 (* Basic properties *********************************************************)
31 lemma lsuby_refl: ∀L. L ⊆ L.
32 #L elim L -L /2 width=1 by lsuby_pair/
35 lemma lsuby_sym: ∀L1,L2. L1 ⊆ L2 → |L1| = |L2| → L2 ⊆ L1.
36 #L1 #L2 #H elim H -L1 -L2
37 [ #L1 #H >(length_inv_zero_dx … H) -L1 //
38 | #I1 #I2 #L1 #L2 #V #_ #IHL12 #H lapply (injective_plus_l … H) -H
39 /3 width=1 by lsuby_pair/
43 (* Basic inversion lemmas ***************************************************)
45 fact lsuby_inv_atom1_aux: ∀L1,L2. L1 ⊆ L2 → L1 = ⋆ → L2 = ⋆.
47 #I1 #I2 #L1 #L2 #V #_ #H destruct
50 lemma lsuby_inv_atom1: ∀L2. ⋆ ⊆ L2 → L2 = ⋆.
51 /2 width=3 by lsuby_inv_atom1_aux/ qed-.
53 fact lsuby_inv_pair1_aux: ∀L1,L2. L1 ⊆ L2 → ∀J1,K1,W. L1 = K1.ⓑ{J1}W →
54 L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆ K2 & L2 = K2.ⓑ{I2}W.
56 [ #L #J1 #K1 #W #H destruct /2 width=1 by or_introl/
57 | #I1 #I2 #L1 #L2 #V #HL12 #J1 #K1 #W #H destruct /3 width=4 by ex2_2_intro, or_intror/
61 lemma lsuby_inv_pair1: ∀I1,K1,L2,W. K1.ⓑ{I1}W ⊆ L2 →
62 L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆ K2 & L2 = K2.ⓑ{I2}W.
63 /2 width=4 by lsuby_inv_pair1_aux/ qed-.
65 fact lsuby_inv_pair2_aux: ∀L1,L2. L1 ⊆ L2 → ∀J2,K2,W. L2 = K2.ⓑ{J2}W →
66 ∃∃I1,K1. K1 ⊆ K2 & L1 = K1.ⓑ{I1}W.
68 [ #L #J2 #K2 #W #H destruct
69 | #I1 #I2 #L1 #L2 #V #HL12 #J2 #K2 #W #H destruct /2 width=4 by ex2_2_intro/
73 lemma lsuby_inv_pair2: ∀I2,L1,K2,W. L1 ⊆ K2.ⓑ{I2}W →
74 ∃∃I1,K1. K1 ⊆ K2 & L1 = K1.ⓑ{I1}W.
75 /2 width=4 by lsuby_inv_pair2_aux/ qed-.
77 (* Basic forward lemmas *****************************************************)
79 lemma lsuby_fwd_length: ∀L1,L2. L1 ⊆ L2 → |L2| ≤ |L1|.
80 #L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
83 lemma lsuby_ldrop_trans: ∀L1,L2. L1 ⊆ L2 →
84 ∀I2,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
85 ∃∃I1,K1. K1 ⊆ K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
86 #L1 #L2 #H elim H -L1 -L2
87 [ #L #J2 #K2 #W #s #i #H
88 elim (ldrop_inv_atom1 … H) -H #H destruct
89 | #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #s #i #H
90 elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
91 [ /3 width=4 by ldrop_pair, ex2_2_intro/
92 | elim (IHL12 … HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_drop_lt, ex2_2_intro/