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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/relocation/lleq_lleq.ma".
16 include "basic_2/relocation/fqu.ma".
18 (* SUPCLOSURE ***************************************************************)
20 (* Properties on lazy equivalence for local environments ********************)
22 lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃ ⦃G2, K2, U⦄ →
24 ∃∃K1. ⦃G1, L1, T⦄ ⊃ ⦃G2, K1, U⦄ & K1 ⋕[0, U] K2.
25 #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
26 [ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H //
27 #K1 #H1 #H2 lapply (ldrop_inv_O2 … H1) -H1
28 #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
29 | * [ #a ] #I #G #L2 #V #T #L1 #H
30 [ elim (lleq_inv_bind … H)
31 | elim (lleq_inv_flat … H)
33 /2 width=3 by fqu_pair_sn, ex2_intro/
34 | #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H
35 #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
36 | #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H
37 /2 width=3 by fqu_flat_dx, ex2_intro/
38 | #G #L2 #K2 #T #U #e #HLK2 #HTU #L1 #HL12
39 elim (ldrop_O1_le (e+1) L1)
40 [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
41 | lapply (ldrop_fwd_length_le2 … HLK2) -K2
42 lapply (lleq_fwd_length … HL12) -T -U //