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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/substitution/fqus_alt.ma".
16 include "basic_2/substitution/lleq_ext.ma".
18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
20 (* Properties on supclosure and derivatives *********************************)
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 ⋕[U, 0] 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 #I1 #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 //
47 lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃⸮ ⦃G2, K2, U⦄ →
49 ∃∃K1. ⦃G1, L1, T⦄ ⊃⸮ ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
50 #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H
51 [ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
52 | * #HG #HL #HT destruct /2 width=3 by ex2_intro/
56 lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃+ ⦃G2, K2, U⦄ →
58 ∃∃K1. ⦃G1, L1, T⦄ ⊃+ ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
59 #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
60 [ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2
61 /3 width=3 by fqu_fqup, ex2_intro/
62 | #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2
63 #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K
64 /3 width=5 by fqup_strap1, ex2_intro/
68 lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊃* ⦃G2, K2, U⦄ →
70 ∃∃K1. ⦃G1, L1, T⦄ ⊃* ⦃G2, K1, U⦄ & K1 ⋕[U, 0] K2.
71 #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H
72 [ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
73 | * #HG #HL #HT destruct /2 width=3 by ex2_intro/