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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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15 include "basic_2/multiple/llpx_sn_drop.ma".
16 include "basic_2/multiple/lleq.ma".
18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
20 (* Advanced properties ******************************************************)
22 lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
23 ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W.
24 /2 width=7 by llpx_sn_bind_repl_O/ qed-.
26 lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R →
27 ∀L1,L2,T,l. L1 ≡[T, l] L2 →
28 ∀L. llpx_sn R l T L2 L → llpx_sn R l T L1 L.
29 #R #HR #L1 #L2 #T #l #H @(lleq_ind … H) -L1 -L2 -T -l
30 [1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/
31 |4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/
32 | #I #L1 #L2 #K1 #K2 #V #l #i #Hli #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2
33 /3 width=11 by llpx_sn_lref/
34 | #a #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H
35 /3 width=1 by llpx_sn_bind/
36 | #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H
37 /3 width=1 by llpx_sn_flat/
41 lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R →
42 ∀L1,L2,T,l. L1 ≡[T, l] L2 →
43 ∀L. llpx_sn R l T L1 L → llpx_sn R l T L2 L.
44 /3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-.
46 (* Advanced inversion lemmas ************************************************)
48 lemma lleq_inv_lref_ge_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
49 ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V →
50 ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2.
51 #L1 #L2 #l #i #H #Hli #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2
52 /2 width=3 by ex2_intro/
55 lemma lleq_inv_lref_ge_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
56 ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V →
57 ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2.
58 #L1 #L2 #l #i #H #Hli #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1
59 /2 width=3 by ex2_intro/
62 lemma lleq_inv_lref_ge_bi: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
64 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
65 ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2.
66 /2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
68 lemma lleq_inv_lref_ge: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
69 ∀I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
71 #L1 #L2 #l #i #HL12 #Hli #I #K1 #K2 #V #HLK1 #HLK2
72 elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) //
75 lemma lleq_inv_S: ∀L1,L2,T,l. L1 ≡[T, l+1] L2 →
76 ∀I,K1,K2,V. ⬇[l] L1 ≡ K1.ⓑ{I}V → ⬇[l] L2 ≡ K2.ⓑ{I}V →
77 K1 ≡[V, 0] K2 → L1 ≡[T, l] L2.
78 /2 width=9 by llpx_sn_inv_S/ qed-.
80 (* Advanced forward lemmas **************************************************)
82 lemma lleq_fwd_lref_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
83 ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V →
85 ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i.
86 #L1 #L2 #l #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
87 [ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
90 lemma lleq_fwd_lref_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
91 ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V →
93 ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i.
94 #L1 #L2 #l #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
95 [ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/