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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/computation/csx_lpxs.ma".
16 include "basic_2/computation/lcosx_cpxs.ma".
18 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
20 (* Advanced properties ******************************************************)
22 lemma lsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,d. d ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
23 ∀K2. G ⊢ ⋕⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g] K2 →
24 ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L2.
25 #h #g #I #G #K1 #V #i #d #Hdi #H @(csx_ind_alt … H) -V
26 #V0 #_ #IHV0 #K2 #H @(lsx_ind … H) -K2
27 #K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @lsx_intro
28 #L2 #HL02 #HnL02 elim (lpxs_ldrop_conf … HLK0 … HL02) -HL02
29 #Y #H #HLK2 elim (lpxs_inv_pair1 … H) -H
30 #K2 #V2 #HK02 #HV02 #H destruct
31 lapply (lpxs_trans … HK10 … HK02) #HK12
32 elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 -HK10 | -IHK0 -HnL02 -HLK0 ]
33 [ /4 width=8 by lleq_lref/
34 | @(IHV0 … HnV02 … HK12 … HLK2) -IHV0 -HnV02 -HK12 -HLK2
35 /3 width=4 by lsx_cpxs_trans_O, lsx_lpxs_trans, lpxs_cpxs_trans/ (**) (* full auto too slow *)
39 lemma lsx_lref_be: ∀h,g,I,G,K,V,i,d. d ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, g] V →
40 G ⊢ ⋕⬊*[h, g, V, 0] K →
41 ∀L. ⇩[i] L ≡ K.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L.
42 /2 width=8 by lsx_lref_be_lpxs/ qed.
44 (* Main properties **********************************************************)
46 theorem csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀d. G ⊢ ⋕⬊*[h, g, T, d] L.
47 #h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
48 #Z #Y #X #IH #G #L * * //
49 [ #i #HG #HL #HT #H #d destruct
50 elim (lt_or_ge i (|L|)) /2 width=1 by lsx_lref_free/
51 elim (ylt_split i d) /2 width=1 by lsx_lref_skip/
52 #Hdi #Hi elim (ldrop_O1_lt … Hi) -Hi
53 #I #K #V #HLK lapply (csx_inv_lref_bind … HLK … H) -H
54 /4 width=6 by lsx_lref_be, fqup_lref/
55 | #a #I #V #T #HG #HL #HT #H #d destruct
56 elim (csx_fwd_bind … H) -H /3 width=1 by lsx_bind/
57 | #I #V #T #HG #HL #HT #H #d destruct
58 elim (csx_fwd_flat … H) -H /3 width=1 by lsx_flat/