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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/notation/relations/lazysn_5.ma".
16 include "basic_2/relocation/lleq.ma".
17 include "basic_2/computation/lpxs.ma".
19 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
21 definition lsx: ∀h. sd h → relation3 term genv lenv ≝
22 λh,g,T,G. SN … (lpxs h g G) (lleq 0 T).
25 "extended strong normalization (local environment)"
26 'LazySN h g T G L = (lsx h g T G L).
28 (* Basic eliminators ********************************************************)
30 lemma lsx_ind: ∀h,g,T,G. ∀R:predicate lenv.
31 (∀L1. G ⊢ ⋕⬊*[h, g, T] L1 →
32 (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[0, T] L2 → ⊥) → R L2) →
35 ∀L. G ⊢ ⋕⬊*[h, g, T] L → R L.
36 #h #g #T #G #R #H0 #L1 #H elim H -L1
37 /5 width=1 by lleq_sym, SN_intro/
40 (* Basic properties *********************************************************)
42 lemma lsx_intro: ∀h,g,T,G,L1.
43 (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[0, T] L2 → ⊥) → G ⊢ ⋕⬊*[h, g, T] L2) →
45 /5 width=1 by lleq_sym, SN_intro/ qed.
47 lemma lsx_atom: ∀h,g,T,G. G ⊢ ⋕⬊*[h, g, T] ⋆.
48 #h #g #T #G @lsx_intro
49 #X #H #HT lapply (lpxs_inv_atom1 … H) -H
50 #H destruct elim HT -HT //
53 lemma lsx_sort: ∀h,g,G,L,k. G ⊢ ⋕⬊*[h, g, ⋆k] L.
54 #h #g #G #L1 #k @lsx_intro
55 #L2 #HL12 #H elim H -H
56 /3 width=4 by lpxs_fwd_length, lleq_sort/
59 lemma lsx_gref: ∀h,g,G,L,p. G ⊢ ⋕⬊*[h, g, §p] L.
60 #h #g #G #L1 #p @lsx_intro
61 #L2 #HL12 #H elim H -H
62 /3 width=4 by lpxs_fwd_length, lleq_gref/