update in lambdadelta

[helm.git] / matita / matita / contribs / lambdadelta / basic_2A / computation / lsx.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/sn_6.ma".
+include "basic_2A/multiple/lleq.ma".
+include "basic_2A/reduction/lpx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+definition lsx: ∀h. sd h → relation4 ynat term genv lenv ≝
+                λh,g,l,T,G. SN … (lpx h g G) (lleq l T).
+
+interpretation
+   "extended strong normalization (local environment)"
+   'SN h g l T G L = (lsx h g T l G L).
+
+(* Basic eliminators ********************************************************)
+
+lemma lsx_ind: ∀h,g,G,T,l. ∀R:predicate lenv.
+               (∀L1. G ⊢ ⬊*[h, g, T, l] L1 →
+                     (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+                     R L1
+               ) →
+               ∀L. G ⊢ ⬊*[h, g, T, l] L → R L.
+#h #g #G #T #l #R #H0 #L1 #H elim H -L1
+/5 width=1 by lleq_sym, SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsx_intro: ∀h,g,G,L1,T,l.
+                 (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, g, T, l] L2) →
+                 G ⊢ ⬊*[h, g, T, l] L1.
+/5 width=1 by lleq_sym, SN_intro/ qed.
+
+lemma lsx_atom: ∀h,g,G,T,l. G ⊢ ⬊*[h, g, T, l] ⋆.
+#h #g #G #T #l @lsx_intro
+#X #H #HT lapply (lpx_inv_atom1 … H) -H
+#H destruct elim HT -HT //
+qed.
+
+lemma lsx_sort: ∀h,g,G,L,l,k. G ⊢ ⬊*[h, g, ⋆k, l] L.
+#h #g #G #L1 #l #k @lsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=4 by lpx_fwd_length, lleq_sort/
+qed.
+
+lemma lsx_gref: ∀h,g,G,L,l,p. G ⊢ ⬊*[h, g, §p, l] L.
+#h #g #G #L1 #l #p @lsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=4 by lpx_fwd_length, lleq_gref/
+qed.
+
+lemma lsx_ge_up: ∀h,g,G,L,T,U,lt,l,m. lt ≤ yinj l + yinj m →
+                 ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L → G ⊢ ⬊*[h, g, U, l] L.
+#h #g #G #L #T #U #lt #l #m #Hltlm #HTU #H @(lsx_ind … H) -L
+/5 width=7 by lsx_intro, lleq_ge_up/
+qed-.
+
+lemma lsx_ge: ∀h,g,G,L,T,l1,l2. l1 ≤ l2 →
+              G ⊢ ⬊*[h, g, T, l1] L → G ⊢ ⬊*[h, g, T, l2] L.
+#h #g #G #L #T #l1 #l2 #Hl12 #H @(lsx_ind … H) -L
+/5 width=7 by lsx_intro, lleq_ge/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsx_fwd_bind_sn: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, V, l] L.
+#h #g #a #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=4 by lleq_fwd_bind_sn/
+qed-.
+
+lemma lsx_fwd_flat_sn: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, V, l] L.
+#h #g #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_sn/
+qed-.
+
+lemma lsx_fwd_flat_dx: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, T, l] L.
+#h #g #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_dx/
+qed-.
+
+lemma lsx_fwd_pair_sn: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ②{I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, V, l] L.
+#h #g * /2 width=4 by lsx_fwd_bind_sn, lsx_fwd_flat_sn/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lsx_inv_flat: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
+                    G ⊢ ⬊*[h, g, V, l] L ∧ G ⊢ ⬊*[h, g, T, l] L.
+/3 width=3 by lsx_fwd_flat_sn, lsx_fwd_flat_dx, conj/ qed-.