refactoring to park the notions:

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / lsx_alt.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsx_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsx_alt.ma
new file mode 100644 (file)
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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_2/notation/relations/snalt_6.ma".
+include "basic_2/computation/lpxs_lleq.ma".
+include "basic_2/computation/lsx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* alternative definition of lsx *)
+definition lsxa: ∀h. sd h → relation4 ynat term genv lenv ≝
+                 λh,o,l,T,G. SN … (lpxs h o G) (lleq l T).
+
+interpretation
+   "extended strong normalization (local environment) alternative"
+   'SNAlt h o l T G L = (lsxa h o T l G L).
+
+(* Basic eliminators ********************************************************)
+
+lemma lsxa_ind: ∀h,o,G,T,l. ∀R:predicate lenv.
+                (∀L1. G ⊢ ⬊⬊*[h, o, T, l] L1 →
+                      (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+                      R L1
+                ) →
+                ∀L. G ⊢ ⬊⬊*[h, o, T, l] L → R L.
+#h #o #G #T #l #R #H0 #L1 #H elim H -L1
+/5 width=1 by lleq_sym, SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsxa_intro: ∀h,o,G,L1,T,l.
+                  (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
+                  G ⊢ ⬊⬊*[h, o, T, l] L1.
+/5 width=1 by lleq_sym, SN_intro/ qed.
+
+fact lsxa_intro_aux: ∀h,o,G,L1,T,l.
+                     (∀L,L2. ⦃G, L⦄ ⊢ ➡*[h, o] L2 → L1 ≡[T, l] L → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
+                     G ⊢ ⬊⬊*[h, o, T, l] L1.
+/4 width=3 by lsxa_intro/ qed-.
+
+lemma lsxa_lleq_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊⬊*[h, o, T, l] L1 →
+                       ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊⬊*[h, o, T, l] L2.
+#h #o #T #G #L1 #l #H @(lsxa_ind … H) -L1
+#L1 #_ #IHL1 #L2 #HL12 @lsxa_intro
+#K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HLK2 … HL12) -HLK2
+/5 width=4 by lleq_canc_sn, lleq_trans/
+qed-.
+
+lemma lsxa_lpxs_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊⬊*[h, o, T, l] L1 →
+                       ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → G ⊢ ⬊⬊*[h, o, T, l] L2.
+#h #o #T #G #L1 #l #H @(lsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+elim (lleq_dec T L1 L2 l) /3 width=4 by lsxa_lleq_trans/
+qed-.
+
+lemma lsxa_intro_lpx: ∀h,o,G,L1,T,l.
+                      (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
+                      G ⊢ ⬊⬊*[h, o, T, l] L1.
+#h #o #G #L1 #T #l #IH @lsxa_intro_aux
+#L #L2 #H @(lpxs_ind_dx … H) -L
+[ #H destruct #H elim H //
+| #L0 #L elim (lleq_dec T L1 L l) /3 width=1 by/
+  #HnT #HL0 #HL2 #_ #HT #_ elim (lleq_lpx_trans … HL0 … HT) -L0
+  #L0 #HL10 #HL0 @(lsxa_lpxs_trans … HL2) -HL2
+  /5 width=3 by lsxa_lleq_trans, lleq_trans/
+]
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lsx_lsxa: ∀h,o,G,L,T,l. G ⊢ ⬊*[h, o, T, l] L → G ⊢ ⬊⬊*[h, o, T, l] L.
+#h #o #G #L #T #l #H @(lsx_ind … H) -L
+/4 width=1 by lsxa_intro_lpx/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lsxa_inv_lsx: ∀h,o,G,L,T,l. G ⊢ ⬊⬊*[h, o, T, l] L → G ⊢ ⬊*[h, o, T, l] L.
+#h #o #G #L #T #l #H @(lsxa_ind … H) -L
+/4 width=1 by lsx_intro, lpx_lpxs/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lsx_intro_alt: ∀h,o,G,L1,T,l.
+                     (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, o, T, l] L2) →
+                     G ⊢ ⬊*[h, o, T, l] L1.
+/6 width=1 by lsxa_inv_lsx, lsx_lsxa, lsxa_intro/ qed.
+
+lemma lsx_lpxs_trans: ∀h,o,G,L1,T,l. G ⊢ ⬊*[h, o, T, l] L1 →
+                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → G ⊢ ⬊*[h, o, T, l] L2.
+/4 width=3 by lsxa_inv_lsx, lsx_lsxa, lsxa_lpxs_trans/ qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma lsx_ind_alt: ∀h,o,G,T,l. ∀R:predicate lenv.
+                   (∀L1. G ⊢ ⬊*[h, o, T, l] L1 →
+                         (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+                         R L1
+                   ) →
+                   ∀L. G ⊢ ⬊*[h, o, T, l] L → R L.
+#h #o #G #T #l #R #IH #L #H @(lsxa_ind h o G T l … L)
+/4 width=1 by lsxa_inv_lsx, lsx_lsxa/
+qed-.