more additions and corrections for the article

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / rsx.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_2/notation/relations/predtysnstrong_4.ma".
+include "static_2/static/rdeq.ma".
+include "basic_2/rt_transition/lpx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+definition rsx (h) (G) (T): predicate lenv ≝
+           SN … (lpx h G) (rdeq T).
+
+interpretation
+   "strong normalization for unbound context-sensitive parallel rt-transition on referred entries (local environment)"
+   'PRedTySNStrong h T G L = (rsx h G T L).
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_2A1: uses: lsx_ind *)
+lemma rsx_ind (h) (G) (T) (Q:predicate lenv):
+      (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+            (∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+            Q L1
+      ) →
+      ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ →  Q L.
+#h #G #T #Q #H0 #L1 #H elim H -L1
+/5 width=1 by SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: uses: lsx_intro *)
+lemma rsx_intro (h) (G) (T):
+      ∀L1.
+      (∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) →
+      G ⊢ ⬈*[h,T] 𝐒⦃L1⦄.
+/5 width=1 by SN_intro/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+(* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
+lemma rsx_fwd_pair_sn (h) (G):
+      ∀I,L,V,T. G ⊢ ⬈*[h,②{I}V.T] 𝐒⦃L⦄ →
+      G ⊢ ⬈*[h,V] 𝐒⦃L⦄.
+#h #G #I #L #V #T #H
+@(rsx_ind … H) -L #L1 #_ #IHL1
+@rsx_intro #L2 #HL12 #HnL12
+/4 width=3 by rdeq_fwd_pair_sn/
+qed-.
+
+(* Basic_2A1: uses: lsx_fwd_flat_dx *)
+lemma rsx_fwd_flat_dx (h) (G):
+      ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ →
+      G ⊢ ⬈*[h,T] 𝐒⦃L⦄.
+#h #G #I #L #V #T #H 
+@(rsx_ind … H) -L #L1 #_ #IHL1
+@rsx_intro #L2 #HL12 #HnL12
+/4 width=3 by rdeq_fwd_flat_dx/
+qed-.
+
+fact rsx_fwd_pair_aux (h) (G):
+     ∀L. G ⊢ ⬈*[h,#0] 𝐒⦃L⦄ →
+     ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+#h #G #L #H
+@(rsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
+/5 width=5 by lpx_pair, rsx_intro, rdeq_fwd_zero_pair/
+qed-.
+
+lemma rsx_fwd_pair (h) (G):
+      ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+/2 width=4 by rsx_fwd_pair_aux/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: uses: lsx_inv_flat *)
+lemma rsx_inv_flat (h) (G):
+      ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ →
+      ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃L⦄ & G ⊢ ⬈*[h,T] 𝐒⦃L⦄.
+/3 width=3 by rsx_fwd_pair_sn, rsx_fwd_flat_dx, conj/ qed-.
+
+(* Basic_2A1: removed theorems 9:
+              lsx_ge_up lsx_ge
+              lsxa_ind lsxa_intro lsxa_lleq_trans lsxa_lpxs_trans lsxa_intro_lpx lsx_lsxa lsxa_inv_lsx
+*)