--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/predtysnstar_5.ma".
+include "basic_2/rt_transition/lfpx.ma".
+
+(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+
+definition lfpxs: ∀h. relation4 genv term lenv lenv ≝
+ λh,G,T. TC … (lfpx h G T).
+
+interpretation
+ "uncounted parallel rt-computation on referred entries (local environment)"
+ 'PRedTySnStar h T G L1 L2 = (lfpxs h G T L1 L2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: was just: lpx_lpxs *)
+lemma lfpx_lfpxs: ∀h,G,T,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=1 by inj/ qed.
+
+(* Basic_2A1: was just: lpxs_strap1 *)
+lemma lfpxs_strap1: ∀h,G,T,L1,L,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L → ⦃G, L⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=3 by step/ qed.
+
+(* Basic_2A1: was just: lpxs_strap2 *)
+lemma lfpxs_strap2: ∀h,G,T,L1,L,L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L → ⦃G, L⦄ ⊢ ⬈*[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=3 by TC_strap/ qed.
+(*
+(* Basic_2A1: was just: lpxs_pair_refl *)
+lemma lfpxs_pair_refl: ∀h,G,T,L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈*[h, T] L2.ⓑ{I}V.
+/2 width=1 by TC_lpx_sn_pair_refl/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: was just: lpxs_inv_atom1 *)
+lemma lfpxs_inv_atom1: ∀h,G,L2.T. ⦃G, ⋆⦄ ⊢ ⬈*[h, T] L2 → L2 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
+
+(* Basic_2A1: was just: lpxs_inv_atom2 *)
+lemma lfpxs_inv_atom2: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] ⋆ → L1 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
+*)
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