+(**************************************************************************)
+(* ___ *)
+(* ||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/predsn_5.ma".
+include "basic_2/static/lfxs.ma".
+include "basic_2/rt_transition/cpm.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+definition lfpr: sh → genv → relation3 term lenv lenv ≝
+ λh,G. lfxs (cpm 0 h G).
+
+interpretation
+ "parallel r-transition on referred entries (local environment)"
+ 'PRedSn h T G L1 L2 = (lfpr h G T L1 L2).
+
+(* Basic properties ***********************************************************)
+
+lemma lfpr_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] ⋆.
+/2 width=1 by lfxs_atom/ qed.
+
+lemma lfpr_sort: ∀h,I,G,L1,L2,V1,V2,s.
+ ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2.
+/2 width=1 by lfxs_sort/ qed.
+
+lemma lfpr_zero: ∀h,I,G,L1,L2,V.
+ ⦃G, L1⦄ ⊢ ➡[h, V] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, #0] L2.ⓑ{I}V.
+/2 width=1 by lfxs_zero/ qed.
+
+lemma lfpr_lref: ∀h,I,G,L1,L2,V1,V2,i.
+ ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, #⫯i] L2.ⓑ{I}V2.
+/2 width=1 by lfxs_lref/ qed.
+
+lemma lfpr_gref: ∀h,I,G,L1,L2,V1,V2,l.
+ ⦃G, L1⦄ ⊢ ➡[h, §l] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, §l] L2.ⓑ{I}V2.
+/2 width=1 by lfxs_gref/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lfpr_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] Y2 → Y2 = ⋆.
+/2 width=3 by lfxs_inv_atom_sn/ qed-.
+
+lemma lfpr_inv_atom_dx: ∀h,I,G,Y1. ⦃G, Y1⦄ ⊢ ➡[h, ⓪{I}] ⋆ → Y1 = ⋆.
+/2 width=3 by lfxs_inv_atom_dx/ qed-.
+
+lemma lfpr_inv_zero: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ➡[h, #0] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 &
+ ⦃G, L1⦄ ⊢ V1 ➡[h] V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+/2 width=1 by lfxs_inv_zero/ qed-.
+
+lemma lfpr_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ➡[h, #⫯i] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+/2 width=1 by lfxs_inv_lref/ qed-.
+
+lemma lfpr_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 →
+ ⦃G, L1⦄ ⊢ ➡[h, V] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V.
+/2 width=2 by lfxs_inv_bind/ qed-.
+
+lemma lfpr_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2 →
+ ⦃G, L1⦄ ⊢ ➡[h, V] L2 ∧ ⦃G, L1⦄ ⊢ ➡[h, T] L2.
+/2 width=2 by lfxs_inv_flat/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lfpr_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, #0] Y2 →
+ ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 &
+ Y2 = L2.ⓑ{I}V2.
+/2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
+
+lemma lfpr_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ➡[h, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 &
+ Y1 = L1.ⓑ{I}V1.
+/2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
+
+lemma lfpr_inv_lref_pair_sn: ∀h,I,G,Y2,L1,V1,i. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, #⫯i] Y2 →
+ ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 & Y2 = L2.ⓑ{I}V2.
+/2 width=2 by lfxs_inv_lref_pair_sn/ qed-.
+
+lemma lfpr_inv_lref_pair_dx: ∀h,I,G,Y1,L2,V2,i. ⦃G, Y1⦄ ⊢ ➡[h, #⫯i] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 & Y1 = L1.ⓑ{I}V1.
+/2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfpr_fwd_bind_sn: ∀h,p,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2.
+/2 width=4 by lfxs_fwd_bind_sn/ qed-.
+
+lemma lfpr_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V.
+/2 width=2 by lfxs_fwd_bind_dx/ qed-.
+
+lemma lfpr_fwd_flat_sn: ∀h,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2.
+/2 width=3 by lfxs_fwd_flat_sn/ qed-.
+
+lemma lfpr_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, T] L2.
+/2 width=3 by lfxs_fwd_flat_dx/ qed-.
+
+lemma lfpr_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ➡[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2.
+/2 width=3 by lfxs_fwd_pair_sn/ qed-.
+
+(* Basic_2A1: removed theorems 11:
+ lpr_inv_atom1 lpr_inv_pair1 lpr_inv_atom2 lpr_inv_pair2
+ lpr_refl lpr_pair
+ lpr_fwd_length lpr_lpx
+ lpr_drop_conf drop_lpr_trans lpr_drop_trans_O1
+*)
+(* Basic_1: removed theorems 7: wcpr0_gen_sort wcpr0_gen_head
+ wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+ wcpr0_drop wcpr0_drop_back
+*)