(* *)
(**************************************************************************)
-include "basic_2/unfold/lpqs.ma".
+include "basic_2/notation/relations/predsn_2.ma".
+include "basic_2/grammar/lpx_sn.ma".
include "basic_2/reduction/cpr.ma".
(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
(* Basic properties *********************************************************)
(* Note: lemma 250 *)
-lemma lpr_refl: ∀L. L ⊢ ➡ L.
+lemma lpr_refl: ∀L. ⦃G, L⦄ ⊢ ➡ L.
/2 width=1 by lpx_sn_refl/ qed.
lemma lpr_pair: ∀I,K1,K2,V1,V2. K1 ⊢ ➡ K2 → K1 ⊢ V1 ➡ V2 →
L1 @@ K1 ⊢ ➡ L2 @@ K2.
/3 width=1 by lpx_sn_append, cpr_append/ qed.
-lemma lpqs_lpr: ∀L1,L2. L1 ⊢ ➤* L2 → L1 ⊢ ➡ L2.
-#L1 #L2 #H elim H -L1 -L2 // /3 width=1/
-qed.
-
-lemma lpss_lpr: ∀L1,L2. L1 ⊢ ▶* L2 → L1 ⊢ ➡ L2.
-/3 width=1/ qed.
-
(* Basic forward lemmas *****************************************************)
lemma lpr_fwd_length: ∀L1,L2. L1 ⊢ ➡ L2 → |L1| = |L2|.
∃∃K2,L2. K1 ⊢ ➡ K2 & L = K2 @@ L2.
/2 width=2 by lpx_sn_fwd_append1/ qed-.
-lemma lpr_fwd_append2: ∀L,K2,L2. L ⊢ ➡ K2 @@ L2 →
+lemma lpr_fwd_append2: ∀L,K2,L2. ⦃G, L⦄ ⊢ ➡ K2 @@ L2 →
∃∃K1,L1. K1 ⊢ ➡ K2 & L = K1 @@ L1.
/2 width=2 by lpx_sn_fwd_append2/ qed-.