(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
definition lpx: ∀h. sd h → relation3 genv lenv lenv ≝
- λh,g,G. lpx_sn (cpx h g G).
+ λh,g,G. lpx_sn (λ_.cpx h g G).
interpretation "extended parallel reduction (local environment, sn variant)"
'PRedSn h g G L1 L2 = (lpx h g G L1 L2).
L1 = K1. ⓑ{I} V1.
/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+lemma lpx_inv_pair: ∀h,g,I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, g] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ➡[h, g] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, g] V2 & I1 = I2.
+/2 width=1 by lpx_sn_inv_pair/ qed-.
+
(* Basic properties *********************************************************)
lemma lpx_refl: ∀h,g,G,L. ⦃G, L⦄ ⊢ ➡[h, g] L.
lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 →
⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2.
-/2 width=1/ qed.
-
-lemma lpx_append: ∀h,g,G,K1,K2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
- ⦃G, L1 @@ K1⦄ ⊢ ➡[h, g] L2 @@ K2.
-/3 width=1 by lpx_sn_append, cpx_append/ qed.
+/2 width=1 by lpx_sn_pair/ qed.
lemma lpr_lpx: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g] L2.
-#h #g #G #L1 #L2 #H elim H -L1 -L2 // /3 width=1/
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
qed.
(* Basic forward lemmas *****************************************************)
lemma lpx_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → |L1| = |L2|.
/2 width=2 by lpx_sn_fwd_length/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpx_fwd_append1: ∀h,g,G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ➡[h, g] L →
- ∃∃K2,L2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K2 @@ L2.
-/2 width=2 by lpx_sn_fwd_append1/ qed-.
-
-lemma lpx_fwd_append2: ∀h,g,G,L,K2,L2. ⦃G, L⦄ ⊢ ➡[h, g] K2 @@ L2 →
- ∃∃K1,L1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K1 @@ L1.
-/2 width=2 by lpx_sn_fwd_append2/ qed-.