-lemma lsx_lref_be_lpxs: ∀h,o,I,G,K1,V,i,l. l ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, o] V →
- ∀K2. G ⊢ ⬊*[h, o, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, o] K2 →
- ∀L2. ⬇[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⬊*[h, o, #i, l] L2.
-#h #o #I #G #K1 #V #i #l #Hli #H @(csx_ind_alt … H) -V
-#V0 #_ #IHV0 #K2 #H @(lsx_ind … H) -K2
-#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @lsx_intro
-#L2 #HL02 #HnL02 elim (lpx_drop_conf … HLK0 … HL02) -HL02
-#Y #H #HLK2 elim (lpx_inv_pair1 … H) -H
-#K2 #V2 #HK02 #HV02 #H destruct
-elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0 ]
-[ /4 width=8 by lpxs_strap1, lleq_lref/
-| @(IHV0 … HnV02 … HLK2) -IHV0 -HnV02 -HLK2
- /3 width=4 by lsx_cpx_trans_O, lsx_lpx_trans, lpxs_cpx_trans, lpxs_strap1/ (**) (* full auto too slow *)
-]
-qed.
+(* Basic_2A1: uses: lsx_lref_be_lpxs *)
+lemma lfsx_pair_lpxs: ∀h,o,G,K1,V. ⦃G, K1⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
+ ∀K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ → ⦃G, K1⦄ ⊢ ⬈*[h] K2 →
+ ∀I. G ⊢ ⬈*[h, o, #0] 𝐒⦃K2.ⓑ{I}V⦄.
+#h #o #G #K1 #V #H
+@(csx_ind_cpxs … H) -V #V0 #_ #IHV0 #K2 #H
+@(lfsx_ind_lpxs … H) -K2 #K0 #HK0 #IHK0 #HK10 #I
+@lfsx_intro_lpxs #Y #HY #HnY
+elim (lpxs_inv_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
+elim (tdeq_dec h o V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnY ]
+[ /5 width=5 by lfsx_lfdeq_trans, lpxs_trans, lfdeq_pair/
+| @(IHV0 … HnV02) -IHV0 -HnV02
+ [
+ | /3 width=3 by lfsx_lpxs_trans, lfsx_cpxs_trans/
+ | /2 width=3 by lpxs_trans/
+ ]