-lemma lsx_flat_lpxs: ∀h,o,I,G,L1,V,l. G ⊢ ⬈*[h, o, V, l] L1 →
- ∀L2,T. G ⊢ ⬈*[h, o, T, l] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, o] L2 →
- G ⊢ ⬈*[h, o, ⓕ{I}V.T, l] L2.
-#h #o #I #G #L1 #V #l #H @(lsx_ind_lfpxs … H) -L1
-#L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind_lfpxs … H) -L2
-#L2 #HL2 #IHL2 #HL12 @lsx_intro_lfpxs
-#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
-#HL10 #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 l)
- [ #HV @(IHL1 … L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
- | -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
+(* Basic_2A1: uses: lsx_flat_lpxs *)
+lemma lfsx_flat_lfpxs: ∀h,o,I,G,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
+ ∀L2,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄ → ⦃G, L1⦄ ⊢ ⬈*[h, ⓕ{I}V.T] L2 →
+ G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L2⦄.
+#h #o #I #G #L1 #V #H @(lfsx_ind_lfpxs … H) -L1
+#L1 #HL1 #IHL1 #L2 #T #H @(lfsx_ind_lfpxs … H) -L2
+#L2 #HL2 #IHL2 #HL12 @lfsx_intro_lfpxs
+#L0 #HL20 lapply (lfpxs_trans … HL12 … HL20)
+#HL10 #H elim (lfdneq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (lfdeq_dec h o L1 L2 V)
+ [ #HV @(IHL1 … L0) -IHL1 -HL12
+ /3 width=5 by lfsx_lfpxs_trans, lfpxs_fwd_flat_dx, lfpxs_fwd_pair_sn, lfdeq_canc_sn/ (**) (* full auto too slow: 47s *)
+ | -HnV -HL10 /4 width=4 by lfsx_lfpxs_trans, lfpxs_fwd_pair_sn/