-lemma lpxs_cpys_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
- ∀T. ⦃G, L1⦄ ⊢ T ▶*[0, ∞] U →
- G ⊢ ⋕⬊*[h, g, T] L1.
-#h #g #G #L1 #U #H @(csx_ind_alt … H) -U
-#U #_ #IHU #T #HTU @lsx_intro
-#L2 #HL02 #HnT elim (lpxs_nlleq_fwd_cpxs … HL02 HnT) -HnT
-#U0 #U2 #H0 #H2 #HU02 #HnU02 elim (cpys_conf_eq … HTU … H0)
-#X #HUX #H0X elim (eq_term_dec U X) #HnUX destruct
-[ -HUX
-| -HnU02 @(lsx_lpxs_trans … HL02) @(IHU … HnUX)
- [ /3 width=3 by cpys_cpx, cpx_cpxs/
- | /2 width=3 by cpys_trans_eq/
- ]
-]
-
-lemma lpxs_cpys_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀T. ⦃G, L2⦄ ⊢ T ▶*[0, ∞] U →
- G ⊢ ⋕⬊*[h, g, T] L2.
-#h #g #G #L1 #U #H @(csx_ind_alt … H) -U
-#U #_ #IHU #L0 #HL10 #T #HTU @lsx_intro
-#L2 #HL02 #HnT elim (lpxs_nlleq_fwd_cpxs … HL02 HnT) -HnT
-#U0 #U2 #H0 #H2 #HU02 #HnU02 elim (cpys_conf_eq … HTU … H0)
-#X #HUX #H0X elim (eq_term_dec U X) #HnUX destruct
-[ -HUX
-| -HnU02 @(IHU … HnUX)
-
-
--HnT /4 width=9 by lpxs_trans, lpxs_cpxs_trans, cpx_cpye_fwd_lpxs/
+lemma lsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,d. d ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
+ ∀K2. G ⊢ ⋕⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g] K2 →
+ ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L2.
+#h #g #I #G #K1 #V #i #d #Hdi #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 (lpxs_ldrop_conf … HLK0 … HL02) -HL02
+#Y #H #HLK2 elim (lpxs_inv_pair1 … H) -H
+#K2 #V2 #HK02 #HV02 #H destruct
+lapply (lpxs_trans … HK10 … HK02) #HK12
+elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 -HK10 | -IHK0 -HnL02 -HLK0 ]
+[ /4 width=8 by lleq_lref/
+| @(IHV0 … HnV02 … HK12 … HLK2) -IHV0 -HnV02 -HK12 -HLK2
+ /3 width=4 by lsx_cpxs_trans_O, lsx_lpxs_trans, lpxs_cpxs_trans/ (**) (* full auto too slow *)