-fact dxprs_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
- ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
-elim (sstas_cprs_lpr_aux … IH3 IH2 IH1 … H01 … HTW1 … HT12 … HL12) // -L0 -T0 -T1 #W2 #HTW2 #HW12
-lapply (cprs_lpr_conf … HL12 … HWU1) -L1 #HWU1
-lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
-elim (cpcs_inv_cprs … H) -H /3 width=3/
+fact snv_ssta_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsstas h g G1 L1 T1) →
+ ∀G,L,T. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
+ ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 →
+ ∀U. ⦃G, L⦄ ⊢ T •[h, g] U → ⦃G, L⦄ ⊢ U ¡[h, g].
+/3 width=8 by lsstas_inv_SO, ssta_lsstas/ qed-.
+
+fact lsstas_cpds_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsstas h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L⦄ ⊢ T1 •*[h, g, l2] U1 → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
+ ∃∃U2,l. l ≤ l2 & ⦃G, L⦄ ⊢ T2 •*[h, g, l] U2 & ⦃G, L⦄ ⊢ U1 •*⬌*[h, g] U2.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 * #T #l0 #l #Hl0 #H #HT1T #HTT2
+lapply (da_mono … H … Hl1) -H #H destruct
+lapply (lsstas_da_conf … HTU1 … Hl1) #Hl12
+elim (le_or_ge l2 l) #Hl2
+[ lapply (lsstas_conf_le … HTU1 … HT1T) -HT1T // #HU1T
+ /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/
+| lapply (lsstas_da_conf … HT1T … Hl1) #Hl1l
+ lapply (lsstas_conf_le … HT1T … HTU1) -HTU1 // #HTU1
+ elim (lsstas_cprs_lpr_aux … IH3 IH2 IH1 … Hl1l … HTU1 … HTT2 L)
+ /3 width=8 by fpbg_fpbs_trans, lsstas_fpbs, monotonic_le_minus_l/ -T #U2 #HTU2 #HU12
+ /3 width=5 by cpcs_cpes, ex3_2_intro/
+]