(* Properties on sn parallel substitution for local environments ************)
-lemma cpcs_lpss_conf: ∀L1,L2. L1 ⊢ ▶* L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #HL12 #T1 #T2 #H
+lemma cpcs_lpss_conf: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
+ ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T1 ⬌* T2.
+#L1 #T1 #T2 #H #L2 #HL12
elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
elim (cprs_lpss_conf_dx … HT1 … HL12) -HT1 #U1 #H1 #HTU1
elim (cprs_lpss_conf_dx … HT2 … HL12) -L1 #U2 #H2 #HTU2
lapply (cprs_cpss_trans … HTU2 … HU2) -U2 /2 width=3/
qed-.
-lemma cpcs_lpss_cpss_conf: ∀L1,L2. L1 ⊢ ▶* L2 →
- ∀T,T2. L1 ⊢ T ⬌* T2 →
- ∀T1. L2 ⊢ T ▶* T1 →
- L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #HL12 #T #T2 #HT2 #T1 #HT1
-lapply (cpcs_lpss_conf … HL12 … HT2) -L1 #HT2
-lapply (cpcs_cpss_conf … HT1 … HT2) -T //
+lemma cpcs_cpss_lpss_conf: ∀L1,T,T2. L1 ⊢ T ⬌* T2 → ∀T1. L1 ⊢ T ▶* T1 →
+ ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T1 ⬌* T2.
+#L1 #T #T2 #HT2 #T1 #HT1 #L2 #HL12
+lapply (cpcs_lpss_conf … HT2 … HL12) -HT2 #HT2
+elim (lpss_cpss_conf_dx … HT1 … HL12) -L1 #T0 #HT0 #HT10
+lapply (cpcs_cpss_conf … HT0 … HT2) -T #HT02
+lapply (cpcs_cpss_strap2 … HT10 … HT02) -T0 //
qed-.
-lemma cpcs_lpss_cpss2_conf: ∀L1,L2. L1 ⊢ ▶* L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 →
- ∀T3. L2 ⊢ T1 ▶* T3 → ∀T4. L2 ⊢ T2 ▶* T4 →
- L2 ⊢ T3 ⬌* T4.
-#L1 #L2 #HL12 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24
-lapply (cpcs_lpss_cpss_conf … HL12 … HT12 … HT13) -L1 -T1 #HT32
-lapply (cpcs_cpss_strap1 … HT32 … HT24) -T2 //
+lemma cpcs_cpss2_lpss_conf: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
+ ∀T3. L1 ⊢ T1 ▶* T3 → ∀T4. L1 ⊢ T2 ▶* T4 →
+ ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T3 ⬌* T4.
+#L1 #T1 #T2 #HT12 #T3 #HT13 #T4 #HT24 #L2 #HL12
+lapply (cpcs_cpss_lpss_conf … HT12 … HT13 … HL12) -T1 #HT32
+elim (lpss_cpss_conf_dx … HT24 … HL12) -L1 #T #HT2 #HT4
+lapply (cpcs_cpss_strap1 … HT32 … HT2) -T2 #HT3
+lapply (cpcs_cpss_div … HT3 … HT4) -T //
qed-.