-lemma lifts_simple_dx: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#T1 #T2 #t #H elim H -T1 -T2 -t //
-#a #I #V1 #V2 #T1 #T2 #t #_ #_ #_ #_ #H elim (simple_inv_bind … H)
+lemma lifts_simple_dx: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#f #T1 #T2 #H elim H -f -T1 -T2 //
+#f #p #I #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H elim (simple_inv_bind … H)
-lemma lifts_simple_sn: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-#T1 #T2 #t #H elim H -T1 -T2 -t //
-#a #I #V1 #V2 #T1 #T2 #t #_ #_ #_ #_ #H elim (simple_inv_bind … H)
+lemma lifts_simple_sn: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+#f #T1 #T2 #H elim H -f -T1 -T2 //
+#f #p #I #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H elim (simple_inv_bind … H)