+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma drops_inv_atom2: ∀b,L,f. ⬇*[b, f] L ≡ ⋆ →
+ ∃∃n,f1. ⬇*[b, 𝐔❴n❵] L ≡ ⋆ & 𝐔❴n❵ ⊚ f1 ≡ f.
+#b #L elim L -L
+[ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/
+| #L #I #IH #f #H elim (pn_split f) * #g #H0 destruct
+ [ elim (drops_inv_skip1 … H) -H #J #K #_ #_ #H destruct
+ | lapply (drops_inv_drop1 … H) -H #HL
+ elim (IH … HL) -IH -HL /3 width=8 by drops_drop, after_next, ex2_2_intro/
+ ]
+]
+qed-.
+
+lemma drops_inv_succ: ∀L1,L2,i. ⬇*[⫯i] L1 ≡ L2 →
+ ∃∃I,K. ⬇*[i] K ≡ L2 & L1 = K.ⓘ{I}.
+#L1 #L2 #i #H elim (drops_inv_isuni … H) -H // *
+[ #H elim (isid_inv_next … H) -H //
+| /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Properties with uniform relocations **************************************)
+
+lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≡ K.ⓘ{I}.
+#L elim L -L /2 width=1 by or_introl/
+#L #I #IH * /4 width=3 by drops_refl, ex1_2_intro, or_intror/
+#i elim (IH i) -IH /3 width=1 by drops_drop, or_introl/
+* /4 width=3 by drops_drop, ex1_2_intro, or_intror/
+qed-.
+
+(* Basic_2A1: includes: drop_split *)
+lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f → 𝐔⦃f1⦄ →
+ ∃∃L. ⬇*[b, f1] L1 ≡ L & ⬇*[b, f2] L ≡ L2.
+#b #f #L1 #L2 #H elim H -f -L1 -L2
+[ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
+ #H lapply (H0f H) -b
+ #H elim (after_inv_isid3 … Hf H) -f //
+| #f #I #L1 #L2 #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ]
+ [ #g1 #g2 #Hf #H1 #H2 destruct
+ lapply (isuni_inv_push … Hf1 ??) -Hf1 [1,2: // ] #Hg1
+ elim (IHL12 … Hf) -f
+ /4 width=5 by drops_drop, drops_skip, liftsb_refl, isuni_isid, ex2_intro/
+ | #g1 #Hf #H destruct elim (IHL12 … Hf) -f
+ /3 width=5 by ex2_intro, drops_drop, isuni_inv_next/
+ ]
+| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … Hf) -Hf [2,3: // ]
+ #g1 #g2 #Hf #H1 #H2 destruct elim (liftsb_split_trans … HI21 … Hf) -HI21
+ elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, isuni_fwd_push/
+]
+qed-.
+
+lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≡ L → ∀f2,f. f1 ⊚ f2 ≡ f → 𝐔⦃f2⦄ →
+ ∃∃L2. ⬇*[Ⓕ, f2] L ≡ L2 & ⬇*[Ⓕ, f] L1 ≡ L2.
+#b #f1 #L1 #L #H elim H -f1 -L1 -L
+[ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
+| #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
+ #g #Hg #H destruct elim (IH … Hg) -IH -Hg /3 width=5 by drops_drop, ex2_intro/
+| #f1 #I1 #I #L1 #L #HL1 #HI1 #IH #f2 #f #Hf #Hf2
+ elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
+ #g2 #g #Hg #H2 #H0 destruct
+ [ lapply (isuni_inv_push … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -IH
+ lapply (after_isid_inv_dx … Hg … Hg2) -Hg #Hg
+ /5 width=7 by drops_eq_repl_back, drops_F, drops_refl, drops_skip, liftsb_eq_repl_back, isid_push, ex2_intro/
+ | lapply (isuni_inv_next … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -HL1 -HI1
+ elim (IH … Hg) -f1 /3 width=3 by drops_drop, ex2_intro/
+ ]
+]
+qed-.
+
+(* Properties with application **********************************************)
+
+lemma drops_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≡ i2 →
+ ∀b,L1,L2. ⬇*[b,⫱*[i2]f] L1 ≡ L2 →
+ ⬇*[b,↑⫱*[⫯i2]f] L1 ≡ L2.
+/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
+
+lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b, f] L ≡ K0.ⓘ{I} → ∀i. @⦃O, f⦄ ≡ i →
+ ∃∃J,K. ⬇*[i]L ≡ K.ⓘ{J} & ⬇*[b, ⫱*[⫯i]f] K ≡ K0 & ⬆*[⫱*[⫯i]f] I ≡ J.
+#b #f #I #L #K0 #H #i #Hf
+elim (drops_split_trans … H) -H [ |5: @(after_uni_dx … Hf) |2,3: skip ] /2 width=1 by after_isid_dx/ #Y #HLY #H
+lapply (drops_tls_at … Hf … H) -H #H
+elim (drops_inv_skip2 … H) -H #J #K #HK0 #HIJ #H destruct
+/3 width=5 by drops_inv_gen, ex3_2_intro/
+qed-.
+
+(* Properties with context-sensitive equivalence for terms ******************)
+
+lemma ceq_lift_sn: d_liftable2_sn … liftsb ceq_ext.
+#K #I1 #I2 #H <(ceq_ext_inv_eq … H) -I2
+/2 width=3 by ex2_intro/ qed-.
+
+lemma ceq_inv_lift_sn: d_deliftable2_sn … liftsb ceq_ext.
+#L #J1 #J2 #H <(ceq_ext_inv_eq … H) -J2
+/2 width=3 by ex2_intro/ qed-.
+
+(* Note: d_deliftable2_sn cfull does not hold *)
+lemma cfull_lift_sn: d_liftable2_sn … liftsb cfull.
+#K #I1 #I2 #_ #b #f #L #_ #J1 #_ -K -I1 -b
+elim (liftsb_total I2 f) /2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: removed theorems 12: