+lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
+ ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
+ ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
+#b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
+[ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+ >(lifts_inv_sort1 … H2) -U /2 width=1 by frees_sort/
+| #f1 #i #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
+ elim (lifts_inv_lref1 … H2) -H2 #j #Hij #H destruct
+ elim (coafter_fwd_xnx_pushs … Hij H3) -H3 #g2 #Hg2 #H2 destruct
+ lapply (coafter_isid_inv_dx … Hg2 … Hf1) -f1 #Hf2
+ elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf
+ elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i
+ lapply (at_inv_uni … Hj) -Hj #H destruct
+ /3 width=8 by frees_atom_drops, drops_trans/
+| #f1 #I #K #V #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+ lapply (isfin_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+ lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
+ elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #H
+ elim (liftsb_inv_pair_sn … H) -H #W #HVW #H destruct
+ elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
+ lapply (IH … HYK … HVW … H3) -IH -H3 -HYK -HVW //
+ /2 width=5 by frees_pair_drops/
+| #f1 #I #K #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
+ lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
+ elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hg2
+ elim (drops_split_trans_bind2 … H1 … Hf) -H1 -Hf #Z #Y #HLY #_ #H
+ lapply (liftsb_inv_unit_sn … H) -H #H destruct
+ /2 width=3 by frees_unit_drops/
+| #f1 #I #K #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+ lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+ lapply (lifts_inv_lref1 … H2) -H2 * #x #Hf #H destruct
+ elim (at_inv_nxx … Hf) -Hf [ |*: // ] #j #Hf #H destruct
+ elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #_
+ elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H3 #H2 destruct
+ lapply (drops_isuni_fwd_drop2 … HLY) -HLY // #HLY
+ lapply (IH … HYK … H3) -IH -H3 -HYK [4: |*: /2 width=2 by lifts_lref/ ]
+ >plus_S1 /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *)
+| #f1 #K #l #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+ >(lifts_inv_gref1 … H2) -U /2 width=1 by frees_gref/
+| #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
+ elim (sor_inv_isfin3 … H1f1) // #Hf1V #H
+ lapply (isfin_inv_tl … H) -H
+ elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H
+ elim (coafter_inv_tl1 … H) -H
+ /5 width=5 by frees_bind, drops_skip, ext2_pair/
+| #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
+ elim (sor_inv_isfin3 … H1f1) //
+ elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ elim (coafter_sor … H3 … H1f1)
+ /3 width=5 by coafter_isfin2_fwd, frees_flat/
+]
+qed-.
+
+lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
+ ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
+#b #L #K #HLK #T #U #HTU #f #Hf
+@(frees_lifts b … Hf … HTU) // (**) (* auto fails *)
+qed.
+
+(* Forward lemmas with generic slicing for local environments ***************)
+
+lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
+/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
+