--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/drops.ma".
+include "basic_2/rt_conversion/cpce.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR TERMS **********************)
+
+(* Properties with uniform slicing for local environments *******************)
+
+lemma cpce_eta_drops (h) (n) (G) (K):
+ ∀p,W,V1,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U →
+ ∀V2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 →
+ ∀i,L. ⬇*[i] L ≘ K.ⓛW →
+ ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬌η[h] +ⓛW2.ⓐ#0.#↑i.
+#h #n #G #K #p #W #V1 #U #HWU #V2 #HV12 #i elim i -i
+[ #L #HLK #W2 #HVW2
+ >(drops_fwd_isid … HLK) -L [| // ] /2 width=8 by cpce_eta/
+| #i #IH #L #HLK #W2 #HVW2
+ elim (drops_inv_succ … HLK) -HLK #I #Y #HYK #H destruct
+ elim (lifts_split_trans … HVW2 (𝐔❴↑i❵) (𝐔❴1❵)) [| // ] #X2 #HVX2 #HXW2
+ /5 width=7 by cpce_lref, lifts_push_lref, lifts_bind, lifts_flat/
+]
+qed.
+
+lemma cpce_zero_drops (h) (G):
+ ∀i,L. (∀n,p,K,W,V,U. ⬇*[i] L ≘ K.ⓛW → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
+ ⦃G,L⦄ ⊢ #i ⬌η[h] #i.
+#h #G #i elim i -i
+[ * [ #_ // ] #L #I #Hi
+ /4 width=8 by cpce_zero, drops_refl/
+| #i #IH * [ -IH #_ // ] #L #I #Hi
+ /5 width=8 by cpce_lref, drops_drop/
+]
+qed.