--- /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 "basic_2/rt_computation/cpms_drops.ma".
+include "basic_2/dynamic/cnv.ma".
+
+(* CONTEXT_SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Advanced dproperties *****************************************************)
+
+(* Basic_2A1: uses: snv_lref *)
+lemma cnv_lref_drops (a) (h) (G): ∀I,K,V,i,L. ⦃G, K⦄ ⊢ V ![a, h] →
+ ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L⦄ ⊢ #i ![a, h].
+#a #h #G #I #K #V #i elim i -i
+[ #L #HV #H
+ lapply (drops_fwd_isid … H ?) -H // #H destruct
+ /2 width=1 by cnv_zero/
+| #i #IH #L #HV #H
+ elim (drops_inv_succ … H) -H #J0 #K0 #HK0 #H destruct
+ /3 width=1 by cnv_lref/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_2A1: uses: snv_inv_lref *)
+lemma cnv_inv_lref_drops (a) (h) (G):
+ ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ![a, h].
+#a #h #G #i elim i -i
+[ #L #H
+ elim (cnv_inv_zero … H) -H #I #K #V #HV #H destruct
+ /3 width=5 by drops_refl, ex2_3_intro/
+| #i #IH #X #H
+ elim (cnv_inv_lref … H) -H #I #L #HL #H destruct
+ elim (IH … HL) -IH -HL #J #K #V #HLK #HV
+ /3 width=5 by drops_drop, ex2_3_intro/
+]
+qed-.
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: uses: snv_lift *)
+lemma csv_lifts (a) (h): ∀G. d_liftable1 (cnv a h G).
+#a #h #G #K #T
+@(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * * [|||| * ]
+[ #s #HG #HK #HT #_ #b #f #L #_ #X #H2 destruct
+ >(lifts_inv_sort1 … H2) -X -K -f //
+| #i #HG #HK #HT #H1 #b #f #L #HLK #X #H2 destruct
+ elim (cnv_inv_lref_drops … H1) -H1 #I0 #K0 #V #HK0 #HV
+ elim (lifts_inv_lref1 … H2) -H2 #j #Hf #H destruct
+(**) (* this should be a lemma *)
+ lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
+ elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL #HY
+ lapply (drops_tls_at … Hf … HY) -HY #HY
+ elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct
+ elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct
+(**) (* end of the lemma *)
+ /4 width=8 by cnv_lref_drops, fqup_lref, drops_inv_gen/
+| #l #HG #HK #HT #H1 #b #f #L #_ #X #_ destruct
+ elim (cnv_inv_gref … H1)
+| #p #I #V #T #HG #HK #HT #H1 #b #f #L #HLK #X #H2 destruct
+ elim (cnv_inv_bind … H1) -H1 #HV #HT
+ elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ /5 width=8 by cnv_bind, drops_skip, ext2_pair/
+| #V #T #HG #HK #HT #H1 #b #f #L #HLK #X #H2 destruct
+ elim (cnv_inv_appl … H1) #n #p #W0 #U0 #Ha #HV #HT #HVW0 #HTW0
+ elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ elim (lifts_total W0 f)
+ elim (lifts_total U0 (⫯f))
+ /4 width=17 by cnv_appl, cpms_lifts_bi, lifts_bind/
+| #V #T #HG #HK #HT #H1 #b #f #L #HLK #X #H2 destruct
+ elim (cnv_inv_cast … H1) #U0 #HV #HT #HVU0 #HTU0
+ elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ elim (lifts_total U0 f)
+ /3 width=12 by cnv_cast, cpms_lifts_bi/
+]
+qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: uses: snv_inv_lift *)
+lemma cnv_inv_lifts (a) (h): ∀G. d_deliftable1 (cnv a h G).
+#a #h #G #L #U
+@(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * * [|||| * ]
+[ #s #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
+ >(lifts_inv_sort2 … H2) -X -L -f //
+| #j #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
+ elim (cnv_inv_lref_drops … H1) -H1 #I0 #L0 #W #HL0 #HW
+ elim (lifts_inv_lref2 … H2) -H2 #i #Hf #H destruct
+(**) (* this should be a lemma *)
+ lapply (drops_split_div … HLK (𝐔❴i❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK #HLY0
+ lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
+ lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
+ elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct
+ elim (liftsb_inv_pair_dx … HZ) -HZ #V #HVW #H destruct
+(**) (* end of the lemma *)
+ /4 width=8 by cnv_lref_drops, fqup_lref, drops_inv_F/
+| #l #HG #HL #HU #H1 #b #f #K #_ #X #_ destruct
+ elim (cnv_inv_gref … H1)
+| #p #I #W #U #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
+ elim (cnv_inv_bind … H1) -H1 #HW #HU
+ elim (lifts_inv_bind2 … H2) -H2 #V #T #HVW #HTU #H destruct
+ /5 width=8 by cnv_bind, drops_skip, ext2_pair/
+| #W #U #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
+ elim (cnv_inv_appl … H1) #n #p #W0 #U0 #Ha #HW #HU #HW0 #HU0
+ elim (lifts_inv_flat2 … H2) -H2 #V #T #HVW #HTU #H destruct
+ elim (cpms_inv_lifts_sn … HW0 … HLK … HVW) -HW0 #V0 #HVW0 #HV0
+ elim (cpms_inv_lifts_sn … HU0 … HLK … HTU) -HU0 #X0 #H #HT0
+ elim (lifts_inv_bind2 … H) -H #X #T0 #HX #HTU0 #H destruct
+ lapply (lifts_inj … HX … HVW0) -HX #H destruct
+ /3 width=8 by cnv_appl/
+| #W #U #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
+ elim (cnv_inv_cast … H1) #U0 #HW #HU #HWU0 #HU0
+ elim (lifts_inv_flat2 … H2) -H2 #V #T #HVW #HTU #H destruct
+ elim (cpms_inv_lifts_sn … HWU0 … HLK … HVW) -HWU0 #V0 #HVU0 #HV0
+ elim (cpms_inv_lifts_sn … HU0 … HLK … HTU) -HU0 #X #HX #HTV0
+ lapply (lifts_inj … HX … HVU0) -HX #H destruct
+ /3 width=8 by cnv_cast/
+]
+qed-.
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