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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/notation/relations/exclaim_5.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* activate genv *)
+(* Basic_2A1: uses: snv *)
+inductive cnv (a) (h): relation3 genv lenv term ≝
+| cnv_sort: ∀G,L,s. cnv a h G L (⋆s)
+| cnv_zero: ∀I,G,K,V. cnv a h G K V → cnv a h G (K.ⓑ{I}V) (#0)
+| cnv_lref: ∀I,G,K,i. cnv a h G K (#i) → cnv a h G (K.ⓘ{I}) (#↑i)
+| cnv_bind: ∀p,I,G,L,V,T. cnv a h G L V → cnv a h G (L.ⓑ{I}V) T → cnv a h G L (ⓑ{p,I}V.T)
+| cnv_appl: ∀n,p,G,L,V,W0,T,U0. (a = Ⓣ → n = 1) → cnv a h G L V → cnv a h G L T →
+            ⦃G, L⦄ ⊢ V ➡*[1, h] W0 → ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
+| cnv_cast: ∀G,L,U,T,U0. cnv a h G L U → cnv a h G L T →
+            ⦃G, L⦄ ⊢ U ➡*[h] U0 → ⦃G, L⦄ ⊢ T ➡*[1, h] U0 → cnv a h G L (ⓝU.T)
+.
+
+interpretation "context-sensitive native validity (term)"
+   'Exclaim a h G L T = (cnv a h G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 →
+                               ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
+#a #h #G #L #X * -G -L -X
+[ #G #L #s #H destruct
+| #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/
+| #I #G #K #i #_ #H destruct
+| #p #I #G #L #V #T #_ #_ #H destruct
+| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #H destruct
+| #G #L #U #T #U0 #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma cnv_inv_zero (a) (h): ∀G,L. ⦃G, L⦄ ⊢ #0 ![a, h] →
+                            ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
+/2 width=3 by cnv_inv_zero_aux/ qed-.
+
+fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X = #(↑i) →
+                              ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
+#a #h #G #L #X * -G -L -X
+[ #G #L #s #j #H destruct
+| #I #G #K #V #_ #j #H destruct
+| #I #G #L #i #Hi #j #H destruct /2 width=4 by ex2_2_intro/
+| #p #I #G #L #V #T #_ #_ #j #H destruct
+| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #j #H destruct
+| #G #L #U #T #U0 #_ #_ #_ #_ #j #H destruct
+]
+qed-.
+
+lemma cnv_inv_lref (a) (h): ∀G,L,i. ⦃G, L⦄ ⊢ #↑i ![a, h] →
+                            ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
+/2 width=3 by cnv_inv_lref_aux/ qed-.
+
+fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X = §l → ⊥.
+#a #h #G #L #X * -G -L -X
+[ #G #L #s #l #H destruct
+| #I #G #K #V #_ #l #H destruct
+| #I #G #K #i #_ #l #H destruct
+| #p #I #G #L #V #T #_ #_ #l #H destruct
+| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #l #H destruct
+| #G #L #U #T #U0 #_ #_ #_ #_ #l #H destruct
+]
+qed-.
+
+(* Basic_2A1: uses: snv_inv_gref *)
+lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G, L⦄ ⊢ §l ![a, h] → ⊥.
+/2 width=8 by cnv_inv_gref_aux/ qed-.
+
+fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] →
+                               ∀p,I,V,T. X = ⓑ{p,I}V.T →
+                               ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
+                                & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
+#a #h #G #L #X * -G -L -X
+[ #G #L #s #q #Z #X1 #X2 #H destruct
+| #I #G #K #V #_ #q #Z #X1 #X2 #H destruct
+| #I #G #K #i #_ #q #Z #X1 #X2 #H destruct
+| #p #I #G #L #V #T #HV #HT #q #Z #X1 #X2 #H destruct /2 width=1 by conj/
+| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #q #Z #X1 #X2 #H destruct
+| #G #L #U #T #U0 #_ #_ #_ #_ #q #Z #X1 #X2 #H destruct
+]
+qed-.
+
+(* Basic_2A1: uses: snv_inv_bind *)
+lemma cnv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T ![a, h] →
+                            ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
+                             & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
+/2 width=4 by cnv_inv_bind_aux/ qed-.
+
+fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X = ⓐV.T →
+                               ∃∃n,p,W0,U0. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
+                                            ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+#a #h #G #L #X * -L -X
+[ #G #L #s #X1 #X2 #H destruct
+| #I #G #K #V #_ #X1 #X2 #H destruct
+| #I #G #K #i #_ #X1 #X2 #H destruct
+| #p #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
+| #n #p #G #L #V #W0 #T #U0 #Ha #HV #HT #HVW0 #HTU0 #X1 #X2 #H destruct /3 width=7 by ex5_4_intro/
+| #G #L #U #T #U0 #_ #_ #_ #_ #X1 #X2 #H destruct
+]
+qed-.
+
+(* Basic_2A1: uses: snv_inv_appl *)
+lemma cnv_inv_appl (a) (h): ∀G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
+                            ∃∃n,p,W0,U0. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
+                                         ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+/2 width=3 by cnv_inv_appl_aux/ qed-.
+
+fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X = ⓝU.T →
+                               ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
+                                     ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
+#a #h #G #L #X * -G -L -X
+[ #G #L #s #X1 #X2 #H destruct
+| #I #G #K #V #_ #X1 #X2 #H destruct
+| #I #G #K #i #_ #X1 #X2 #H destruct
+| #p #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
+| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #X1 #X2 #H destruct
+| #G #L #U #T #U0 #HV #HT #HU0 #HTU0 #X1 #X2 #H destruct /2 width=3 by ex4_intro/
+]
+qed-.
+
+(* Basic_2A1: uses: snv_inv_appl *)
+lemma cnv_inv_cast (a) (h): ∀G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] →
+                            ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
+                                  ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
+/2 width=3 by cnv_inv_cast_aux/ qed-.