(* activate genv *)
inductive snv (h) (g): relation3 genv lenv term ≝
| snv_sort: ∀G,L,k. snv h g G L (⋆k)
-| snv_lref: â\88\80I,G,L,K,V,i. â\87©[i] L ≡ K.ⓑ{I}V → snv h g G K V → snv h g G L (#i)
+| snv_lref: â\88\80I,G,L,K,V,i. â¬\87[i] L ≡ K.ⓑ{I}V → snv h g G K V → snv h g G L (#i)
| snv_bind: ∀a,I,G,L,V,T. snv h g G L V → snv h g G (L.ⓑ{I}V) T → snv h g G L (ⓑ{a,I}V.T)
| snv_appl: ∀a,G,L,V,W0,T,U0,l. snv h g G L V → snv h g G L T →
⦃G, L⦄ ⊢ V •*➡*[h, g, 1] W0 → ⦃G, L⦄ ⊢ T •*➡*[h, g, l] ⓛ{a}W0.U0 → snv h g G L (ⓐV.T)
(* Basic inversion lemmas ***************************************************)
fact snv_inv_lref_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀i. X = #i →
- â\88\83â\88\83I,K,V. â\87©[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ¡[h, g].
+ â\88\83â\88\83I,K,V. â¬\87[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ¡[h, g].
#h #g #G #L #X * -G -L -X
[ #G #L #k #i #H destruct
| #I #G #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5 by ex2_3_intro/
qed-.
lemma snv_inv_lref: ∀h,g,G,L,i. ⦃G, L⦄ ⊢ #i ¡[h, g] →
- â\88\83â\88\83I,K,V. â\87©[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ¡[h, g].
+ â\88\83â\88\83I,K,V. â¬\87[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ¡[h, g].
/2 width=3 by snv_inv_lref_aux/ qed-.
fact snv_inv_gref_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀p. X = §p → ⊥.