+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/lenv.ma".
+include "apps_2/models/model_push.ma".
+include "apps_2/notation/models/inwbrackets_4.ma".
+
+(* LOCAL ENVIRONMENT INTERPRETATION ****************************************)
+
+inductive li (M) (gv): relation2 lenv (evaluation M) ≝
+| li_atom: ∀lv. li M gv (⋆) lv
+| li_abbr: ∀lv,d,L,V. li M gv L lv → ⟦V⟧[gv, lv] ≗ d → li M gv (L.ⓓV) (⫯[d]lv)
+| li_abst: ∀lv,d,L,W. li M gv L lv → li M gv (L.ⓛW) (⫯[d]lv)
+| li_unit: ∀lv,d,I,L. li M gv L lv → li M gv (L.ⓤ{I}) (⫯[d]lv)
+.
+
+interpretation "local environment interpretation (model)"
+ 'InWBrackets M gv L lv = (li M gv L lv).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact li_inv_abbr_aux (M) (gv): ∀v,Y. v ϵ ⟦Y⟧{M}[gv] → ∀L,V. Y = L.ⓓV →
+ ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⟦V⟧{M}[gv, lv] ≗ d & ⫯{M}[d]lv = v.
+#M #gv #v #Y * -v -Y
+[ #lv #K #W #H destruct
+| #lv #d #L #V #HL #HV #K #W #H destruct /2 width=5 by ex3_2_intro/
+| #lv #d #L #V #_ #K #W #H destruct
+| #lv #d #I #L #_ #K #W #H destruct
+]
+qed-.
+
+lemma li_inv_abbr (M) (gv): ∀v,L,V. v ϵ ⟦L.ⓓV⟧{M}[gv] →
+ ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⟦V⟧{M}[gv, lv] ≗ d & ⫯{M}[d]lv = v.
+/2 width=3 by li_inv_abbr_aux/ qed-.
+
+fact li_inv_abst_aux (M) (gv): ∀v,Y. v ϵ ⟦Y⟧{M}[gv] → ∀L,W. Y = L.ⓛW →
+ ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⫯{M}[d]lv = v.
+#M #gv #v #Y * -v -Y
+[ #lv #K #U #H destruct
+| #lv #d #L #V #_ #_ #K #U #H destruct
+| #lv #d #L #V #HL #K #U #H destruct /2 width=4 by ex2_2_intro/
+| #lv #d #I #L #_ #K #U #H destruct
+]
+qed-.
+
+lemma li_inv_abst (M) (gv): ∀v,L,W. v ϵ ⟦L.ⓛW⟧{M}[gv] →
+ ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⫯{M}[d]lv = v.
+/2 width=4 by li_inv_abst_aux/ qed-.
+
+fact li_inv_unit_aux (M) (gv): ∀v,Y. v ϵ ⟦Y⟧{M}[gv] → ∀I,L. Y = L.ⓤ{I} →
+ ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⫯{M}[d]lv = v.
+#M #gv #v #Y * -v -Y
+[ #lv #J #K #H destruct
+| #lv #d #L #V #_ #_ #J #K #H destruct
+| #lv #d #L #V #_ #J #K #H destruct
+| #lv #d #I #L #HL #J #K #H destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma li_inv_unit (M) (gv): ∀v,I,L. v ϵ ⟦L.ⓤ{I}⟧{M}[gv] →
+ ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⫯{M}[d]lv = v.
+/2 width=4 by li_inv_unit_aux/ qed-.