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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/lazyor_4.ma".
+include "basic_2/substitution/cofrees.ma".
+
+(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
+
+definition llor: relation4 term lenv lenv lenv ≝ λT,L2,L1,L.
+                 ∧∧ |L1| ≤ |L2| & |L1| = |L|
+                  & (∀I1,I2,I,K1,K2,K,V1,V2,V,i.
+                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → ⇩[i] L ≡ K.ⓑ{I}V →
+                       (∧∧ L1 ⊢ i ~ϵ 𝐅*[yinj 0]⦃T⦄ & I1 = I & V1 = V) ∨
+                       (∧∧ (L1 ⊢ i ~ϵ 𝐅*[yinj 0]⦃T⦄ → ⊥) & I1 = I & V2 = V)
+                    ).
+
+interpretation
+   "lazy union (local environment)"
+   'LazyOr L1 T L2 L = (llor T L2 L1 L).
+
+(* Basic properties *********************************************************)
+
+lemma llor_atom: ∀T,L2. ⋆ ⩖[T] L2 ≡ ⋆.
+#T #L2 @and3_intro //
+#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1
+elim (ldrop_inv_atom1 … HLK1) -HLK1 #H destruct
+qed.
+
+axiom llor_total: ∀L1,L2,T. |L1| ≤ |L2| → ∃L. L1 ⩖[T] L2 ≡ L.