(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lazyor_4.ma".
+include "basic_2/notation/relations/lazyor_5.ma".
include "basic_2/multiple/frees.ma".
(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
-definition llor: relation4 term lenv lenv lenv ≝ λT,L2,L1,L.
- ∧∧ |L1| ≤ |L2| & |L1| = |L|
+definition llor: ynat → relation4 term lenv lenv lenv ≝ λl,T,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)
+ ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → ⬇[i] L ≡ K.ⓑ{I}V → ∨∨
+ (∧∧ yinj i < l & I1 = I & V1 = V) |
+ (∧∧ (L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → ⊥) & I1 = I & V1 = V) |
+ (∧∧ l ≤ yinj i & L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ & I2 = I & V2 = V)
).
interpretation
"lazy union (local environment)"
- 'LazyOr L1 T L2 L = (llor T L2 L1 L).
+ 'LazyOr L1 T l L2 L = (llor l T L2 L1 L).
(* Basic properties *********************************************************)
-lemma llor_atom: ∀T,L2. ⋆ ⩖[T] L2 ≡ ⋆.
-#T #L2 @and3_intro //
+(* Note: this can be proved by llor_skip *)
+lemma llor_atom: ∀T,l. ⋆ ⋓[T, l] ⋆ ≡ ⋆.
+#T #l @and3_intro //
#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1
-elim (ldrop_inv_atom1 … HLK1) -HLK1 #H destruct
+elim (drop_inv_atom1 … HLK1) -HLK1 #H destruct
qed.