(* *)
(**************************************************************************)
-include "basic_2/notation/relations/crsubeqa_2.ma".
+include "basic_2/notation/relations/lrsubeqa_3.ma".
+include "basic_2/substitution/lsubr.ma".
include "basic_2/static/aaa.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
-inductive lsuba: relation lenv ≝
-| lsuba_atom: lsuba (⋆) (⋆)
-| lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1. ⓑ{I} V) (L2. ⓑ{I} V)
-| lsuba_abbr: ∀L1,L2,V,W,A. L1 ⊢ V ⁝ A → L2 ⊢ W ⁝ A →
- lsuba L1 L2 → lsuba (L1. ⓓV) (L2. ⓛW)
+inductive lsuba (G:genv): relation lenv ≝
+| lsuba_atom: lsuba G (⋆) (⋆)
+| lsuba_pair: ∀I,L1,L2,V. lsuba G L1 L2 → lsuba G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsuba_abbr: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
+ lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
.
interpretation
"local environment refinement (atomic arity assigment)"
- 'CrSubEqA L1 L2 = (lsuba L1 L2).
+ 'LRSubEqA G L1 L2 = (lsuba G L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ⁝⊑ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2
+fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
]
-qed.
+qed-.
+
+lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⁝⊑ L2 → L2 = ⋆.
+/2 width=4 by lsuba_inv_atom1_aux/ qed-.
-lemma lsuba_inv_atom1: ∀L2. ⋆ ⁝⊑ L2 → L2 = ⋆.
-/2 width=3/ qed-.
-
-fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
- (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
- L2 = K2. ⓛW & I = Abbr.
-#L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
+fact lsuba_inv_pair1_aux: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+ (∃∃K2. G ⊢ K1 ⁝⊑ K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⁝⊑ K2 & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+#G #L1 #L2 * -L1 -L2
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9/
]
-qed.
+qed-.
-lemma lsuba_inv_pair1: ∀I,K1,L2,V. K1. ⓑ{I} V ⁝⊑ L2 →
- (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
- L2 = K2. ⓛW & I = Abbr.
-/2 width=3/ qed-.
+lemma lsuba_inv_pair1: ∀I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⁝⊑ L2 →
+ (∃∃K2. G ⊢ K1 ⁝⊑ K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⁝⊑ K2 &
+ I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+/2 width=3 by lsuba_inv_pair1_aux/ qed-.
-fact lsuba_inv_atom2_aux: ∀L1,L2. L1 ⁝⊑ L2 → L2 = ⋆ → L1 = ⋆.
-#L1 #L2 * -L1 -L2
+fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → L2 = ⋆ → L1 = ⋆.
+#G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
]
-qed.
+qed-.
-lemma lsubc_inv_atom2: ∀L1. L1 ⁝⊑ ⋆ → L1 = ⋆.
-/2 width=3/ qed-.
-
-fact lsuba_inv_pair2_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
- (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
- L1 = K1. ⓓV & I = Abst.
-#L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
+lemma lsubc_inv_atom2: ∀G,L1. G ⊢ L1 ⁝⊑ ⋆ → L1 = ⋆.
+/2 width=4 by lsuba_inv_atom2_aux/ qed-.
+
+fact lsuba_inv_pair2_aux: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
+ (∃∃K1. G ⊢ K1 ⁝⊑ K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⁝⊑ K2 & I = Abst & L1 = K1.ⓓⓝW.V.
+#G #L1 #L2 * -L1 -L2
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7/
]
-qed.
+qed-.
+
+lemma lsuba_inv_pair2: ∀I,G,L1,K2,W. G ⊢ L1 ⁝⊑ K2.ⓑ{I}W →
+ (∃∃K1. G ⊢ K1 ⁝⊑ K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⁝⊑ K2 &
+ I = Abst & L1 = K1.ⓓⓝW.V.
+/2 width=3 by lsuba_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
-lemma lsuba_inv_pair2: ∀I,L1,K2,W. L1 ⁝⊑ K2. ⓑ{I} W →
- (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
- L1 = K1. ⓓV & I = Abst.
-/2 width=3/ qed-.
+lemma lsuba_fwd_lsubr: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → L1 ⊑ L2.
+#G #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
(* Basic properties *********************************************************)
-lemma lsuba_refl: ∀L. L ⁝⊑ L.
-#L elim L -L // /2 width=1/
+lemma lsuba_refl: ∀G,L. G ⊢ L ⁝⊑ L.
+#G #L elim L -L // /2 width=1/
qed.