-(**************************************************************************)
-(* ___ *)
-(* ||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/crsubeqt_2.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED REDUCTION **********************)
-
-inductive lsubx: relation lenv ≝
-| lsubx_sort: ∀L. lsubx L (⋆)
-| lsubx_bind: ∀I,L1,L2,V. lsubx L1 L2 → lsubx (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubx_abst: ∀L1,L2,V,W. lsubx L1 L2 → lsubx (L1.ⓓⓝW.V) (L2.ⓛW)
-.
-
-interpretation
- "local environment refinement (reduction)"
- 'CrSubEqT L1 L2 = (lsubx L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsubx_refl: ∀L. L ⓝ⊑ L.
-#L elim L -L // /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubx_inv_atom1_aux: ∀L1,L2. L1 ⓝ⊑ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2 //
-[ #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #_ #H destruct
-]
-qed-.
-
-lemma lsubx_inv_atom1: ∀L2. ⋆ ⓝ⊑ L2 → L2 = ⋆.
-/2 width=3 by lsubx_inv_atom1_aux/ qed-.
-
-fact lsubx_inv_abst1_aux: ∀L1,L2. L1 ⓝ⊑ L2 → ∀K1,W. L1 = K1.ⓛW →
- L2 = ⋆ ∨ ∃∃K2. K1 ⓝ⊑ K2 & L2 = K2.ⓛW.
-#L1 #L2 * -L1 -L2
-[ #L #K1 #W #H destruct /2 width=1/
-| #I #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #V2 #_ #K1 #W #H destruct
-]
-qed-.
-
-lemma lsubx_inv_abst1: ∀K1,L2,W. K1.ⓛW ⓝ⊑ L2 →
- L2 = ⋆ ∨ ∃∃K2. K1 ⓝ⊑ K2 & L2 = K2.ⓛW.
-/2 width=3 by lsubx_inv_abst1_aux/ qed-.
-
-fact lsubx_inv_abbr2_aux: ∀L1,L2. L1 ⓝ⊑ L2 → ∀K2,W. L2 = K2.ⓓW →
- ∃∃K1. K1 ⓝ⊑ K2 & L1 = K1.ⓓW.
-#L1 #L2 * -L1 -L2
-[ #L #K2 #W #H destruct
-| #I #L1 #L2 #V #HL12 #K2 #W #H destruct /2 width=3/
-| #L1 #L2 #V1 #V2 #_ #K2 #W #H destruct
-]
-qed-.
-
-lemma lsubx_inv_abbr2: ∀L1,K2,W. L1 ⓝ⊑ K2.ⓓW →
- ∃∃K1. K1 ⓝ⊑ K2 & L1 = K1.ⓓW.
-/2 width=3 by lsubx_inv_abbr2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsubx_fwd_length: ∀L1,L2. L1 ⓝ⊑ L2 → |L2| ≤ |L1|.
-#L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-lemma lsubx_fwd_ldrop2_bind: ∀L1,L2. L1 ⓝ⊑ L2 →
- ∀I,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I}W →
- (∃∃K1. K1 ⓝ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I}W) ∨
- ∃∃K1,V. K1 ⓝ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓓⓝW.V & I = Abst.
-#L1 #L2 #H elim H -L1 -L2
-[ #L #I #K2 #W #i #H
- elim (ldrop_inv_atom1 … H) -H #H destruct
-| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #i #H
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=3/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 * /4 width=3/ /4 width=4/
- ]
-| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #i #H
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=4/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 * /4 width=3/ /4 width=4/
- ]
-]
-qed-.
-
-lemma lsubx_fwd_ldrop2_abbr: ∀L1,L2. L1 ⓝ⊑ L2 →
- ∀K2,V,i. ⇩[0, i] L2 ≡ K2.ⓓV →
- ∃∃K1. K1 ⓝ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓓV.
-#L1 #L2 #HL12 #K2 #V #i #HLK2 elim (lsubx_fwd_ldrop2_bind … HL12 … HLK2) -L2 // *
-#K1 #W #_ #_ #H destruct
-qed-.