-(**************************************************************************)
-(* ___ *)
-(* ||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/static/ssta.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******)
-
-(* Note: may not be transitive *)
-inductive lsubss (h:sh) (g:sd h): relation lenv ≝
-| lsubss_atom: lsubss h g (⋆) (⋆)
-| lsubss_pair: ∀I,L1,L2,W. lsubss h g L1 L2 →
- lsubss h g (L1. ⓑ{I} W) (L2. ⓑ{I} W)
-| lsubss_abbr: ∀L1,L2,V,W,l. ⦃h, L1⦄ ⊢ V •[g, l+1] W → ⦃h, L2⦄ ⊢ V •[g, l+1] W →
- lsubss h g L1 L2 → lsubss h g (L1. ⓓV) (L2. ⓛW)
-.
-
-interpretation
- "local environment refinement (stratified static type assigment)"
- 'CrSubEqS h g L1 L2 = (lsubss h g L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubss_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 = ⋆ → L2 = ⋆.
-#h #g #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #l #_ #_ #_ #H destruct
-]
-qed.
-
-lemma lsubss_inv_atom1: ∀h,g,L2. h ⊢ ⋆ •⊑[g] L2 → L2 = ⋆.
-/2 width=5/ qed-.
-
-fact lsubss_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
- ∀I,K1,V. L1 = K1. ⓑ{I} V →
- (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
- h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr.
-#h #g #L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/
-]
-qed.
-
-lemma lsubss_inv_pair1: ∀h,g,I,K1,L2,V. h ⊢ K1. ⓑ{I} V •⊑[g] L2 →
- (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
- h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr.
-/2 width=3/ qed-.
-
-fact lsubss_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
-#h #g #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #l #_ #_ #_ #H destruct
-]
-qed.
-
-lemma lsubss_inv_atom2: ∀h,g,L1. h ⊢ L1 •⊑[g] ⋆ → L1 = ⋆.
-/2 width=5/ qed-.
-
-fact lsubss_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
- ∀I,K2,W. L2 = K2. ⓑ{I} W →
- (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
- h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst.
-#h #g #L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/
-]
-qed.
-
-lemma lsubss_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 •⊑[g] K2. ⓑ{I} W →
- (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
- h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst.
-/2 width=3/ qed-.
-
-(* Basic_forward lemmas *****************************************************)
-
-lemma lsubss_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L1|] L2.
-#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-lemma lsubss_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L2|] L2.
-#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsubss_refl: ∀h,g,L. h ⊢ L •⊑[g] L.
-#h #g #L elim L -L // /2 width=1/
-qed.