- new definition of subst based on drop

[helm.git] / matita / matita / lib / lambda-delta / substitution / subst_defs.ma

diff --git a/matita/matita/lib/lambda-delta/substitution/subst_defs.ma b/matita/matita/lib/lambda-delta/substitution/subst_defs.ma
index 5ef343857bb0aee3f39ed98e71cd50e0ad2e19c9..27de6a5ce4fbd9b1fa1e91f15bf2892ff93f141a 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/subst_defs.ma
+++ b/matita/matita/lib/lambda-delta/substitution/subst_defs.ma
@@ -9,19 +9,17 @@
      \ /
       V_______________________________________________________________ *)
 
-include "lambda-delta/syntax/lenv.ma".
-include "lambda-delta/substitution/lift_defs.ma".
+include "lambda-delta/substitution/drop_defs.ma".
 
 (* TELESCOPIC SUBSTITUTION **************************************************)
 
 inductive subst: lenv → term → nat → nat → term → Prop ≝
 | subst_sort   : ∀L,k,d,e. subst L (⋆k) d e (⋆k)
 | subst_lref_lt: ∀L,i,d,e. i < d → subst L (#i) d e (#i)
-| subst_lref_O : ∀L,V1,V2,e. subst L V1 0 e V2 →
-                 subst (L. 𝕓{Abbr} V1) #0 0 (e + 1) V2
-| subst_lref_S : ∀L,I,V,i,T1,T2,d,e.
-                 d ≤ i → i < d + e → subst L #i d e T1 → ↑[d,1] T1 ≡ T2 →
-                 subst (L. 𝕓{I} V) #(i + 1) (d + 1) e T2
+| subst_lref_be: ∀L,K,V,U,i,d,e.
+                 d ≤ i → i < d + e →
+                 ↑[0, i] K. 𝕓{Abbr} V ≡ L → subst K V d (d + e - i - 1) U →
+                 subst L (#i) d e U
 | subst_lref_ge: ∀L,i,d,e. d + e ≤ i → subst L (#i) d e (#(i - e))
 | subst_bind   : ∀L,I,V1,V2,T1,T2,d,e.
                  subst L V1 d e V2 → subst (L. 𝕓{I} V1) T1 (d + 1) e T2 →
@@ -33,9 +31,42 @@ inductive subst: lenv → term → nat → nat → term → Prop ≝
 
 interpretation "telescopic substritution" 'RSubst L T1 d e T2 = (subst L T1 d e T2).
 
+(* The basic inversion lemmas ***********************************************)
+
+lemma subst_inv_lref1_be_aux: ∀d,e,L,T,U. L ⊢ ↓[d, e] T ≡ U →
+                              ∀i. d ≤ i → i < d + e → T = #i →
+                              ∃∃K,V. ↑[0, i] K. 𝕓{Abbr} V ≡ L &
+                                     K ⊢ ↓[d, d + e - i - 1] V ≡ U.
+#d #e #L #T #U #H elim H -H d e L T U
+[ #L #k #d #e #i #_ #_ #H destruct
+| #L #j #d #e #Hid #i #Hdi #_ #H destruct -j;
+  lapply (le_to_lt_to_lt … Hdi … Hid) -Hdi Hid #Hdd
+  elim (lt_false … Hdd)
+| #L #K #V #U #j #d #e #_ #_ #HLK #HVU #_ #i #Hdi #Hide #H destruct -j /2/
+| #L #j #d #e #Hdei #i #_ #Hide #H destruct -j;
+  lapply (le_to_lt_to_lt … Hdei … Hide) -Hdei Hide #Hdede
+  elim (lt_false … Hdede)
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #_ #_ #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #_ #_ #H destruct
+]
+qed.
+
+lemma subst_inv_lref1_be: ∀d,e,i,L,U. L ⊢ ↓[d, e] #i ≡ U →
+                          d ≤ i → i < d + e →
+                          ∃∃K,V. ↑[0, i] K. 𝕓{Abbr} V ≡ L &
+                                 K ⊢ ↓[d, d + e - i - 1] V ≡ U.
+/2/ qed.
+
 (* The basic properties *****************************************************)
 
 lemma subst_lift_inv: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀L. L ⊢ ↓[d,e] T2 ≡ T1.
 #d #e #T1 #T2 #H elim H -H d e T1 T2 /2/
-#i #d #e #Hdi #L >(minus_plus_m_m i e) in ⊢ (? ? ? ? ? %) /3/ (**) (* use \ldots *)
+#i #d #e #Hdi #L >(minus_plus_m_m i e) in ⊢ (? ? ? ? ? %) /3/
 qed.
+(*
+| subst_lref_O : ∀L,V1,V2,e. subst L V1 0 e V2 →
+                 subst (L. 𝕓{Abbr} V1) #0 0 (e + 1) V2
+| subst_lref_S : ∀L,I,V,i,T1,T2,d,e.
+                 d ≤ i → i < d + e → subst L #i d e T1 → ↑[d,1] T2 ≡ T1 →
+                 subst (L. 𝕓{I} V) #(i + 1) (d + 1) e T2
+*)