- we introduced the pointer_step rc in the perspective of proving

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / substitution / ldrop_sfr.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_sfr.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_sfr.ma
index e9befe700f343d46404040719aaac433826718fb..78a15d70dcdc6731047d8e8a7f4cf1aae80bb9b1 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_sfr.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/substitution/ldrop_sfr.ma
@@ -13,27 +13,14 @@
 (**************************************************************************)
 
 include "basic_2/substitution/lsubs_sfr.ma".
-include "basic_2/substitution/ldrop.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
 
 (* DROPPING *****************************************************************)
 
-(* Properties about local env. full refinement for substitution *************)
-
-lemma sfr_ldrop: ∀L,d,e.
-                 (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
-                 ≼ [d, e] L.
-#L elim L -L //
-#L #I #V #IHL #d @(nat_ind_plus … d) -d
-[ #e @(nat_ind_plus … e) -e //
-  #e #_ #HH
-  >(HH I L V 0 ? ? ?) // /5 width=6/
-| /5 width=6/
-]
-qed.
-
 (* Inversion lemmas about local env. full refinement for substitution *******)
 
-lemma sfr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ≼ [d, e] L →
+(* Note: ldrop_ldrop not needed *)
+lemma sfr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ≽ [d, e] L →
                      d ≤ i → i < d + e → I = Abbr.
 #I #L elim L -L
 [ #K #V #i #H
@@ -43,7 +30,7 @@ lemma sfr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ≼ [d
   [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct
     lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct
     lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H
-    elim (lsubs_inv_abbr1 … H ?) -H // -Hide #K #_ #H destruct //
+    elim (lsubs_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct //
   | #Hi #HLK #d @(nat_ind_plus … d) -d
     [ #e #H #_ #Hide
       elim (sfr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct
@@ -55,3 +42,51 @@ lemma sfr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ≼ [d
   ]
 ]
 qed-.
+
+(* Properties about local env. full refinement for substitution *************)
+
+(* Note: ldrop_ldrop not needed *)
+lemma sfr_ldrop: ∀L,d,e.
+                 (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
+                 ≽ [d, e] L.
+#L elim L -L //
+#L #I #V #IHL #d @(nat_ind_plus … d) -d
+[ #e @(nat_ind_plus … e) -e //
+  #e #_ #HH
+  >(HH I L V 0 ? ? ?) // /5 width=6/
+| /5 width=6/
+]
+qed.
+
+lemma sfr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 → 
+                          dd + ee ≤ d → ≽ [dd, ee] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
+@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
+lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid
+elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H
+elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct
+@(sfr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 //
+qed.
+
+lemma sfr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
+                             ∀dd,ee. ≽ [dd, ee] L1 →
+                             dd ≤ d + e → d + e ≤ dd + ee →
+                             ≽ [d, dd + ee - d - e] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee
+@sfr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2
+lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie
+>commutative_plus in Hiddee; >minus_minus_comm <plus_minus_m_m /2 width=1/ -Hddee #Hiddee
+lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hdi  #HL1K2
+@(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus // -Hddie /2 width=1/
+qed.
+
+lemma sfr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 → 
+                          d + e ≤ dd → ≽ [dd - e, ee] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
+@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
+elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd
+>plus_minus in Hiddee; // #Hiddee
+lapply (transitive_le … Hdde Hddi) -Hdde #Hid
+lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2
+@(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/
+qed.