X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2Fbasic_2%2Funfold%2Fdelift_lift.ma;h=26c4e3e8a6e27cdcd704685a1622acce0b194b61;hb=439b6ec33d749ba4e6ae0938e973a85bc23e306e;hp=a7f31fb809643badd7894803607de1d22ed3d1d1;hpb=fc5a0d62ece398d8547dda0f429b9f1e24bca306;p=helm.git

diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma
index a7f31fb80..26c4e3e8a 100644
--- a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma
@@ -12,25 +12,58 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/substitution/ldrop_sfr.ma".
 include "basic_2/unfold/tpss_lift.ma".
 include "basic_2/unfold/delift.ma".
 
-(* INVERSE TERM RELOCATION  *************************************************)
+(* INVERSE BASIC TERM RELOCATION  *******************************************)
 
 (* Advanced properties ******************************************************)
 
 lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e →
-                      ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 [0, d + e - i - 1] ≡ V2 →
-                      ⇧[0, d] V2 ≡ U2 → L ⊢ #i [d, e] ≡ U2.
+                      ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 →
+                      ⇧[0, d] V2 ≡ U2 → L ⊢ ▼*[d, e] #i ≡ U2.
 #L #K #V1 #V2 #U2 #i #d #e #Hdi #Hide #HLK * #V #HV1 #HV2 #HVU2
 elim (lift_total V 0 (i+1)) #U #HVU
-lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus <plus_minus_m_m
-/2 width=1/ /3 width=6/
+lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus <plus_minus_m_m /2 width=1/ #HV2U 
+lapply (lift_conf_be … HVU2 … HV2U ?) //
+>commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/ 
 qed.
- 
-(* Advanced forward lemmas **************************************************)
 
-lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 → i < d → U2 = #i.
+fact sfr_delift_aux: ∀L,T,T1,d,e. d + e ≤ |L| → ≽ [d, e] L → T = T1 →
+                     ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T @(cw_wf_ind … L T) -L -T #L #T #IH * * /2 width=2/
+[ #i #d #e #Hde #HL #H destruct
+  elim (lt_or_ge i d) #Hdi [ /3 width=2/ ]
+  elim (lt_or_ge i (d+e)) #Hide [2: /3 width=2/ ]
+  lapply (lt_to_le_to_lt … Hide Hde) #Hi
+  elim (ldrop_O1 … Hi) -Hi #I #K #V1 #HLK
+  lapply (sfr_inv_ldrop … HLK … HL ? ?) // #H destruct
+  lapply (ldrop_pair2_fwd_cw … HLK (#i)) #HKL
+  lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
+  lapply (ldrop_fwd_O1_length … HLK0) #H
+  lapply (sfr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
+  [1,2: /2 width=1/ | <minus_n_O <minus_plus ] #HK
+  elim (IH … HKL … HK ?) -IH -HKL -HK
+  [3: // |2: skip |4: >H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *)
+  elim (lift_total V2 0 d) /3 width=7/
+| #a #I #V1 #T1 #d #e #Hde #HL #H destruct
+  elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12
+  elim (IH (L.ⓑ{I}V1) T1 ? ? (d+1) e ? ? ?) -IH [3,6: // |2: skip |4,5: /2 width=1/ ] -Hde -HL #T2 #HT12
+  lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/
+| #I #V1 #T1 #d #e #Hde #HL #H destruct
+  elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12
+  elim (IH … T1 … Hde HL ?) -IH -Hde -HL [3,4: // |2: skip ] /3 width=2/
+]
+qed.
+
+lemma sfr_delift: ∀L,T1,d,e. d + e ≤ |L| → ≽ [d, e] L →
+                  ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
+/2 width=2/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → i < d → U2 = #i.
 #L #U2 #i #d #e * #U #HU #HU2 #Hid
 elim (tpss_inv_lref1 … HU) -HU
 [ #H destruct >(lift_inv_lref2_lt … HU2) //
@@ -40,10 +73,10 @@ elim (tpss_inv_lref1 … HU) -HU
 ]
 qed-.
 
-lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ #i [d, e] ≡ U2 →
+lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ ▼*[d, e] #i ≡ U2 →
                            d ≤ i → i < d + e →
                            ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 &
-                                      K ⊢ V1 [0, d + e - i - 1] ≡ V2 &
+                                      K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
                                       ⇧[0, d] V2 ≡ U2.
 #L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide
 elim (tpss_inv_lref1 … HU) -HU
@@ -53,7 +86,7 @@ elim (tpss_inv_lref1 … HU) -HU
 ]
 qed-.
 
-lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 →
+lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
                            d + e ≤ i → U2 = #(i - e).
 #L #U2 #i #d #e * #U #HU #HU2 #Hdei
 elim (tpss_inv_lref1 … HU) -HU
@@ -64,11 +97,11 @@ elim (tpss_inv_lref1 … HU) -HU
 ]
 qed-.
 
-lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 →
+lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
                         ∨∨ (i < d ∧ U2 = #i) 
                         |  (∃∃K,V1,V2. d ≤ i & i < d + e &
                                        ⇩[0, i] L ≡ K. ⓓV1 &
-                                       K ⊢ V1 [0, d + e - i - 1] ≡ V2 &
+                                       K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
                                        ⇧[0, d] V2 ≡ U2
                            )
                         |  (d + e ≤ i ∧ U2 = #(i - e)).
@@ -82,12 +115,12 @@ elim (lt_or_ge i d) #Hdi
 ]
 qed-.
 
-(* Relocation properties ****************************************************)
+(* Properties on basic term relocation **************************************)
 
-lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≡ T2 →
+lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
                       ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
                       ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 →
-                      L ⊢ U1 [dt, et] ≡ U2.
+                      L ⊢ ▼*[dt, et] U1 ≡ U2.
 #K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2
 elim (lift_total T d e) #U #HTU
 lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
@@ -95,23 +128,40 @@ elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU
 >(lift_mono … HTU2 … HT2) -T2 /2 width=3/
 qed.
 
-lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≡ T2 →
+lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
                       ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
                       ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
-                      L ⊢ U1 [dt, et + e] ≡ T2.
+                      L ⊢ ▼*[dt, et + e] U1 ≡ T2.
 #K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1
 elim (lift_total T d e) #U #HTU
 lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
 lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/
 qed.
 
-lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≡ T2 →
+lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
                       ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
                       ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
-                      L ⊢ U1 [dt + e, et] ≡ U2.
+                      L ⊢ ▼*[dt + e, et] U1 ≡ U2.
 #K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2
 elim (lift_total T d e) #U #HTU
 lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
 elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU
 >(lift_mono … HTU2 … HT2) -T2 /2 width=3/
 qed.
+
+lemma delift_inv_lift1_eq: ∀L,U1,T2,d,e. L ⊢ ▼*[d, e] U1 ≡ T2 →
+                           ∀K. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → T1 = T2.
+#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1
+lapply (tpss_inv_lift1_eq … HU12 … HTU1) -L -K #H destruct
+lapply (lift_inj … HTU1 … HTU2) -U2 //
+qed-.
+
+lemma delift_lift_div_be: ∀L,T1,T,d,e,i. L ⊢ ▼*[i, d + e - i] T1 ≡ T →
+                          ∀T2. ⇧[d, i - d] T2 ≡ T → d ≤ i → i ≤ d + e →
+                          L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide
+lapply (tpss_weak … HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10
+lapply (lift_trans_be … HT2 … HT0 ? ?) -T //
+>commutative_plus >commutative_plus in ⊢ (? ? (? % ?) ? ? → ?);
+<minus_le_minus_minus_comm // <plus_minus_m_m [ /2 width=3/ | /2 width=1/ ]
+qed.