(* *)
(**************************************************************************)
+include "basic_2/substitution/ldrop_sfr.ma".
include "basic_2/unfold/tpss_lift.ma".
include "basic_2/unfold/delift.ma".
-(* DELIFT ON TERMS **********************************************************)
+(* INVERSE BASIC TERM RELOCATION *******************************************)
-(* Advanced forward lemmas **************************************************)
+(* Advanced properties ******************************************************)
-lemma delift_fwd_lref1_lt: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 → i < d → U2 = #i.
+lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e →
+ ⇩[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/ #HV2U
+lapply (lift_conf_be … HVU2 … HV2U ?) //
+>commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/
+qed.
+
+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 @(fw_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_lt … Hi) -Hi #I #K #V1 #HLK
+ lapply (sfr_inv_ldrop … HLK … HL ? ?) // #H destruct
+ lapply (ldrop_pair2_fwd_fw … 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) //
]
qed-.
-lemma delift_fwd_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
]
qed-.
-lemma delift_fwd_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
elim (lt_refl_false … Hi)
]
qed-.
+
+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 ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
+ ⇧[0, d] V2 ≡ U2
+ )
+ | (d + e ≤ i ∧ U2 = #(i - e)).
+#L #U2 #i #d #e #H
+elim (lt_or_ge i d) #Hdi
+[ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/
+| elim (lt_or_ge i (d+e)) #Hide
+ [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/
+ | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/
+ ]
+]
+qed-.
+
+(* Properties on basic term relocation **************************************)
+
+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 ⊢ ▼*[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
+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 ⊢ ▼*[dt, et] T1 ≡ T2 →
+ ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+ 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 ⊢ ▼*[dt, et] T1 ≡ T2 →
+ ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ 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.
projects / helm.git / blobdiff