(* *)
(**************************************************************************)
-include "basic_2/unfold/tpss_alt.ma".
-include "basic_2/unfold/ltpss_ltpss.ma".
-include "basic_2/unfold/delift_alt.ma".
+include "basic_2/unfold/delift_tpss.ma".
+include "basic_2/unfold/delift_ltpss.ma".
include "basic_2/unfold/thin.ma".
-(* DELIFT ON LOCAL ENVIRONMENTS *********************************************)
+(* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************)
-(* Properties on deflift on terms *******************************************)
+(* Inversion lemmas on inverse basic term relocation ************************)
-lemma thin_delift1: ∀L1,L2,d,e. L1 [d, e] ≡ L2 → ∀V1,V2. L1 ⊢ V1 [d, e] ≡ V2 →
- ∀I. L1.ⓑ{I}V1 [d + 1, e] ≡ L2.ⓑ{I}V2.
+lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 &
+ K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 &
+ L2 = K2. ⓑ{I} V2.
+#I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e
+elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct
+elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
+qed-.
+
+(* Properties on inverse basic term relocation ******************************)
+
+lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 →
+ ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
#L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I
-elim (ltpss_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
-elim (tpss_inv_lift1_be … HV0 … HL2 … HV2 ? ?) -HV0 // <minus_n_n #X #H1 #H2
-lapply (tpss_inv_refl_O2 … H1) -H1 #H destruct
-lapply (lift_mono … H2 … HV2) -H2 #H destruct /3 width=5/
+elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
+lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct
+lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/
+qed.
+
+lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
+
+lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *)
+
+lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
qed.
-axiom delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
- ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 → ∀K. L [dd, ee] ≡ K →
- d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 [dd - d, e - ee] ▶* T2 & L ⊢ U2 [dd, ee] ≡ T2.
-(*
-#L #U1 #U2 #d #e #H @(tpss_ind_alt … H) -L -U1 -U2 -d -e
-[ #L * #i #d #e #X #dd #ee #H
- [ >(delift_inv_sort1 … H) -X /2 width=3/
- | elim (delift_inv_lref1 … H) -H * [1,3: /3 width=3/ | /3 width=6/ ]
- | >(delift_inv_gref1 … H) -X /2 width=3/
- ]
-| #L #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #T1 #dd #ee #H #K2 #HLK2 #Hdd #Hddee
- lapply
-
- @(ex2_1_intro … X) // /2 width=6/
-*)
+lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
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