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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/unfold/delift_tpss.ma".
16 include "basic_2/unfold/delift_ltpss.ma".
17 include "basic_2/unfold/thin.ma".
19 (* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************)
21 (* Inversion lemmas on inverse basic term relocation ************************)
23 lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
24 ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 &
25 K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 &
27 #I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e
28 elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct
29 elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
32 (* Properties on inverse basic term relocation ******************************)
34 lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 →
35 ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
36 #L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I
37 elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
38 lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct
39 lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/
42 lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
43 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
44 ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
45 ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
46 L ⊢ ▼*[dd, ee] U2 ≡ T2.
47 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd
48 lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
49 elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
50 elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT
51 lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
52 lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
55 lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
56 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
57 ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
58 ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
59 L ⊢ ▼*[dd, ee] U2 ≡ T2.
62 lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
63 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
64 ∀K. ▼*[dd, ee] L ≡ K →
65 d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
66 ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
67 L ⊢ ▼*[dd, ee] U2 ≡ T2.
68 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee
69 lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
70 elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
71 elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT
72 lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
73 lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
76 lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
77 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
78 ∀K. ▼*[dd, ee] L ≡ K →
79 d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
80 ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
81 L ⊢ ▼*[dd, ee] U2 ≡ T2.
82 /3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *)
84 lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
85 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
86 ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
87 ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
88 L ⊢ ▼*[dd, ee] U2 ≡ T2.
89 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee
90 lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
91 elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
92 elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT
93 lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
94 lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
97 lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
98 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
99 ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
100 ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
101 L ⊢ ▼*[dd, ee] U2 ≡ T2.