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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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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. K1. ⓑ{I} V1 [d, e] ≡ L2 → 0 < d →
24 ∃∃K2,V2. K1 [d - 1, e] ≡ K2 &
25 K1 ⊢ V1 [d - 1, e] ≡ V2 &
27 #I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e
28 elim (ltpss_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct
29 elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct
30 lapply (ltpss_tpss_trans_eq … HV1 … HK1) -HV1 /3 width=5/
33 (* Properties on inverse basic term relocation ******************************)
35 lemma thin_delift1: ∀L1,L2,d,e. L1 [d, e] ≡ L2 → ∀V1,V2. L1 ⊢ V1 [d, e] ≡ V2 →
36 ∀I. L1.ⓑ{I}V1 [d + 1, e] ≡ L2.ⓑ{I}V2.
37 #L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I
38 elim (ltpss_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
39 elim (tpss_inv_lift1_be … HV0 … HL2 … HV2 ? ?) -HV0 // <minus_n_n #X #H1 #H2
40 lapply (tpss_inv_refl_O2 … H1) -H1 #H destruct
41 lapply (lift_mono … H2 … HV2) -H2 #H destruct /3 width=5/
44 lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
45 ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
46 ∀K. L [dd, ee] ≡ K → d + e ≤ dd →
47 ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
49 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd
50 lapply (delift_ltpss_conf_eq … HUT1 … HLY) -HUT1 #HUT1
51 elim (ltpss_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
52 elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 // -Hdedd #T #HT1 #HUT
53 lapply (tpss_delift_trans_eq … HU2 … HUT) -U #HU2T
54 lapply (ltpss_delift_trans_eq … HLY … HU2T) -Y /2 width=3/
57 lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
58 ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
59 ∀K. L [dd, ee] ≡ K → d + e ≤ dd →
60 ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
64 lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
65 ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
67 d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
68 ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
70 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee
71 lapply (delift_ltpss_conf_eq … HUT1 … HLY) -HUT1 #HUT1
72 elim (ltpss_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
73 elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 // -Hdd -Hdde -Hddee #T #HT1 #HUT
74 lapply (tpss_delift_trans_eq … HU2 … HUT) -U #HU2T
75 lapply (ltpss_delift_trans_eq … HLY … HU2T) -Y /2 width=3/
78 lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
79 ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
81 d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
82 ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
84 /3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap/ qed. (**) (* too slow without trace *)
86 lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
87 ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
88 ∀K. L [dd, ee] ≡ K → d ≤ dd → dd + ee ≤ d + e →
89 ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
91 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee
92 lapply (delift_ltpss_conf_eq … HUT1 … HLY) -HUT1 #HUT1
93 elim (ltpss_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
94 elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 // -Hdd -Hddee #T #HT1 #HUT
95 lapply (tpss_delift_trans_eq … HU2 … HUT) -U #HU2T
96 lapply (ltpss_delift_trans_eq … HLY … HU2T) -Y /2 width=3/
99 lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
100 ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
101 ∀K. L [dd, ee] ≡ K → d ≤ dd → dd + ee ≤ d + e →
102 ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
103 L ⊢ U2 [dd, ee] ≡ T2.