matita/matita/contribs/lambdadelta/basic_2A/etc/delift/thin_delift.etc

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  14
  15 include "basic_2/unfold/delift_tpss.ma".
  16 include "basic_2/unfold/delift_ltpss.ma".
  17 include "basic_2/unfold/thin.ma".
  18
  19 (* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************)
  20
  21 (* Inversion lemmas on inverse basic term relocation ************************)
  22
  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 &
  26                                  L2 = K2. ⓑ{I} 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/
  30 qed-.
  31
  32 (* Properties on inverse basic term relocation ******************************)
  33
  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/
  40 qed.
  41
  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/
  53 qed.
  54
  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.
  60 /3 width=3/ qed.
  61
  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/
  74 qed.
  75
  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 *)
  83
  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/
  95 qed.
  96
  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.
 102 /3 width=3/ qed.