matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_lift.etc

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  14
  15 include "basic_2/relocation/ldrop_ldrop.ma".
  16 include "basic_2/static/sta.ma".
  17
  18 (* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
  19
  20 (* Properties on relocation *************************************************)
  21
  22 (* Basic_1: was: sty0_lift *)
  23 lemma sta_lift: ∀h,G. l_liftable (sta h G).
  24 #h #G #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1
  25 [ #G #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2
  26   >(lift_inv_sort1 … H1) -X1
  27   >(lift_inv_sort1 … H2) -X2 //
  28 | #G #L1 #K1 #V1 #W1 #W #i #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2
  29   elim (lift_inv_lref1 … H) * #Hid #H destruct
  30   [ elim (lift_trans_ge … HW1 … HWU2) -W // #W2 #HW12 #HWU2
  31     elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
  32     elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
  33     /3 width=8/
  34   | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
  35     lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
  36   ]
  37 | #G #L1 #K1 #W1 #V1 #W #i #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2
  38   elim (lift_inv_lref1 … H) * #Hid #H destruct
  39   [ elim (lift_trans_ge … HW1 … HWU2) -W // <minus_plus #W #HW1 #HWU2
  40     elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
  41     elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
  42     lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
  43     elim (lift_total V1 (d-i-1) e) /3 width=8/
  44   | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
  45     lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
  46   ]
  47 | #a #I #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
  48   elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
  49   elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
  50   lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=5/
  51 | #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
  52   elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
  53   elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
  54   lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=5/
  55 | #G #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X #H #U2 #HU12
  56   elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=5/
  57 ]
  58 qed.
  59
  60 (* Note: apparently this was missing in basic_1 *)
  61 lemma sta_inv_lift1: ∀h,G. l_deliftable_sn (sta h G).
  62 #h #G #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2
  63 [ #G #L2 #k #L1 #d #e #_ #X #H
  64   >(lift_inv_sort2 … H) -X /2 width=3/
  65 | #G #L2 #K2 #V2 #W2 #W #i #HLK2 #HVW2 #HW2 #IHVW2 #L1 #d #e #HL21 #X #H
  66   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
  67   [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
  68     elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
  69     elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
  70   | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
  71     elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
  72     elim (lift_split … HW2 d (i-e+1)) -HW2 // [3: /2 width=1/ ]
  73     [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=8/
  74     | <le_plus_minus_comm //
  75     ]
  76   ]
  77 | #G #L2 #K2 #W2 #V2 #W #i #HLK2 #HWV2 #HW2 #IHWV2 #L1 #d #e #HL21 #X #H
  78   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
  79   [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
  80     elim (IHWV2 … HK21 … HW12) -K2 #V1 #_ #HWV1
  81     elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
  82   | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
  83     elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
  84     elim (lift_split … HW2 d (i-e+1)) -HW2 // [3: /2 width=1/ ]
  85     [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=8/
  86     | <le_plus_minus_comm //
  87     ]
  88   ]
  89 | #a #I #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
  90   elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
  91   elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /2 width=1/ -HL21 /3 width=5/
  92 | #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
  93   elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
  94   elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5/
  95 | #G #L2 #W2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
  96   elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct
  97   elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3/
  98 ]
  99 qed-.
 100
 101 (* Advanced forvard lemmas **************************************************)
 102
 103 (* Basic_1: was: sty0_correct *)
 104 lemma sta_fwd_correct: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ∃T0. ⦃G, L⦄ ⊢ U •[h] T0.
 105 #h #G #L #T #U #H elim H -G -L -T -U
 106 [ /2 width=2/
 107 | #G #L #K #V #W #W0 #i #HLK #_ #HW0 * #V0 #HWV0
 108   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
 109   elim (lift_total V0 0 (i+1)) /3 width=10/
 110 | #G #L #K #W #V #V0 #i #HLK #HWV #HWV0 #_
 111   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
 112   elim (lift_total V 0 (i+1)) /3 width=10/
 113 | #a #I #G #L #V #T #U #_ * /3 width=2/
 114 | #G #L #V #T #U #_ * #T0 #HUT0 /3 width=2/
 115 | #G #L #W #T #U #_ * /2 width=2/
 116 ]
 117 qed-.