matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/substitution/ldrop_ldrop.ma".
  16 include "basic_2/static/ssta.ma".
  17
  18 (* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
  19
  20 (* Properties on relocation *************************************************)
  21
  22 (* Basic_1: was just: sty0_lift *)
  23 lemma ssta_lift: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
  24                  ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
  25                  ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄.
  26 #h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
  27 [ #L1 #k #l #Hkl #L2 #d #e #HL21 #X1 #H1 #X2 #H2
  28   >(lift_inv_sort1 … H1) -X1
  29   >(lift_inv_sort1 … H2) -X2 /2 width=1/
  30 | #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2
  31   elim (lift_inv_lref1 … H) * #Hid #H destruct
  32   [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2
  33     elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
  34     elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
  35     /3 width=8/
  36   | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
  37     lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
  38   ]
  39 | #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2
  40   elim (lift_inv_lref1 … H) * #Hid #H destruct
  41   [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // <minus_plus #W #HW1 #HWU2
  42     elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
  43     elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
  44     lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
  45     elim (lift_total V1 (d-i-1) e) /3 width=8/
  46   | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
  47     lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
  48   ]
  49 | #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
  50   elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
  51   elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
  52   lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=5/
  53 | #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
  54   elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
  55   elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
  56   lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=5/
  57 | #L1 #W1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL21 #X #H #U2 #HU12
  58   elim (lift_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct /3 width=5/
  59 ]
  60 qed.
  61
  62 (* Note: apparently this was missing in basic_1 *)
  63 lemma ssta_inv_lift1: ∀h,g,L2,T2,U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ →
  64                       ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
  65                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ & ⇧[d, e] U1 ≡ U2.
  66 #h #g #L2 #T2 #U2 #l #H elim H -L2 -T2 -U2 -l
  67 [ #L2 #k #l #Hkl #L1 #d #e #_ #X #H
  68   >(lift_inv_sort2 … H) -X /3 width=3/
  69 | #L2 #K2 #V2 #W2 #W #i #l #HLK2 #HVW2 #HW2 #IHVW2 #L1 #d #e #HL21 #X #H
  70   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
  71   [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
  72     elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HVW1 #HW12
  73     elim (lift_trans_le … HW12 … HW2 ?) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=6/
  74   | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
  75     elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
  76     elim (lift_split … HW2 d (i-e+1) ? ? ?) -HW2 // [3: /2 width=1/ ]
  77     [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=6/
  78     | <le_plus_minus_comm //
  79     ]
  80   ]
  81 | #L2 #K2 #W2 #V2 #W #i #l #HLK2 #HWV2 #HW2 #IHWV2 #L1 #d #e #HL21 #X #H
  82   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
  83   [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
  84     elim (IHWV2 … HK21 … HW12) -K2 #V1 #HWV1 #_
  85     elim (lift_trans_le … HW12 … HW2 ?) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=6/
  86   | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
  87     elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
  88     elim (lift_split … HW2 d (i-e+1) ? ? ?) -HW2 // [3: /2 width=1/ ]
  89     [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=6/
  90     | <le_plus_minus_comm //
  91     ]
  92   ]
  93 | #a #I #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #d #e #HL21 #X #H
  94   elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
  95   elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /2 width=1/ -HL21 /3 width=5/
  96 | #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #d #e #HL21 #X #H
  97   elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
  98   elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5/
  99 | #L2 #W2 #T2 #U2 #l #_ #IHTU2 #L1 #d #e #HL21 #X #H
 100   elim (lift_inv_flat2 … H) -H #W1 #T1 #HW12 #HT12 #H destruct
 101   elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3/
 102 ]
 103 qed.
 104
 105 (* Advanced forvard lemmas **************************************************)
 106
 107 (* Basic_1: was just: sty0_correct *)
 108 lemma ssta_fwd_correct: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
 109                         ∃T0. ⦃h, L⦄ ⊢ U •[g] ⦃l-1, T0⦄.
 110 #h #g #L #T #U #l #H elim H -L -T -U -l
 111 [ /4 width=2/
 112 | #L #K #V #W #W0 #i #l #HLK #_ #HW0 * #V0 #HWV0
 113   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
 114   elim (lift_total V0 0 (i+1)) /3 width=10/
 115 | #L #K #W #V #V0 #i #l #HLK #HWV #HWV0 #_
 116   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
 117   elim (lift_total V 0 (i+1)) /3 width=10/
 118 | #a #I #L #V #T #U #l #_ * /3 width=2/
 119 | #L #V #T #U #l #_ * #T0 #HUT0 /3 width=2/
 120 | #L #W #T #U #l #_ * /2 width=2/
 121 ]
 122 qed-.