matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  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/relocation/lifts.ma".
  16
  17 (* GENERIC RELOCATION *******************************************************)
  18
  19 (* Main properties **********************************************************)
  20
  21 (* Basic_2A1: includes: lift_inj *)
  22 theorem lifts_inj: ∀t,T1,U. ⬆*[t] T1 ≡ U → ∀T2. ⬆*[t] T2 ≡ U → T1 = T2.
  23 #t #T1 #U #H elim H -t -T1 -U
  24 [ /2 width=2 by lifts_inv_sort2/
  25 | #i1 #j #t #Hi1j #X #HX elim (lifts_inv_lref2 … HX) -HX
  26   /4 width=4 by at_inj, eq_f/
  27 | /2 width=2 by lifts_inv_gref2/
  28 | #a #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_bind2 … HX) -HX
  29   #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
  30 | #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_flat2 … HX) -HX
  31   #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
  32 ]
  33 qed-.
  34
  35 (* Basic_1: includes: lift_gen_lift *)
  36 (* Basic_2A1: includes: lift_div_le lift_div_be *)
  37 theorem lifts_div: ∀T,T2,t2. ⬆*[t2] T2 ≡ T → ∀T1,t. ⬆*[t] T1 ≡ T →
  38                    ∀t1. t2 ⊚ t1 ≡ t → ⬆*[t1] T1 ≡ T2.
  39 #T #T2 #t2 #H elim H -T -T2 -t2
  40 [ #k #t2 #T1 #t #H >(lifts_inv_sort2 … H) -T1 //
  41 | #i2 #i #t2 #Hi2 #T1 #t #H #t1 #Ht21 elim (lifts_inv_lref2 … H) -H
  42   #i1 #Hi1 #H destruct /3 width=6 by lifts_lref, after_fwd_at1/
  43 | #p #t2 #T1 #t #H >(lifts_inv_gref2 … H) -T1 //
  44 | #a #I #W2 #W #U2 #U #t2 #_ #_ #IHW #IHU #T1 #t #H
  45   elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
  46   /4 width=3 by lifts_bind, after_true/
  47 | #I #W2 #W #U2 #U #t2 #_ #_ #IHW #IHU #T1 #t #H
  48   elim (lifts_inv_flat2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
  49   /3 width=3 by lifts_flat/
  50 ]
  51 qed-.
  52
  53 (* Basic_2A1: includes: lift_mono *)
  54 theorem lifts_mono: ∀t,T,U1. ⬆*[t] T ≡ U1 → ∀U2. ⬆*[t] T ≡ U2 → U1 = U2.
  55 #t #T #U1 #H elim H -t -T -U1
  56 [ /2 width=2 by lifts_inv_sort1/
  57 | #i1 #j #t #Hi1j #X #HX elim (lifts_inv_lref1 … HX) -HX
  58   /4 width=4 by at_mono, eq_f/
  59 | /2 width=2 by lifts_inv_gref1/
  60 | #a #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_bind1 … HX) -HX
  61   #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
  62 | #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_flat1 … HX) -HX
  63   #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
  64 ]
  65 qed-.
  66
  67 (* Basic_1: was: lift1_lift1 (left to right) *)
  68 (* Basic_1: includes: lift_free (left to right) lift_d lift1_xhg (right to left) lift1_free (right to left) *)
  69 (* Basic_2A1: includes: lift_trans_be lift_trans_le lift_trans_ge lifts_lift_trans_le lifts_lift_trans *)
  70 theorem lifts_trans: ∀T1,T,t1. ⬆*[t1] T1 ≡ T → ∀T2,t2. ⬆*[t2] T ≡ T2 →
  71                      ∀t. t2 ⊚ t1 ≡ t → ⬆*[t] T1 ≡ T2.
  72 #T1 #T #t1 #H elim H -T1 -T -t1
  73 [ #k #t1 #T2 #t2 #H >(lifts_inv_sort1 … H) -T2 //
  74 | #i1 #i #t1 #Hi1 #T2 #t2 #H #t #Ht21 elim (lifts_inv_lref1 … H) -H
  75   #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at/
  76 | #p #t1 #T2 #t2 #H >(lifts_inv_gref1 … H) -T2 //
  77 | #a #I #W1 #W #U1 #U #t1 #_ #_ #IHW #IHU #T2 #t2 #H
  78   elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
  79   /4 width=3 by lifts_bind, after_true/
  80 | #I #W1 #W #U1 #U #t1 #_ #_ #IHW #IHU #T2 #t2 #H
  81   elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
  82   /3 width=3 by lifts_flat/
  83 ]
  84 qed-.
  85
  86 (* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
  87 theorem lifts_conf: ∀T,T1,t1. ⬆*[t1] T ≡ T1 → ∀T2,t. ⬆*[t] T ≡ T2 →
  88                     ∀t2. t2 ⊚ t1 ≡ t → ⬆*[t2] T1 ≡ T2.
  89 #T #T1 #t1 #H elim H -T -T1 -t1
  90 [ #k #t1 #T2 #t #H >(lifts_inv_sort1 … H) -T2 //
  91 | #i #i1 #t1 #Hi1 #T2 #t #H #t2 #Ht21 elim (lifts_inv_lref1 … H) -H
  92   #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at2/
  93 | #p #t1 #T2 #t #H >(lifts_inv_gref1 … H) -T2 //
  94 | #a #I #W #W1 #U #U1 #t1 #_ #_ #IHW #IHU #T2 #t #H
  95   elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
  96   /4 width=3 by lifts_bind, after_true/
  97 | #I #W #W1 #U #U1 #t1 #_ #_ #IHW #IHU #T2 #t #H
  98   elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
  99   /3 width=3 by lifts_flat/
 100 ]
 101 qed-.