matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts_bind.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_lifts.ma".
  16 include "basic_2/relocation/lifts_bind.ma".
  17
  18 (* GENERIC RELOCATION FOR BINDERS *******************************************)
  19
  20 (* Main properties **********************************************************)
  21
  22 theorem liftsb_div3: ∀f2,I,I2. ⬆*[f2] I2 ≘ I → ∀f,I1. ⬆*[f] I1 ≘ I →
  23                      ∀f1. f2 ⊚ f1 ≘ f → ⬆*[f1] I1 ≘ I2.
  24 #f2 #I #I2 * -I -I2 #I [2: #V #V2 #HV2 ] #f #I1 #H
  25 [ elim (liftsb_inv_pair_dx … H) | lapply (liftsb_inv_unit_dx … H) ] -H
  26 /3 width=6 by lifts_div3, ext2_pair, ext2_unit/
  27 qed-.
  28
  29 theorem liftsb_trans: ∀f1,I1,I. ⬆*[f1] I1 ≘ I → ∀f2,I2. ⬆*[f2] I ≘ I2 →
  30                       ∀f. f2 ⊚ f1 ≘ f → ⬆*[f] I1 ≘ I2.
  31 #f1 #I1 #I * -I1 -I #I1 [2: #V1 #V #HV1 ] #f2 #I2 #H
  32 [ elim (liftsb_inv_pair_sn … H) | lapply (liftsb_inv_unit_sn … H) ] -H
  33 /3 width=6 by lifts_trans, ext2_pair, ext2_unit/
  34 qed-.
  35
  36 theorem liftsb_conf: ∀f1,I,I1. ⬆*[f1] I ≘ I1 → ∀f,I2. ⬆*[f] I ≘ I2 →
  37                      ∀f2. f2 ⊚ f1 ≘ f → ⬆*[f2] I1 ≘ I2.
  38 #f1 #I #I1 * -I -I1 #I [2: #V #V1 #HV1 ] #f2 #I2 #H
  39 [ elim (liftsb_inv_pair_sn … H) | lapply (liftsb_inv_unit_sn … H) ] -H
  40 /3 width=6 by lifts_conf, ext2_pair, ext2_unit/
  41 qed-.
  42
  43 (* Advanced proprerties *****************************************************)
  44
  45 lemma liftsb_inj: ∀f. is_inj2 … (liftsb f).
  46 #f #T1 #U #H1 #T2 #H2 lapply (after_isid_dx 𝐈𝐝  … f)
  47 /3 width=6 by liftsb_div3, liftsb_fwd_isid/
  48 qed-.
  49
  50 lemma liftsb_mono: ∀f,T. is_mono … (liftsb f T).
  51 #f #T #U1 #H1 #U2 #H2 lapply (after_isid_sn 𝐈𝐝  … f)
  52 /3 width=6 by liftsb_conf, liftsb_fwd_isid/
  53 qed-.