matita/matita/contribs/lambdadelta/ground/relocation/tr_uni_compose.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 "ground/relocation/tr_uni_pap.ma".
  16 include "ground/relocation/tr_id_compose.ma".
  17 include "ground/relocation/tr_compose_pn.ma".
  18 include "ground/lib/stream_hdtl_eq.ma".
  19
  20 (* UNIFORM ELEMENTS FOR TOTAL RELOCATION MAPS *******************************)
  21
  22 (* Constructions with tr_compose and tr_next ********************************)
  23
  24 lemma tr_compose_uni_unit_sn (f):
  25       ↑f ≗ 𝐮❨𝟏❩∘f.
  26 #f >nsucc_zero <tr_uni_succ //
  27 qed.
  28
  29 (* Constructions with tr_compose and tr_tl **********************************)
  30
  31 lemma tr_tl_compose_uni_sn (n) (f):
  32       ⇂f ≗ ⇂(𝐮❨n❩∘f).
  33 #n @(nat_ind_succ … n) -n //
  34 /2 width=1 by stream_tl_eq_repl/
  35 qed.
  36
  37 (* Constructions with tr_compose and tr_tls *********************************)
  38
  39 lemma tr_tls_compose_uni_sn (f) (n) (p:pnat):
  40       ⇂*[p]f ≗ ⇂*[p](𝐮❨n❩∘f).
  41 #f #n #p elim p -p //
  42 #p #IH /2 width=1 by stream_tl_eq_repl/
  43 qed.
  44
  45 lemma tr_tl_compose_uni_dx (f) (n):
  46       ⇂*[↑n]f ≗ ⇂(f∘𝐮❨n❩).
  47 // qed.
  48
  49 lemma tr_tls_compose_uni_dx (f) (p) (n):
  50       ⇂*[p+n]f ≗ ⇂*[p](f∘𝐮❨n❩).
  51 #f #p elim p -p [| #p #IH ] #n
  52 [ <nrplus_unit_sn //
  53 | <nrplus_succ_sn >nsucc_inj >nsucc_inj
  54   /2 width=3 by stream_tl_eq_repl/
  55 ]
  56 qed.
  57
  58 (* Main constructions with tr_compose and tr_tls ****************************)
  59
  60 theorem tr_compose_uni_dx_pap (f) (p):
  61         (𝐮❨f＠⧣❨p❩❩∘⇂*[p]f) ≗ f∘𝐮❨p❩.
  62 #f #p
  63 @nstream_eq_inv_ext #q
  64 <tr_compose_pap <tr_compose_pap
  65 <tr_uni_pap <tr_uni_pap
  66 <tr_pap_plus //
  67 qed.