matita/matita/contribs/lambdadelta/ground/relocation/tr_compose.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   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/lib/stream_tls.ma".
  16 include "ground/relocation/tr_pap.ma".
  17
  18 (* COMPOSITION FOR TOTAL RELOCATION MAPS ************************************)
  19
  20 corec definition tr_compose: tr_map → tr_map → tr_map.
  21 #f2 * #p1 #f1
  22 @(stream_cons … (f2＠⧣❨p1❩))
  23 @(tr_compose ? f1) -tr_compose -f1
  24 @(⇂*[p1]f2)
  25 defined.
  26
  27 interpretation
  28   "composition (total relocation maps)"
  29   'compose f2 f1 = (tr_compose f2 f1).
  30
  31 (* Basic constructions ******************************************************)
  32
  33 (*** compose_rew *)
  34 lemma tr_compose_unfold (f2) (f1):
  35       ∀p1. f2＠⧣❨p1❩⨮(⇂*[p1]f2)∘f1 = f2∘(p1⨮f1).
  36 #f2 #f1 #p1 <(stream_unfold … (f2∘(p1⨮f1))) //
  37 qed.
  38
  39 (* Basic inversions *********************************************************)
  40
  41 (*** compose_inv_rew *)
  42 lemma tr_compose_inv_unfold (f2) (f1):
  43       ∀f,p1,p. f2∘(p1⨮f1) = p⨮f →
  44       ∧∧ f2＠⧣❨p1❩ = p & (⇂*[p1]f2)∘f1 = f.
  45 #f2 #f1 #f #p1 #p
  46 <tr_compose_unfold #H destruct
  47 /2 width=1 by conj/
  48 qed-.
  49
  50 (*** compose_inv_S2 *)
  51 lemma tr_compose_inv_succ_dx (f2) (f1):
  52       ∀f,p2,p1,p. (p2⨮f2)∘(↑p1⨮f1) = p⨮f →
  53       ∧∧ f2＠⧣❨p1❩+p2 = p & f2∘(p1⨮f1) = f2＠⧣❨p1❩⨮f.
  54 #f2 #f1 #f #p2 #p1 #p
  55 <tr_compose_unfold #H destruct
  56 >nsucc_inj <stream_tls_swap
  57 /2 width=1 by conj/
  58 qed-.