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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   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/arith/nat_le_pred.ma".
  16 include "ground/relocation/rtmap_after_uni.ma".
  17 include "ground/relocation/rtmap_basic.ma".
  18
  19 (* RELOCATION MAP ***********************************************************)
  20
  21 (* Properties with composition **********************************************)
  22
  23 lemma after_basic_rc (m2,m1):
  24       m1 ≤ m2 → ∀n2,n1.m2 ≤ n1+m1 → 𝐁❨m2,n2❩ ⊚ 𝐁❨m1,n1❩ ≘ 𝐁❨m1,n1+n2❩.
  25 #m2 #m1 @(nat_ind_2_succ … m2 m1) -m2 -m1
  26 [ #m1 #H #n2 #n1 #_
  27   <(nle_inv_zero_dx … H) -m1 //
  28 | #m2 #IH #_ #n2 #n1 <nplus_zero_dx #H
  29   elim (nle_inv_succ_sn … H) -H #Hm2 #Hn1
  30   >Hn1 -Hn1 <nplus_succ_sn
  31   /3 width=7 by after_push/
  32 | #m2 #m1 #IH #H1 #n2 #n1 <nplus_succ_dx #H2
  33   lapply (nle_inv_succ_bi … H1) -H1 #H1
  34   lapply (nle_inv_succ_bi … H2) -H2 #H2
  35   /3 width=7 by after_refl/
  36 ]
  37 qed.