matita/matita/contribs/lambdadelta/ground/relocation/pr_after_eq.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/pr_tl_eq.ma".
  16 include "ground/relocation/pr_after.ma".
  17
  18 (* RELATIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
  19
  20 (* Constructions with pr_eq *************************************************)
  21
  22 (*** after_eq_repl_back2 *)
  23 corec lemma pr_after_eq_repl_back_sn:
  24             ∀f1,f. pr_eq_repl_back (λf2. f2 ⊚ f1 ≘ f).
  25 #f1 #f #f2 * -f2 -f1 -f
  26 #f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
  27 [ cases (pr_eq_inv_push_sn …  H0 …  H21) -g21 /3 width=7 by pr_after_refl/
  28 | cases (pr_eq_inv_push_sn …  H0 …  H21) -g21 /3 width=7 by pr_after_push/
  29 | cases (pr_eq_inv_next_sn …  H0 …  H21) -g21 /3 width=5 by pr_after_next/
  30 ]
  31 qed-.
  32
  33 (*** after_eq_repl_fwd2 *)
  34 lemma pr_after_eq_repl_fwd_sn:
  35       ∀f1,f. pr_eq_repl_fwd (λf2. f2 ⊚ f1 ≘ f).
  36 #f1 #f @pr_eq_repl_sym /2 width=3 by pr_after_eq_repl_back_sn/
  37 qed-.
  38
  39 (*** after_eq_repl_back1 *)
  40 corec lemma pr_after_eq_repl_back_dx:
  41             ∀f2,f. pr_eq_repl_back (λf1. f2 ⊚ f1 ≘ f).
  42 #f2 #f #f1 * -f2 -f1 -f
  43 #f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
  44 [ cases (pr_eq_inv_push_sn …  H0 …  H11) -g11 /3 width=7 by pr_after_refl/
  45 | cases (pr_eq_inv_next_sn …  H0 …  H11) -g11 /3 width=7 by pr_after_push/
  46 | @(pr_after_next … H2 H) /2 width=5 by/
  47 ]
  48 qed-.
  49
  50 (*** after_eq_repl_fwd1 *)
  51 lemma pr_after_eq_repl_fwd_dx:
  52       ∀f2,f. pr_eq_repl_fwd (λf1. f2 ⊚ f1 ≘ f).
  53 #f2 #f @pr_eq_repl_sym /2 width=3 by pr_after_eq_repl_back_dx/
  54 qed-.
  55
  56 (*** after_eq_repl_back0 *)
  57 corec lemma pr_after_eq_repl_back:
  58             ∀f1,f2. pr_eq_repl_back (λf. f2 ⊚ f1 ≘ f).
  59 #f2 #f1 #f * -f2 -f1 -f
  60 #f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
  61 [ cases (pr_eq_inv_push_sn …  H0 …  H01) -g01 /3 width=7 by pr_after_refl/
  62 | cases (pr_eq_inv_next_sn …  H0 …  H01) -g01 /3 width=7 by pr_after_push/
  63 | cases (pr_eq_inv_next_sn …  H0 …  H01) -g01 /3 width=5 by pr_after_next/
  64 ]
  65 qed-.
  66
  67 (*** after_eq_repl_fwd0 *)
  68 lemma pr_after_eq_repl_fwd:
  69       ∀f2,f1. pr_eq_repl_fwd (λf. f2 ⊚ f1 ≘ f).
  70 #f2 #f1 @pr_eq_repl_sym /2 width=3 by pr_after_eq_repl_back/
  71 qed-.