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

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "ground/relocation/gr_tl_eq.ma".
  16 include "ground/relocation/gr_coafter.ma".
  17
  18 (* RELATIONAL CO-COMPOSITION FOR GENERIC RELOCATION MAPS ********************)
  19
  20 (* Constructions with gr_eq *************************************************)
  21
  22 (*** coafter_eq_repl_back2 *)
  23 corec lemma gr_coafter_eq_repl_back_sn:
  24             ∀f1,f. gr_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 (gr_eq_inv_push_sn …  H0 …  H21) -g21 /3 width=7 by gr_coafter_refl/
  28 | cases (gr_eq_inv_push_sn …  H0 …  H21) -g21 /3 width=7 by gr_coafter_push/
  29 | cases (gr_eq_inv_next_sn …  H0 …  H21) -g21 /3 width=5 by gr_coafter_next/
  30 ]
  31 qed-.
  32
  33 (*** coafter_eq_repl_fwd2 *)
  34 lemma gr_coafter_eq_repl_fwd_sn:
  35       ∀f1,f. gr_eq_repl_fwd (λf2. f2 ~⊚ f1 ≘ f).
  36 #f1 #f @gr_eq_repl_sym /2 width=3 by gr_coafter_eq_repl_back_sn/
  37 qed-.
  38
  39 (*** coafter_eq_repl_back1 *)
  40 corec lemma gr_coafter_eq_repl_back_dx:
  41             ∀f2,f. gr_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 (gr_eq_inv_push_sn …  H0 …  H11) -g11 /3 width=7 by gr_coafter_refl/
  45 | cases (gr_eq_inv_next_sn …  H0 …  H11) -g11 /3 width=7 by gr_coafter_push/
  46 | @(gr_coafter_next … H2 H) /2 width=5 by/
  47 ]
  48 qed-.
  49
  50 (*** coafter_eq_repl_fwd1 *)
  51 lemma gr_coafter_eq_repl_fwd_dx:
  52       ∀f2,f. gr_eq_repl_fwd (λf1. f2 ~⊚ f1 ≘ f).
  53 #f2 #f @gr_eq_repl_sym /2 width=3 by gr_coafter_eq_repl_back_dx/
  54 qed-.
  55
  56 (*** coafter_eq_repl_back0 *)
  57 corec lemma gr_coafter_eq_repl_back:
  58             ∀f1,f2. gr_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 (gr_eq_inv_push_sn …  H0 …  H01) -g01 /3 width=7 by gr_coafter_refl/
  62 | cases (gr_eq_inv_next_sn …  H0 …  H01) -g01 /3 width=7 by gr_coafter_push/
  63 | cases (gr_eq_inv_push_sn …  H0 …  H01) -g01 /3 width=5 by gr_coafter_next/
  64 ]
  65 qed-.
  66
  67 (*** coafter_eq_repl_fwd0 *)
  68 lemma gr_coafter_eq_repl_fwd:
  69       ∀f2,f1. gr_eq_repl_fwd (λf. f2 ~⊚ f1 ≘ f).
  70 #f2 #f1 @gr_eq_repl_sym /2 width=3 by gr_coafter_eq_repl_back/
  71 qed-.