matita/matita/contribs/lambdadelta/ground/relocation/pr_pat_eq.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/relocation/pr_tl_eq.ma".
  16 include "ground/relocation/pr_pat_lt.ma".
  17
  18 (* POSITIVE APPLICATION FOR PARTIAL RELOCATION MAPS *************************)
  19
  20 (* Constructions with pr_eq *************************************************)
  21
  22 (*** at_eq_repl_back *)
  23 corec lemma pr_pat_eq_repl_back (i1) (i2):
  24             pr_eq_repl_back (λf. @❨i1,f❩ ≘ i2).
  25 #f1 * -f1 -i1 -i2
  26 [ #f1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12
  27   cases (pr_eq_inv_push_sn … H12 … H) -g1 /2 width=2 by pr_pat_refl/
  28 | #f1 #i1 #i2 #Hf1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12
  29   cases (pr_eq_inv_push_sn … H12 … H) -g1 /3 width=7 by pr_pat_push/
  30 | #f1 #i1 #i2 #Hf1 #g1 #j2 #H #H2 #f2 #H12
  31   cases (pr_eq_inv_next_sn … H12 … H) -g1 /3 width=5 by pr_pat_next/
  32 ]
  33 qed-.
  34
  35 (*** at_eq_repl_fwd *)
  36 lemma pr_pat_eq_repl_fwd (i1) (i2):
  37       pr_eq_repl_fwd (λf. @❨i1,f❩ ≘ i2).
  38 #i1 #i2 @pr_eq_repl_sym /2 width=3 by pr_pat_eq_repl_back/
  39 qed-.
  40
  41 lemma pr_pat_eq (f): ⫯f ≡ f → ∀i. @❨i,f❩ ≘ i.
  42 #f #Hf #i elim i -i
  43 [ /3 width=3 by pr_pat_eq_repl_back, pr_pat_refl/
  44 | /3 width=7 by pr_pat_eq_repl_back, pr_pat_push/
  45 ]
  46 qed.
  47
  48 (* Inversions with pr_eq ****************************************************)
  49
  50 corec lemma pr_pat_inv_eq (f):
  51             (∀i. @❨i,f❩ ≘ i) → ⫯f ≡ f.
  52 #Hf
  53 lapply (Hf (𝟏)) #H
  54 lapply (pr_pat_des_id … H) -H #H
  55 cases H in Hf; -H #Hf
  56 @pr_eq_push [3:|*: // ]
  57 /3 width=7 by pr_pat_inv_succ_push_succ/
  58 qed-.