matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_guard.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "delayed_updating/syntax/path_closed.ma".
  16 include "delayed_updating/syntax/path_guard.ma".
  17
  18 (* CLOSED CONDITION FOR PATH ************************************************)
  19
  20 axiom list_ind_pippo (A) (Q:predicate …):
  21       (∀l. (∀m. (∃p. p⨁{A}m = l) → Q m) → Q l) → ∀l. Q l.
  22
  23 (* Destructions with pgc ****************************************************)
  24
  25 lemma path_closed_des_guard (x) (n):
  26       x ϵ 𝐂❨Ⓕ,n❩ →
  27       ∃∃p,q. p ϵ 𝐆 & q ϵ 𝐂❨Ⓕ,𝟎❩ & p●q = x.
  28 #x @(list_ind_pippo … x) -x
  29 #x #IH #n #H0 @(insert_eq_1 … x … H0) -H0
  30 #y * -y -n
  31 [|*: #y #n [ #k #_ ] #Hy ] #H0 destruct
  32 [ /2 width=5 by pgc_empty, pcc_empty, ex3_2_intro/
  33 | elim (pcc_false_inv_append_bi … Hy) -Hy #r #s #Hr #Hs #H0 destruct
  34   elim (IH … Hr) -IH -Hr [| /2 width=2 by ex_intro/ ]
  35   #p #q #Hp #Hq #H0 destruct
  36   @(ex3_2_intro … Hp) -Hp [1,3: // ]
  37   /3 width=2 by pcc_append_bi, pcc_false_d_dx/
  38 | elim (IH … Hy) -IH -Hy [| /2 width=2 by ex_intro/ ]
  39   #p #q #Hp #Hq #H0 destruct
  40   /3 width=5 by pcc_m_dx, ex3_2_intro/
  41 | @(ex3_2_intro … (y◖𝗟) (𝐞)) //
  42 | elim (IH … Hy) -IH -Hy [| /2 width=2 by ex_intro/ ]
  43   #p #q #Hp #Hq #H0 destruct
  44   /3 width=5 by pcc_A_dx, ex3_2_intro/
  45 | elim (IH … Hy) -IH -Hy [| /2 width=2 by ex_intro/ ]
  46   #p #q #Hp #Hq #H0 destruct
  47   /3 width=5 by pcc_S_dx, ex3_2_intro/
  48 ]
  49 qed-.