matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma

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   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                  *)
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  13 (**************************************************************************)
  14
  15 include "delayed_updating/substitution/lift_eq.ma".
  16 include "delayed_updating/syntax/path_head.ma".
  17 include "delayed_updating/syntax/path_reverse.ma".
  18 include "ground/lib/stream_eq_eq.ma".
  19
  20 (* LIFT MAP FOR PATH ********************************************************)
  21
  22 (* Constructions with path_head *********************************************)
  23
  24 lemma tls_plus_lift_rmap_reverse_closed (f) (q) (p) (n) (k):
  25       q = (↳[n]q)●p →
  26       ⇂*[k]↑[pᴿ]f ≗ ⇂*[n+k]↑[qᴿ]f.
  27 #f #q elim q -q
  28 [ #p #n #k #Hp
  29   elim (eq_inv_list_empty_append … Hp) -Hp #Hn #H0 destruct
  30   <path_head_empty in Hn; #Hn
  31   <(eq_inv_empty_labels … Hn) -n //
  32 | #l #q #IH #p #n @(nat_ind_succ … n) -n //
  33   #n #_ #k cases l [ #m ]
  34   [ <path_head_d_sn <list_append_lcons_sn #Hp
  35     elim (eq_inv_list_lcons_bi ????? Hp) -Hp #_ #Hp <nrplus_inj_sn
  36     <reverse_lcons
  37     @(stream_eq_trans … (tls_lift_rmap_d_dx …))
  38     >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
  39     >nsucc_unfold /2 width=1 by/
  40   | <path_head_m_sn <list_append_lcons_sn #Hp
  41     elim (eq_inv_list_lcons_bi ????? Hp) -Hp #_ #Hp
  42     <reverse_lcons <lift_rmap_m_dx /2 width=1 by/
  43   | <path_head_L_sn <list_append_lcons_sn #Hp
  44     elim (eq_inv_list_lcons_bi ????? Hp) -Hp #_ #Hp
  45     <reverse_lcons <lift_rmap_L_dx <nplus_succ_sn /2 width=1 by/
  46   | <path_head_A_sn <list_append_lcons_sn #Hp
  47     elim (eq_inv_list_lcons_bi ????? Hp) -Hp #_ #Hp
  48     <reverse_lcons <lift_rmap_A_dx /2 width=2 by/
  49   | <path_head_S_sn <list_append_lcons_sn #Hp
  50     elim (eq_inv_list_lcons_bi ????? Hp) -Hp #_ #Hp
  51     <reverse_lcons <lift_rmap_S_dx /2 width=2 by/
  52   ]
  53 ]
  54 qed-.
  55
  56 lemma tls_lift_rmap_append_closed (f) (p) (q) (n):
  57       qᴿ = ↳[n](p●q)ᴿ →
  58       ↑[p]f ≗ ⇂*[n]↑[p●q]f.
  59 #f #p #q #n #H0
  60 >(reverse_reverse p) >(reverse_reverse q)
  61 /2 width=1 by tls_plus_lift_rmap_reverse_closed/
  62 qed.