matita/matita/contribs/lambdadelta/delayed_updating/etc/lift/lift_preterm_eq.etc

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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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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  14
  15 include "ground/lib/subset_equivalence.ma".
  16 include "delayed_updating/syntax/preterm.ma".
  17 include "delayed_updating/substitution/lift_structure.ma".
  18 include "delayed_updating/substitution/lift_prototerm.ma".
  19
  20 lemma pippo (p):
  21       ⊗p ϵ 𝐈.
  22 #p @(list_ind_rcons … p) -p
  23 [ <structure_empty //
  24 | #p * [ #n ] #IH
  25   [ <structure_d_dx //
  26   | <structure_m_dx //
  27   | <structure_L_dx //
  28   | <structure_A_dx //
  29   | <structure_S_dx //
  30   ]
  31 ]
  32 qed.
  33
  34 (* LIFT FOR PRETERM *********************************************************)
  35
  36 (* Constructions with subset_equivalence ************************************)
  37
  38 lemma lift_grafted_sn (f) (t) (p): p ϵ 𝐈 →
  39       ↑[↑[p]f](t⋔p) ⊆ (↑[f]t)⋔(⊗p).
  40 #f #t #p #Hp #q * #r #Hr #H0 destruct
  41 @(ex2_intro … Hr) -Hr
  42 <lift_append_inner_sn //
  43 qed-.
  44
  45 lemma lift_grafted_dx (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
  46       (↑[f]t)⋔(⊗p) ⊆ ↑[↑[p]f](t⋔p).
  47 #f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
  48 elim (lift_inv_append_inner_sn … (sym_eq … H0)) -H0 //
  49 #p0 #q0 #Hp0 #Hq0 #H0 destruct
  50 <(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
  51 /2 width=1 by in_comp_lift_bi/
  52 qed-.
  53
  54 lemma lift_grafted (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
  55       ↑[↑[p]f](t⋔p) ⇔ (↑[f]t)⋔(⊗p).
  56 /3 width=1 by lift_grafted_sn, conj, lift_grafted_dx/ qed.
  57
  58
  59 lemma lift_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
  60       (↑[f]t)⋔((⊗p)◖𝗦) ⊆ ↑[↑[p]f](t⋔(p◖𝗦)).
  61 #f #t #p #Hp #Ht #q * #r #Hr
  62 <list_append_rcons_sn #H0
  63 elim (lift_inv_append_proper_dx … (sym_eq … H0)) -H0 //
  64 #p0 #q0 #Hp0 #Hq0 #H0 destruct
  65 <(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
  66 elim (lift_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
  67 #r1 #r2 #Hr1 #Hr2 #H0 destruct
  68 lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
  69 [ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
  70 /2 width=1 by in_comp_lift_bi/
  71 qed-.
  72
  73 lemma lift_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
  74       ↑[↑[p]f](t⋔(p◖𝗦)) ⇔ (↑[f]t)⋔((⊗p)◖𝗦).
  75 #f #t #p #Hp #Ht
  76 @conj
  77 [ >lift_rmap_S_dx >structure_S_dx
  78   @lift_grafted_sn //
  79 | /2 width=1 by lift_grafted_S_dx/
  80 ]
  81 qed.