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

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   2 (*       ___                                                              *)
   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                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/arith/pnat_le_plus.ma".
  16 include "ground/relocation/pr_pat_lt.ma".
  17 include "ground/relocation/tr_map.ma".
  18
  19 (* TOTAL RELOCATION MAPS ****************************************************)
  20
  21 (* Constructions with pr_pat ***********************************************)
  22
  23 (*** at_O1 *)
  24 lemma tr_pat_unit_sn: ∀i2,f. @❨𝟏,𝐭❨i2⨮f❩❩ ≘ i2.
  25 #i2 elim i2 -i2 /2 width=5 by pr_pat_refl, pr_pat_next/
  26 qed.
  27
  28 (*** at_S1 *)
  29 lemma tr_pat_succ_sn: ∀p,f,i1,i2. @❨i1, 𝐭❨f❩❩ ≘ i2 → @❨↑i1, 𝐭❨p⨮f❩❩ ≘ i2+p.
  30 #p elim p -p /3 width=7 by pr_pat_push, pr_pat_next/
  31 qed.
  32
  33 (*** at_plus2 *)
  34 lemma tr_pat_plus_dx (f):
  35       ∀i1,i,p,q. @❨i1, 𝐭❨p⨮f❩❩ ≘ i → @❨i1, 𝐭❨(p+q)⨮f❩❩ ≘ i+q.
  36 #f #i1 #i #p #q #H elim q -q
  37 /2 width=5 by pr_pat_next/
  38 qed.
  39
  40 (* Inversions with pr_pat ***************************************************)
  41
  42 (*** at_inv_O1 *)
  43 lemma tr_pat_inv_unit_sn (f):
  44       ∀p,i2. @❨𝟏, 𝐭❨p⨮f❩❩ ≘ i2 → p = i2.
  45 #f #p elim p -p /2 width=6 by pr_pat_inv_unit_push/
  46 #p #IH #i2 #H elim (pr_pat_inv_next … H) -H [|*: // ]
  47 #j2 #Hj * -i2 /3 width=1 by eq_f/
  48 qed-.
  49
  50 (*** at_inv_S1 *)
  51 lemma tr_pat_inv_succ_sn (f):
  52       ∀p,j1,i2. @❨↑j1, 𝐭❨p⨮f❩❩ ≘ i2 →
  53       ∃∃j2. @❨j1, 𝐭❨f❩❩ ≘ j2 & j2+p = i2.
  54 #f #p elim p -p /2 width=5 by pr_pat_inv_succ_push/
  55 #p #IH #j1 #i2 #H elim (pr_pat_inv_next … H) -H [|*: // ]
  56 #j2 #Hj * -i2 elim (IH … Hj) -IH -Hj
  57 #i2 #Hi * -j2 /2 width=3 by ex2_intro/
  58 qed-.
  59
  60 (* Destructions with pr_pat *************************************************)
  61
  62 (* Note: a better conclusion would be: "i1 + ↓p ≤ i2" *)
  63 (*** at_increasing_plus *)
  64 lemma tr_pat_increasing_plus (f):
  65       ∀p,i1,i2. @❨i1, 𝐭❨p⨮f❩❩ ≘ i2 → i1 + p ≤ ↑i2.
  66 #f #p *
  67 [ #i2 #H <(tr_pat_inv_unit_sn … H) -i2 //
  68 | #i1 #i2 #H elim (tr_pat_inv_succ_sn … H) -H
  69   #j1 #Ht * -i2 <pplus_succ_sn
  70   /4 width=2 by pr_pat_increasing, ple_plus_bi_dx, ple_succ_bi/
  71 ]
  72 qed-.
  73
  74 (*** at_fwd_id *)
  75 lemma tr_pat_des_id (f):
  76       ∀p,i. @❨i, 𝐭❨p⨮f❩❩ ≘ i → 𝟏 = p.
  77 #f #p #i #H lapply (pr_pat_des_id … H) -H #H
  78 elim (eq_inv_pr_push_cons … H) -H //
  79 qed-.