matita/matita/contribs/lambdadelta/ground/relocation/pr_ist.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                  *)
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  13 (**************************************************************************)
  14
  15 include "ground/notation/relations/predicate_t_1.ma".
  16 include "ground/relocation/pr_pat.ma".
  17
  18 (* TOTALITY CONDITION FOR PARTIAL RELOCATION MAPS ***************************)
  19
  20 (*** istot *)
  21 definition pr_ist: predicate pr_map ≝
  22            λf. ∀i. ∃j. ＠⧣❨i,f❩ ≘ j.
  23
  24 interpretation
  25   "totality condition (partial relocation maps)"
  26   'PredicateT f = (pr_ist f).
  27
  28 (* Basic inversions *********************************************************)
  29
  30 (*** istot_inv_push *)
  31 lemma pr_ist_inv_push (g): 𝐓❨g❩ → ∀f. ⫯f = g → 𝐓❨f❩.
  32 #g #Hg #f #H #i elim (Hg (↑i)) -Hg
  33 #j #Hg elim (pr_pat_inv_succ_push … Hg … H) -Hg -H /2 width=3 by ex_intro/
  34 qed-.
  35
  36 (*** istot_inv_next *)
  37 lemma pr_ist_inv_next (g): 𝐓❨g❩ → ∀f. ↑f = g → 𝐓❨f❩.
  38 #g #Hg #f #H #i elim (Hg i) -Hg
  39 #j #Hg elim (pr_pat_inv_next … Hg … H) -Hg -H /2 width=2 by ex_intro/
  40 qed-.
  41
  42 (* Basic constructions ******************************************************)
  43
  44 lemma pr_ist_push (f): 𝐓❨f❩ → 𝐓❨⫯f❩.
  45 #f #Hf *
  46 [ /3 width=2 by pr_pat_refl, ex_intro/
  47 | #i elim (Hf i) -Hf /3 width=8 by pr_pat_push, ex_intro/
  48 ]
  49 qed.
  50
  51 lemma pr_ist_next (f): 𝐓❨f❩ → 𝐓❨↑f❩.
  52 #f #Hf #i elim (Hf i) -Hf
  53 /3 width=6 by pr_pat_next, ex_intro/
  54 qed.
  55
  56 (* Constructions with pr_tl *************************************************)
  57
  58 (*** istot_tl *)
  59 lemma pr_ist_tl (f): 𝐓❨f❩ → 𝐓❨⫰f❩.
  60 #f cases (pr_map_split_tl f) *
  61 /2 width=3 by pr_ist_inv_next, pr_ist_inv_push/
  62 qed.