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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/notation/functions/uparrowstar_2.ma".
  16 include "ground/relocation/rtmap_eq.ma".
  17
  18 (* RELOCATION MAP ***********************************************************)
  19
  20 rec definition nexts (f:rtmap) (n:nat) on n: rtmap ≝ match n with
  21 [ O ⇒ f | S m ⇒ ↑(nexts f m) ].
  22
  23 interpretation "nexts (rtmap)" 'UpArrowStar n f = (nexts f n).
  24
  25 (* Basic_inversion lemmas *****************************************************)
  26
  27 lemma eq_inv_nexts_sn: ∀n,f1,g2. ↑*[n] f1 ≡ g2 →
  28                        ∃∃f2. f1 ≡ f2 & ↑*[n] f2 = g2.
  29 #n elim n -n /2 width=3 by ex2_intro/
  30 #n #IH #f1 #g2 #H elim (eq_inv_nx … H) -H [|*: // ]
  31 #f0 #Hf10 #H1 elim (IH … Hf10) -IH -Hf10 #f2 #Hf12 #H2 destruct
  32 /2 width=3 by ex2_intro/
  33 qed-.
  34
  35 lemma eq_inv_nexts_dx: ∀n,f2,g1. g1 ≡ ↑*[n] f2 →
  36                        ∃∃f1. f1 ≡ f2 & ↑*[n] f1 = g1.
  37 #n elim n -n /2 width=3 by ex2_intro/
  38 #n #IH #f2 #g1 #H elim (eq_inv_xn … H) -H [|*: // ]
  39 #f0 #Hf02 #H1 elim (IH … Hf02) -IH -Hf02 #f1 #Hf12 #H2 destruct
  40 /2 width=3 by ex2_intro/
  41 qed-.
  42
  43 (* Basic properties *********************************************************)
  44
  45 lemma nexts_O: ∀f. f = ↑*[0] f.
  46 // qed.
  47
  48 lemma nexts_S: ∀f,n. ↑↑*[n] f = ↑*[↑n] f.
  49 // qed.
  50
  51 lemma nexts_eq_repl: ∀n. eq_repl (λf1,f2. ↑*[n] f1 ≡ ↑*[n] f2).
  52 #n elim n -n /3 width=5 by eq_next/
  53 qed.
  54
  55 (* Advanced properties ******************************************************)
  56
  57 lemma nexts_xn: ∀n,f. ↑*[n] ↑f = ↑*[↑n] f.
  58 #n elim n -n //
  59 qed.