matita/matita/contribs/lambdadelta/ground/steps/rtc.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   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/xoa/ex_1_2.ma".
  16 include "ground/notation/functions/tuple_4.ma".
  17 include "ground/notation/functions/zerozero_0.ma".
  18 include "ground/notation/functions/zeroone_0.ma".
  19 include "ground/notation/functions/onezero_0.ma".
  20 include "ground/lib/arith.ma".
  21
  22 (* RT-TRANSITION COUNTER ****************************************************)
  23
  24 record rtc: Type[0] ≝ {
  25    ri: nat; (* Note: inner r-steps *)
  26    rs: nat; (* Note: spine r-steps *)
  27    ti: nat; (* Note: inner t-steps *)
  28    ts: nat  (* Note: spine t-steps *)
  29 }.
  30
  31 interpretation "constructor (rtc)"
  32    'Tuple ri rs ti ts = (mk_rtc ri rs ti ts).
  33
  34 interpretation "one structural step (rtc)"
  35    'ZeroZero = (mk_rtc O O O O).
  36
  37 interpretation "one r-step (rtc)"
  38    'OneZero = (mk_rtc O (S O) O O).
  39
  40 interpretation "one t-step (rtc)"
  41    'ZeroOne = (mk_rtc O O O (S O)).
  42
  43 definition eq_f: relation rtc ≝ λc1,c2. ⊤.
  44
  45 inductive eq_t: relation rtc ≝
  46 | eq_t_intro: ∀ri1,ri2,rs1,rs2,ti,ts.
  47               eq_t (〈ri1,rs1,ti,ts〉) (〈ri2,rs2,ti,ts〉)
  48 .
  49
  50 (* Basic properties *********************************************************)
  51
  52 lemma eq_t_refl: reflexive …  eq_t.
  53 * // qed.
  54
  55 (* Basic inversion lemmas ***************************************************)
  56
  57 fact eq_t_inv_dx_aux: ∀x,y. eq_t x y →
  58                       ∀ri1,rs1,ti,ts. x = 〈ri1,rs1,ti,ts〉 →
  59                       ∃∃ri2,rs2. y = 〈ri2,rs2,ti,ts〉.
  60 #x #y * -x -y
  61 #ri1 #ri #rs1 #rs #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H destruct -ri2 -rs2
  62 /2 width=3 by ex1_2_intro/
  63 qed-.
  64
  65 lemma eq_t_inv_dx: ∀ri1,rs1,ti,ts,y. eq_t (〈ri1,rs1,ti,ts〉) y →
  66                    ∃∃ri2,rs2. y = 〈ri2,rs2,ti,ts〉.
  67 /2 width=5 by eq_t_inv_dx_aux/ qed-.