matita/matita/contribs/lambdadelta/ground_2/steps/rtc_ist_plus.ma

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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_2/xoa/ex_3_2.ma".
  16 include "ground_2/steps/rtc_plus.ma".
  17 include "ground_2/steps/rtc_ist.ma".
  18
  19 (* RT-TRANSITION COUNTER ****************************************************)
  20
  21 (* Properties with test for t-transition counter ****************************)
  22
  23 lemma ist_plus: ∀n1,n2,c1,c2. 𝐓❪n1,c1❫ → 𝐓❪n2,c2❫ → 𝐓❪n1+n2,c1+c2❫.
  24 #n1 #n2 #c1 #c2 #H1 #H2 destruct //
  25 qed.
  26
  27 lemma ist_plus_O1: ∀n,c1,c2. 𝐓❪0,c1❫ → 𝐓❪n,c2❫ → 𝐓❪n,c1+c2❫.
  28 /2 width=1 by ist_plus/ qed.
  29
  30 lemma ist_plus_O2: ∀n,c1,c2. 𝐓❪n,c1❫ → 𝐓❪0,c2❫ → 𝐓❪n,c1+c2❫.
  31 #n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by ist_plus/
  32 qed.
  33
  34 lemma ist_succ: ∀n,c. 𝐓❪n,c❫ → 𝐓❪↑n,c+𝟘𝟙❫.
  35 /2 width=1 by ist_plus/ qed.
  36
  37 (* Inversion properties with test for constrained rt-transition counter *****)
  38
  39 lemma ist_inv_plus:
  40       ∀n,c1,c2. 𝐓❪n,c1 + c2❫ →
  41       ∃∃n1,n2. 𝐓❪n1,c1❫ & 𝐓❪n2,c2❫ & n1 + n2 = n.
  42 #n #c1 #c2 #H
  43 elim (plus_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H1 #H2 #H3 #H4 #H5 #H6 destruct
  44 elim (plus_inv_O3 … H1) -H1 #H11 #H12 destruct
  45 elim (plus_inv_O3 … H2) -H2 #H21 #H22 destruct
  46 elim (plus_inv_O3 … H3) -H3 #H31 #H32 destruct
  47 /3 width=5 by ex3_2_intro/
  48 qed-.
  49
  50 lemma ist_inv_plus_O_dx: ∀n,c1,c2. 𝐓❪n,c1 + c2❫ → 𝐓❪0,c2❫ → 𝐓❪n,c1❫.
  51 #n #c1 #c2 #H #H2
  52 elim (ist_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct //
  53 qed-.
  54
  55 lemma ist_inv_plus_SO_dx:
  56       ∀n,c1,c2. 𝐓❪n,c1 + c2❫ → 𝐓❪1,c2❫ →
  57       ∃∃m. 𝐓❪m,c1❫ & n = ↑m.
  58 #n #c1 #c2 #H #H2 destruct
  59 elim (ist_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
  60 /2 width=3 by ex2_intro/
  61 qed-.
  62
  63 lemma ist_inv_plus_10_dx: ∀n,c. 𝐓❪n,c+𝟙𝟘❫ → ⊥.
  64 #n #c #H
  65 elim (ist_inv_plus … H) -H #n1 #n2 #_ #H #_
  66 /2 width=2 by ist_inv_10/
  67 qed-.