matita/matita/contribs/lambdadelta/static_2/syntax/item.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_2/lib/bool.ma".
  16 include "ground_2/lib/arith.ma".
  17
  18 (* ITEMS ********************************************************************)
  19
  20 (* atomic items *)
  21 inductive item0: Type[0] ≝
  22    | Sort: nat → item0 (* sort: starting at 0 *)
  23    | LRef: nat → item0 (* reference by index: starting at 0 *)
  24    | GRef: nat → item0 (* reference by position: starting at 0 *)
  25 .
  26
  27 (* unary binding items *)
  28 inductive bind1: Type[0] ≝
  29   | Void: bind1 (* exclusion *)
  30 .
  31
  32 (* binary binding items *)
  33 inductive bind2: Type[0] ≝
  34   | Abbr: bind2 (* abbreviation *)
  35   | Abst: bind2 (* abstraction *)
  36 .
  37
  38 (* binary non-binding items *)
  39 inductive flat2: Type[0] ≝
  40   | Appl: flat2 (* application *)
  41   | Cast: flat2 (* explicit type annotation *)
  42 .
  43
  44 (* binary items *)
  45 inductive item2: Type[0] ≝
  46   | Bind2: bool → bind2 → item2 (* polarized binding item *)
  47   | Flat2: flat2 → item2        (* non-binding item *)
  48 .
  49
  50 (* Basic inversion lemmas ***************************************************)
  51
  52 fact destruct_sort_sort_aux: ∀s1,s2. Sort s1 = Sort s2 → s1 = s2.
  53 #s1 #s2 #H destruct //
  54 qed-.
  55
  56 (* Basic properties *********************************************************)
  57
  58 lemma eq_item0_dec: ∀I1,I2:item0. Decidable (I1 = I2).
  59 * #i1 * #i2 [2,3,4,6,7,8: @or_intror #H destruct ]
  60 [2: elim (eq_nat_dec i1 i2) |1,3: elim (eq_nat_dec i1 i2) ] /2 width=1 by or_introl/
  61 #Hni12 @or_intror #H destruct /2 width=1 by/
  62 qed-.
  63
  64 lemma eq_bind1_dec: ∀I1,I2:bind1. Decidable (I1 = I2).
  65 * * /2 width=1 by or_introl/
  66 qed-.
  67
  68 (* Basic_1: was: bind_dec *)
  69 lemma eq_bind2_dec: ∀I1,I2:bind2. Decidable (I1 = I2).
  70 * * /2 width=1 by or_introl/
  71 @or_intror #H destruct
  72 qed-.
  73
  74 (* Basic_1: was: flat_dec *)
  75 lemma eq_flat2_dec: ∀I1,I2:flat2. Decidable (I1 = I2).
  76 * * /2 width=1 by or_introl/
  77 @or_intror #H destruct
  78 qed-.
  79
  80 (* Basic_1: was: kind_dec *)
  81 lemma eq_item2_dec: ∀I1,I2:item2. Decidable (I1 = I2).
  82 * [ #p1 ] #I1 * [1,3: #p2 ] #I2
  83 [2,3: @or_intror #H destruct
  84 | elim (eq_bool_dec p1 p2) #Hp
  85   [ elim (eq_bind2_dec I1 I2) /2 width=1 by or_introl/ #HI ]
  86   @or_intror #H destruct /2 width=1 by/
  87 | elim (eq_flat2_dec I1 I2) /2 width=1 by or_introl/ #HI
  88   @or_intror #H destruct /2 width=1 by/
  89 ]
  90 qed-.
  91
  92 (* Basic_1: removed theorems 21:
  93             s_S s_plus s_plus_sym s_minus minus_s_s s_le s_lt s_inj s_inc
  94             s_arith0 s_arith1
  95             r_S r_plus r_plus_sym r_minus r_dis s_r r_arith0 r_arith1
  96             not_abbr_abst bind_dec_not
  97 *)