matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr.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 "basic_2/notation/relations/predsn_4.ma".
  16 include "static_2/relocation/lex.ma".
  17 include "basic_2/rt_transition/cpr_ext.ma".
  18
  19 (* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
  20
  21 definition lpr (h) (G): relation lenv ≝
  22                         lex (λL. cpm h G L 0).
  23
  24 interpretation
  25    "parallel rt-transition (full local environment)"
  26    'PRedSn h G L1 L2 = (lpr h G L1 L2).
  27
  28 (* Basic properties *********************************************************)
  29
  30 lemma lpr_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 →
  31                         ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ➡[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] K2.ⓘ{I2}.
  32 /2 width=1 by lex_bind/ qed.
  33
  34 (* Note: lemma 250 *)
  35 lemma lpr_refl (h) (G): reflexive … (lpr h G).
  36 /2 width=1 by lex_refl/ qed.
  37
  38 (* Advanced properties ******************************************************)
  39
  40 lemma lpr_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 →
  41                                 ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ➡[h] K2.ⓘ{I}.
  42 /2 width=1 by lex_bind_refl_dx/ qed.
  43
  44 lemma lpr_pair (h) (G): ∀K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h] V2 →
  45                         ∀I. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] K2.ⓑ{I}V2.
  46 /2 width=1 by lex_pair/ qed.
  47
  48 (* Basic inversion lemmas ***************************************************)
  49
  50 (* Basic_2A1: was: lpr_inv_atom1 *)
  51 (* Basic_1: includes: wcpr0_gen_sort *)
  52 lemma lpr_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡[h] L2 → L2 = ⋆.
  53 /2 width=2 by lex_inv_atom_sn/ qed-.
  54
  55 lemma lpr_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] L2 →
  56                                ∃∃I2,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 &
  57                                         L2 = K2.ⓘ{I2}.
  58 /2 width=1 by lex_inv_bind_sn/ qed-.
  59
  60 (* Basic_2A1: was: lpr_inv_atom2 *)
  61 lemma lpr_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡[h] ⋆ → L1 = ⋆.
  62 /2 width=2 by lex_inv_atom_dx/ qed-.
  63
  64 lemma lpr_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓘ{I2} →
  65                                ∃∃I1,K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 &
  66                                         L1 = K1.ⓘ{I1}.
  67 /2 width=1 by lex_inv_bind_dx/ qed-.
  68
  69 (* Advanced inversion lemmas ************************************************)
  70
  71 lemma lpr_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ➡[h] L2 →
  72                                ∃∃K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & L2 = K2.ⓤ{I}.
  73 /2 width=1 by lex_inv_unit_sn/ qed-.
  74
  75 (* Basic_2A1: was: lpr_inv_pair1 *)
  76 (* Basic_1: includes: wcpr0_gen_head *)
  77 lemma lpr_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] L2 →
  78                                ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 &
  79                                         L2 = K2.ⓑ{I}V2.
  80 /2 width=1 by lex_inv_pair_sn/ qed-.
  81
  82 lemma lpr_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓤ{I} →
  83                                ∃∃K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & L1 = K1.ⓤ{I}.
  84 /2 width=1 by lex_inv_unit_dx/ qed-.
  85
  86 (* Basic_2A1: was: lpr_inv_pair2 *)
  87 lemma lpr_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓑ{I}V2 →
  88                                ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 &
  89                                         L1 = K1.ⓑ{I}V1.
  90 /2 width=1 by lex_inv_pair_dx/ qed-.
  91
  92 lemma lpr_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h] L2.ⓑ{I2}V2 →
  93                             ∧∧ ⦃G, L1⦄ ⊢ ➡[h] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 & I1 = I2.
  94 /2 width=1 by lex_inv_pair/ qed-.
  95
  96 (* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
  97                                 pr0_subst1_back
  98 *)