matita/matita/contribs/lambdadelta/basic_2/rt_conversion/lpce.ma

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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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  13 (**************************************************************************)
  14
  15 include "static_2/relocation/lex.ma".
  16 include "basic_2/notation/relations/pconveta_4.ma".
  17 include "basic_2/rt_conversion/cpce_ext.ma".
  18
  19 (* PARALLEL ETA-CONVERSION FOR FULL LOCAL ENVIRONMENTS **********************)
  20
  21 definition lpce (h) (G): relation lenv ≝ lex (cpce h G).
  22
  23 interpretation
  24   "parallel eta-conversion on all entries (local environment)"
  25   'PConvEta h G L1 L2 = (lpce h G L1 L2).
  26
  27 (* Basic properties *********************************************************)
  28
  29 lemma lpce_bind (h) (G):
  30       ∀K1,K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 →
  31       ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬌η[h] K2.ⓘ{I2}.
  32 /2 width=1 by lex_bind/ qed.
  33
  34 (* Advanced properties ******************************************************)
  35
  36 lemma lpce_pair (h) (G):
  37       ∀K1,K2,V1,V2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 → ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 →
  38       ∀I. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬌η[h] K2.ⓑ{I}V2.
  39 /2 width=1 by lex_pair/ qed.
  40
  41 (* Basic inversion lemmas ***************************************************)
  42
  43 lemma lpce_inv_atom_sn (h) (G):
  44       ∀L2. ⦃G,⋆⦄ ⊢ ⬌η[h] L2 → L2 = ⋆.
  45 /2 width=2 by lex_inv_atom_sn/ qed-.
  46
  47 lemma lpce_inv_bind_sn (h) (G):
  48       ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬌η[h] L2 →
  49       ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 & L2 = K2.ⓘ{I2}.
  50 /2 width=1 by lex_inv_bind_sn/ qed-.
  51
  52 lemma lpce_inv_atom_dx (h) (G):
  53       ∀L1. ⦃G,L1⦄ ⊢ ⬌η[h] ⋆ → L1 = ⋆.
  54 /2 width=2 by lex_inv_atom_dx/ qed-.
  55
  56 lemma lpce_inv_bind_dx (h) (G):
  57       ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓘ{I2} →
  58       ∃∃I1,K1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 & L1 = K1.ⓘ{I1}.
  59 /2 width=1 by lex_inv_bind_dx/ qed-.
  60
  61 (* Advanced inversion lemmas ************************************************)
  62
  63 lemma lpce_inv_unit_sn (h) (G):
  64       ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬌η[h] L2 →
  65       ∃∃K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}.
  66 /2 width=1 by lex_inv_unit_sn/ qed-.
  67
  68 lemma lpce_inv_pair_sn (h) (G):
  69       ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬌η[h] L2 →
  70       ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 & L2 = K2.ⓑ{I}V2.
  71 /2 width=1 by lex_inv_pair_sn/ qed-.
  72
  73 lemma lpce_inv_unit_dx (h) (G):
  74       ∀I,L1,K2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓤ{I} →
  75       ∃∃K1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L1 = K1.ⓤ{I}.
  76 /2 width=1 by lex_inv_unit_dx/ qed-.
  77
  78 lemma lpce_inv_pair_dx (h) (G):
  79       ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓑ{I}V2 →
  80       ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 & L1 = K1.ⓑ{I}V1.
  81 /2 width=1 by lex_inv_pair_dx/ qed-.
  82
  83 lemma lpce_inv_pair (h) (G):
  84       ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ⬌η[h] L2.ⓑ{I2}V2 →
  85       ∧∧ ⦃G,L1⦄ ⊢ ⬌η[h] L2 & ⦃G,L1⦄ ⊢ V1 ⬌η[h] V2 & I1 = I2.
  86 /2 width=1 by lex_inv_pair/ qed-.