matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx.ma

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  14
  15 include "basic_2/notation/relations/predtysn_4.ma".
  16 include "static_2/relocation/lex.ma".
  17 include "basic_2/rt_transition/cpx_ext.ma".
  18
  19 (* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
  20
  21 definition lpx (h) (G): relation lenv ≝
  22                         lex (cpx h G).
  23
  24 interpretation
  25    "unbound parallel rt-transition on all entries (local environment)"
  26    'PRedTySn h G L1 L2 = (lpx h G L1 L2).
  27
  28 (* Basic properties *********************************************************)
  29
  30 lemma lpx_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 lemma lpx_refl (h) (G): reflexive … (lpx h G).
  35 /2 width=1 by lex_refl/ qed.
  36
  37 (* Advanced properties ******************************************************)
  38
  39 lemma lpx_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
  40                                 ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}.
  41 /2 width=1 by lex_bind_refl_dx/ qed.
  42
  43 lemma lpx_pair (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → ∀V1,V2. ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 →
  44                         ∀I.⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
  45 /2 width=1 by lex_pair/ qed.
  46
  47 (* Basic inversion lemmas ***************************************************)
  48
  49 (* Basic_2A1: was: lpx_inv_atom1 *)
  50 lemma lpx_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆.
  51 /2 width=2 by lex_inv_atom_sn/ qed-.
  52
  53 lemma lpx_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 →
  54                                ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
  55                                         L2 = K2.ⓘ{I2}.
  56 /2 width=1 by lex_inv_bind_sn/ qed-.
  57
  58 (* Basic_2A1: was: lpx_inv_atom2 *)
  59 lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G, L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆.
  60 /2 width=2 by lex_inv_atom_dx/ qed-.
  61
  62 lemma lpx_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} →
  63                                ∃∃I1,K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
  64                                         L1 = K1.ⓘ{I1}.
  65 /2 width=1 by lex_inv_bind_dx/ qed-.
  66
  67 (* Advanced inversion lemmas ************************************************)
  68
  69 lemma lpx_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ⬈[h] L2 →
  70                                ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L2 = K2.ⓤ{I}.
  71 /2 width=1 by lex_inv_unit_sn/ qed-.
  72
  73 (* Basic_2A1: was: lpx_inv_pair1 *)
  74 lemma lpx_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
  75                                ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
  76                                         L2 = K2.ⓑ{I}V2.
  77 /2 width=1 by lex_inv_pair_sn/ qed-.
  78
  79 lemma lpx_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓤ{I} →
  80                                ∃∃K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L1 = K1.ⓤ{I}.
  81 /2 width=1 by lex_inv_unit_dx/ qed-.
  82
  83 (* Basic_2A1: was: lpx_inv_pair2 *)
  84 lemma lpx_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
  85                                ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
  86                                         L1 = K1.ⓑ{I}V1.
  87 /2 width=1 by lex_inv_pair_dx/ qed-.
  88
  89 lemma lpx_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
  90                             ∧∧ ⦃G, L1⦄ ⊢ ⬈[h] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
  91 /2 width=1 by lex_inv_pair/ qed-.