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 "basic_2/relocation/lex.ma".
  17 include "basic_2/rt_transition/cpx_ext.ma".
  18
  19 (* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENVIRONMENTS ********************)
  20
  21 definition lpx: sh → genv → relation lenv ≝
  22                 λh,G. lex (cpx h G).
  23
  24 interpretation
  25    "unbound parallel rt-transition (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,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 →
  44                 ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
  45 /2 width=1 by lpx_sn_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,I1,G,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,I2,G,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 (* Basic_2A1: was: lpx_inv_pair1 *)
  70 lemma lpx_inv_pair_sn: ∀h,I,G,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
  71                        ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
  72                                 L2 = K2.ⓑ{I}V2.
  73 /2 width=1 by lex_inv_pair_sn/ qed-.
  74
  75 (* Basic_2A1: was: lpx_inv_pair2 *)
  76 lemma lpx_inv_pair_dx: ∀h,I,G,L1,K2,V2.  ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
  77                        ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
  78                                 L1 = K1.ⓑ{I}V1.
  79 /2 width=1 by lex_inv_pair_dx/ qed-.
  80
  81 lemma lpx_inv_pair: ∀h,I1,I2,G,L1,L2,V1,V2.  ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
  82                     ∧∧ ⦃G, L1⦄ ⊢ ⬈[h] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
  83 /2 width=1 by lex_inv_pair/ qed-.