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

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  14
  15 include "basic_2/notation/relations/predtysn_5.ma".
  16 include "basic_2/static/lfxs.ma".
  17 include "basic_2/rt_transition/cpx_ext.ma".
  18
  19 (* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
  20
  21 definition lfpx (h) (G): relation3 term lenv lenv ≝
  22                          lfxs (cpx h G).
  23
  24 interpretation
  25    "unbound parallel rt-transition on referred entries (local environment)"
  26    'PRedTySn h T G L1 L2 = (lfpx h G T L1 L2).
  27
  28 (* Basic properties ***********************************************************)
  29
  30 lemma lfpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆.
  31 /2 width=1 by lfxs_atom/ qed.
  32
  33 lemma lfpx_sort: ∀h,I1,I2,G,L1,L2,s.
  34                  ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2}.
  35 /2 width=1 by lfxs_sort/ qed.
  36
  37 lemma lfpx_pair: ∀h,I,G,L1,L2,V1,V2.
  38                  ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2.
  39 /2 width=1 by lfxs_pair/ qed.
  40
  41 lemma lfpx_lref: ∀h,I1,I2,G,L1,L2,i.
  42                  ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2}.
  43 /2 width=1 by lfxs_lref/ qed.
  44
  45 lemma lfpx_gref: ∀h,I1,I2,G,L1,L2,l.
  46                  ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2}.
  47 /2 width=1 by lfxs_gref/ qed.
  48
  49 lemma lfpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
  50                          ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I1} →
  51                          ∀I2. ⦃G, L1⦄ ⊢ I ⬈[h] I2 →
  52                          ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I2}.
  53 /2 width=2 by lfxs_bind_repl_dx/ qed-.
  54
  55 (* Basic inversion lemmas ***************************************************)
  56
  57 lemma lfpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
  58 /2 width=3 by lfxs_inv_atom_sn/ qed-.
  59
  60 lemma lfpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
  61 /2 width=3 by lfxs_inv_atom_dx/ qed-.
  62
  63 lemma lfpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 →
  64                      ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
  65                       | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 &
  66                                        Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
  67 /2 width=1 by lfxs_inv_sort/ qed-.
  68
  69 lemma lfpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] Y2 →
  70                      ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
  71                       | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
  72                                        Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
  73 /2 width=1 by lfxs_inv_lref/ qed-.
  74
  75 lemma lfpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 →
  76                      ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
  77                       | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 &
  78                                        Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
  79 /2 width=1 by lfxs_inv_gref/ qed-.
  80
  81 lemma lfpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
  82                      ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
  83 /2 width=2 by lfxs_inv_bind/ qed-.
  84
  85 lemma lfpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 →
  86                      ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
  87 /2 width=2 by lfxs_inv_flat/ qed-.
  88
  89 (* Advanced inversion lemmas ************************************************)
  90
  91 lemma lfpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] Y2 →
  92                              ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
  93 /2 width=2 by lfxs_inv_sort_bind_sn/ qed-.
  94
  95 lemma lfpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2} →
  96                              ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
  97 /2 width=2 by lfxs_inv_sort_bind_dx/ qed-.
  98
  99 lemma lfpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 →
 100                              ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
 101                                       Y2 = L2.ⓑ{I}V2.
 102 /2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
 103
 104 lemma lfpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2 →
 105                              ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
 106                                       Y1 = L1.ⓑ{I}V1.
 107 /2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
 108
 109 lemma lfpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] Y2 →
 110                              ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓘ{I2}.
 111 /2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
 112
 113 lemma lfpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2} →
 114                              ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}.
 115 /2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
 116
 117 lemma lfpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] Y2 →
 118                              ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓘ{I2}.
 119 /2 width=2 by lfxs_inv_gref_bind_sn/ qed-.
 120
 121 lemma lfpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2} →
 122                              ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓘ{I1}.
 123 /2 width=2 by lfxs_inv_gref_bind_dx/ qed-.
 124
 125 (* Basic forward lemmas *****************************************************)
 126
 127 lemma lfpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
 128                         ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
 129 /2 width=3 by lfxs_fwd_pair_sn/ qed-.
 130
 131 lemma lfpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
 132                         ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
 133 /2 width=2 by lfxs_fwd_bind_dx/ qed-.
 134
 135 lemma lfpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
 136                         ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
 137 /2 width=3 by lfxs_fwd_flat_dx/ qed-.