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

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  14
  15 include "basic_2/notation/relations/predtysn_5.ma".
  16 include "static_2/static/rex.ma".
  17 include "basic_2/rt_transition/cpx_ext.ma".
  18
  19 (* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
  20
  21 definition rpx (h) (G): relation3 term lenv lenv ≝
  22                         rex (cpx h G).
  23
  24 interpretation
  25    "unbound parallel rt-transition on referred entries (local environment)"
  26    'PRedTySn h T G L1 L2 = (rpx h G T L1 L2).
  27
  28 (* Basic properties ***********************************************************)
  29
  30 lemma rpx_atom: ∀h,I,G. ❪G,⋆❫ ⊢ ⬈[h,⓪[I]] ⋆.
  31 /2 width=1 by rex_atom/ qed.
  32
  33 lemma rpx_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 rex_sort/ qed.
  36
  37 lemma rpx_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 rex_pair/ qed.
  40
  41 lemma rpx_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 rex_lref/ qed.
  44
  45 lemma rpx_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 rex_gref/ qed.
  48
  49 lemma rpx_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 rex_bind_repl_dx/ qed-.
  54
  55 (* Basic inversion lemmas ***************************************************)
  56
  57 lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ❪G,⋆❫ ⊢ ⬈[h,T] Y2 → Y2 = ⋆.
  58 /2 width=3 by rex_inv_atom_sn/ qed-.
  59
  60 lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ❪G,Y1❫ ⊢ ⬈[h,T] ⋆ → Y1 = ⋆.
  61 /2 width=3 by rex_inv_atom_dx/ qed-.
  62
  63 lemma rpx_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 rex_inv_sort/ qed-.
  68
  69 lemma rpx_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 rex_inv_lref/ qed-.
  74
  75 lemma rpx_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 rex_inv_gref/ qed-.
  80
  81 lemma rpx_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 rex_inv_bind/ qed-.
  84
  85 lemma rpx_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 rex_inv_flat/ qed-.
  88
  89 (* Advanced inversion lemmas ************************************************)
  90
  91 lemma rpx_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 rex_inv_sort_bind_sn/ qed-.
  94
  95 lemma rpx_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 rex_inv_sort_bind_dx/ qed-.
  98
  99 lemma rpx_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 rex_inv_zero_pair_sn/ qed-.
 103
 104 lemma rpx_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 rex_inv_zero_pair_dx/ qed-.
 108
 109 lemma rpx_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 rex_inv_lref_bind_sn/ qed-.
 112
 113 lemma rpx_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 rex_inv_lref_bind_dx/ qed-.
 116
 117 lemma rpx_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 rex_inv_gref_bind_sn/ qed-.
 120
 121 lemma rpx_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 rex_inv_gref_bind_dx/ qed-.
 124
 125 (* Basic forward lemmas *****************************************************)
 126
 127 lemma rpx_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 rex_fwd_pair_sn/ qed-.
 130
 131 lemma rpx_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 rex_fwd_bind_dx/ qed-.
 134
 135 lemma rpx_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 rex_fwd_flat_dx/ qed-.