matita/matita/contribs/lambdadelta/basic_2/static/lfeq.ma

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  14
  15 include "basic_2/notation/relations/lazyeq_3.ma".
  16 include "basic_2/static/lfxs.ma".
  17
  18 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
  19
  20 definition lfeq: relation3 term lenv lenv ≝ lfxs ceq.
  21
  22 interpretation
  23    "equivalence on referred entries (local environment)"
  24    'LazyEq T L1 L2 = (lfeq T L1 L2).
  25
  26 definition lfeq_transitive: predicate (relation3 lenv term term) ≝
  27            λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2.
  28
  29 (* Basic properties ***********************************************************)
  30
  31 lemma lfeq_atom: ∀I. ⋆ ≡[⓪{I}] ⋆.
  32 /2 width=1 by lfxs_atom/ qed.
  33
  34 lemma lfeq_sort: ∀I,L1,L2,V1,V2,s.
  35                  L1 ≡[⋆s] L2 → L1.ⓑ{I}V1 ≡[⋆s] L2.ⓑ{I}V2.
  36 /2 width=1 by lfxs_sort/ qed.
  37
  38 lemma lfeq_zero: ∀I,L1,L2,V.
  39                  L1 ≡[V] L2 → L1.ⓑ{I}V ≡[#0] L2.ⓑ{I}V.
  40 /2 width=1 by lfxs_zero/ qed.
  41
  42 lemma lfeq_lref: ∀I,L1,L2,V1,V2,i.
  43                  L1 ≡[#i] L2 → L1.ⓑ{I}V1 ≡[#⫯i] L2.ⓑ{I}V2.
  44 /2 width=1 by lfxs_lref/ qed.
  45
  46 lemma lfeq_gref: ∀I,L1,L2,V1,V2,l.
  47                  L1 ≡[§l] L2 → L1.ⓑ{I}V1 ≡[§l] L2.ⓑ{I}V2.
  48 /2 width=1 by lfxs_gref/ qed.
  49
  50 (* Basic inversion lemmas ***************************************************)
  51
  52 lemma lfeq_inv_atom_sn: ∀I,Y2. ⋆ ≡[⓪{I}] Y2 → Y2 = ⋆.
  53 /2 width=3 by lfxs_inv_atom_sn/ qed-.
  54
  55 lemma lfeq_inv_atom_dx: ∀I,Y1. Y1 ≡[⓪{I}] ⋆ → Y1 = ⋆.
  56 /2 width=3 by lfxs_inv_atom_dx/ qed-.
  57
  58 lemma lfeq_inv_zero: ∀Y1,Y2. Y1 ≡[#0] Y2 →
  59                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  60                      ∃∃I,L1,L2,V. L1 ≡[V] L2 &
  61                                   Y1 = L1.ⓑ{I}V & Y2 = L2.ⓑ{I}V.
  62 #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
  63 /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
  64 qed-.
  65
  66 lemma lfeq_inv_lref: ∀Y1,Y2,i. Y1 ≡[#⫯i] Y2 →
  67                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  68                      ∃∃I,L1,L2,V1,V2. L1 ≡[#i] L2 &
  69                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  70 #Y1 #Y2 #i #H elim (lfxs_inv_lref … H) -H *
  71 /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
  72 qed-.
  73
  74 lemma lfeq_inv_bind: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 →
  75                      L1 ≡[V] L2 ∧ L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
  76 #p #I #L1 #L2 #V #T #H elim (lfxs_inv_bind … H) -H /2 width=3 by conj/
  77 qed-.
  78
  79 lemma lfeq_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 →
  80                      L1 ≡[V] L2 ∧ L1 ≡[T] L2.
  81 #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H /2 width=3 by conj/
  82 qed-.
  83
  84 (* Advanced inversion lemmas ************************************************)
  85
  86 lemma lfeq_inv_zero_pair_sn: ∀I,Y2,L1,V. L1.ⓑ{I}V ≡[#0] Y2 →
  87                              ∃∃L2. L1 ≡[V] L2 & Y2 = L2.ⓑ{I}V.
  88 #I #Y2 #L1 #V #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=3 by ex2_intro/
  89 qed-.
  90
  91 lemma lfeq_inv_zero_pair_dx: ∀I,Y1,L2,V. Y1 ≡[#0] L2.ⓑ{I}V →
  92                              ∃∃L1. L1 ≡[V] L2 & Y1 = L1.ⓑ{I}V.
  93 #I #Y1 #L2 #V #H elim (lfxs_inv_zero_pair_dx … H) -H
  94 #L1 #X #HL12 #HX #H destruct /2 width=3 by ex2_intro/
  95 qed-.
  96
  97 lemma lfeq_inv_lref_pair_sn: ∀I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[#⫯i] Y2 →
  98                              ∃∃L2,V2. L1 ≡[#i] L2 & Y2 = L2.ⓑ{I}V2.
  99 /2 width=2 by lfxs_inv_lref_pair_sn/ qed-.
 100
 101 lemma lfeq_inv_lref_pair_dx: ∀I,Y1,L2,V2,i. Y1 ≡[#⫯i] L2.ⓑ{I}V2 →
 102                              ∃∃L1,V1. L1 ≡[#i] L2 & Y1 = L1.ⓑ{I}V1.
 103 /2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
 104
 105 (* Basic forward lemmas *****************************************************)
 106
 107 lemma lfeq_fwd_bind_sn: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 → L1 ≡[V] L2.
 108 /2 width=4 by lfxs_fwd_bind_sn/ qed-.
 109
 110 lemma lfeq_fwd_bind_dx: ∀p,I,L1,L2,V,T.
 111                         L1 ≡[ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
 112 /2 width=2 by lfxs_fwd_bind_dx/ qed-.
 113
 114 lemma lfeq_fwd_flat_sn: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[V] L2.
 115 /2 width=3 by lfxs_fwd_flat_sn/ qed-.
 116
 117 lemma lfeq_fwd_flat_dx: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[T] L2.
 118 /2 width=3 by lfxs_fwd_flat_dx/ qed-.
 119
 120 lemma lfeq_fwd_pair_sn: ∀I,L1,L2,V,T. L1 ≡[②{I}V.T] L2 → L1 ≡[V] L2.
 121 /2 width=3 by lfxs_fwd_pair_sn/ qed-.
 122
 123 (* Advanceded forward lemmas with generic extension on referred entries *****)
 124
 125 lemma lfex_fwd_lfxs_refl: ∀R. (∀L. reflexive … (R L)) →
 126                           ∀L1,L2,T. L1 ≡[T] L2 → L1 ⦻*[R, T] L2.
 127 /2 width=3 by lfxs_co/ qed-.
 128
 129 (* Basic_2A1: removed theorems 30:
 130               lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref
 131               lleq_fwd_drop_sn lleq_fwd_drop_dx
 132               lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx
 133               lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat
 134               lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl
 135               lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div
 136 *)