matita/matita/contribs/lambdadelta/basic_2/substitution/fsups.ma

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  14
  15 include "basic_2/substitution/fsupp.ma".
  16
  17 (* STAR-ITERATED SUPCLOSURE *************************************************)
  18
  19 definition fsups: bi_relation lenv term ≝ bi_star … fsup.
  20
  21 interpretation "star-iterated structural successor (closure)"
  22    'SupTermStar L1 T1 L2 T2 = (fsups L1 T1 L2 T2).
  23
  24 (* Basic eliminators ********************************************************)
  25
  26 lemma fsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
  27                  (∀L,L2,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ → R L T → R L2 T2) →
  28                  ∀L2,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L2 T2.
  29 #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
  30 @(bi_star_ind … IH1 IH2 ? ? H)
  31 qed-.
  32
  33 lemma fsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
  34                     (∀L1,L,T1,T. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ → R L T → R L1 T1) →
  35                     ∀L1,T1. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L1 T1.
  36 #L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
  37 @(bi_star_ind_dx … IH1 IH2 ? ? H)
  38 qed-.
  39
  40 (* Basic properties *********************************************************)
  41
  42 lemma fsups_refl: bi_reflexive … fsups.
  43 /2 width=1/ qed.
  44
  45 lemma fsupp_fsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
  46 /2 width=1/ qed.
  47
  48 lemma fsup_fsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
  49 /2 width=1/ qed.
  50
  51 lemma fsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ →
  52                     ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
  53 /2 width=4/ qed.
  54
  55 lemma fsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ →
  56                     ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
  57 /2 width=4/ qed.
  58
  59 lemma fsups_fsupp_fsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ →
  60                          ⦃L, T⦄ ⊃+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
  61 /2 width=4/ qed.
  62
  63 lemma fsupp_fsups_fsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃+ ⦃L, T⦄ →
  64                          ⦃L, T⦄ ⊃* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
  65 /2 width=4/ qed.
  66
  67 (* Basic forward lemmas *****************************************************)
  68
  69 lemma fsups_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
  70 #L1 #L2 #T1 #T2 #H @(fsups_ind … H) -L2 -T2 //
  71 /4 width=3 by fsup_fwd_cw, lt_to_le_to_lt, lt_to_le/ (**) (* slow even with trace *)
  72 qed-.
  73 (*
  74 (* Advanced inversion lemmas on plus-iterated supclosure ********************)
  75
  76 lemma fsupp_inv_bind1_fsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
  77                              ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⊃* ⦃L2, T2⦄.
  78 #b #J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
  79 [ #L2 #T2 #H
  80   elim (fsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
  81 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
  82 ]
  83 qed-.
  84
  85 lemma fsupp_inv_flat1_fsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
  86                              ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⊃* ⦃L2, T2⦄.
  87 #J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
  88 [ #L2 #T2 #H
  89   elim (fsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
  90 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
  91 ]
  92 qed-.
  93 *)