matita/matita/contribs/lambdadelta/basic_2/unfold/frsups.ma

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  14
  15 include "basic_2/unfold/frsupp.ma".
  16
  17 (* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
  18
  19 definition frsups: bi_relation lenv term ≝ bi_star … frsup.
  20
  21 interpretation "star-iterated restricted structural predecessor (closure)"
  22    'RestSupTermStar L1 T1 L2 T2 = (frsups L1 T1 L2 T2).
  23
  24 (* Basic eliminators ********************************************************)
  25
  26 lemma frsups_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 frsups_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 frsups_refl: bi_reflexive … frsups.
  43 /2 width=1/ qed.
  44
  45 lemma frsupp_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  46 /2 width=1/ qed.
  47
  48 lemma frsup_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  49 /2 width=1/ qed.
  50
  51 lemma frsups_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 frsups_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 frsups_frsupp_frsupp: ∀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 frsupp_frsups_frsupp: ∀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 inversion lemmas ***************************************************)
  68
  69 lemma frsups_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ →
  70                       (L1 = L2 ∧ T1 = T2) ∨ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
  71 #L1 #L2 #T1 #T2 * /2 width=1/
  72 qed-.
  73
  74 (* Basic forward lemmas *****************************************************)
  75
  76 lemma frsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{L2, T2} ≤ #{L1, T1}.
  77 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  78 /3 width=1 by frsupp_fwd_fw, lt_to_le/
  79 qed-.
  80
  81 lemma frsups_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{L1} ≤ #{L2}.
  82 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  83 /2 width=3 by frsupp_fwd_lw/
  84 qed-.
  85
  86 lemma frsups_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{T2} ≤ #{T1}.
  87 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  88 /3 width=3 by frsupp_fwd_tw, lt_to_le/
  89 qed-.
  90
  91 lemma frsups_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
  92 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H
  93 [ * #H1 #H2 destruct
  94   @(ex_intro … (⋆)) //
  95 | /2 width=3 by frsupp_fwd_append/
  96 qed-.
  97
  98 (* Advanced forward lemmas ***************************************************)
  99
 100 lemma lift_frsups_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
 101                          ∀L,K,U2. ⦃L, U1⦄ ⧁* ⦃L @@ K, U2⦄ →
 102                          ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
 103 #T1 #U1 #d #e #HTU1 #L #K #U2 #H elim (frsups_inv_all … H) -H
 104 [ * #H1 #H2 destruct
 105   >(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/
 106 | /2 width=5 by lift_frsupp_trans/
 107 ]
 108 qed-.
 109
 110 (* Advanced inversion lemmas for frsupp **************************************)
 111
 112 lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥.
 113 #J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 //
 114 #L2 #T2 #H elim (frsup_inv_atom1 … H)
 115 qed-.
 116
 117 lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
 118                                ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄.
 119 #b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
 120 [ #L2 #T2 #H
 121   elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
 122 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
 123 ]
 124 qed-.
 125
 126 lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
 127                                ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄.
 128 #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
 129 [ #L2 #T2 #H
 130   elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
 131 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
 132 ]
 133 qed-.