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

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  14
  15 include "basic_2/substitution/frsup.ma".
  16 include "basic_2/unfold/frsupp.ma".
  17
  18 (* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
  19
  20 definition frsups: bi_relation lenv term ≝ bi_star … frsup.
  21
  22 interpretation "star-iterated restricted structural predecessor (closure)"
  23    'RestSupTermStar L1 T1 L2 T2 = (frsups L1 T1 L2 T2).
  24
  25 (* Basic eliminators ********************************************************)
  26
  27 lemma frsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
  28                   (∀L,L2,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ → R L T → R L2 T2) →
  29                   ∀L2,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L2 T2.
  30 #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
  31 @(bi_star_ind … IH1 IH2 ? ? H)
  32 qed-.
  33
  34 lemma frsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
  35                      (∀L1,L,T1,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → R L T → R L1 T1) →
  36                      ∀L1,T1. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L1 T1.
  37 #L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
  38 @(bi_star_ind_dx … IH1 IH2 ? ? H)
  39 qed-.
  40
  41 (* Basic properties *********************************************************)
  42
  43 lemma frsups_refl: bi_reflexive … frsups.
  44 /2 width=1/ qed.
  45
  46 lemma frsupp_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  47 /2 width=1/ qed.
  48
  49 lemma frsup_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  50 /2 width=1/ qed.
  51
  52 lemma frsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ →
  53                      ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  54 /2 width=4/ qed.
  55
  56 lemma frsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ →
  57                      ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  58 /2 width=4/ qed.
  59
  60 lemma frsups_frsupp_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ →
  61                             ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
  62 /2 width=4/ qed.
  63
  64 lemma frsupp_frsups_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ →
  65                             ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
  66 /2 width=4/ qed.
  67
  68 (* Basic inversion lemmas ***************************************************)
  69
  70 lemma frsups_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ →
  71                       (L1 = L2 ∧ T1 = T2) ∨ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
  72 #L1 #L2 #T1 #T2 * /2 width=1/
  73 qed-.
  74
  75 (* Basic forward lemmas *****************************************************)
  76
  77 lemma frsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{L2, T2} ≤ #{L1, T1}.
  78 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  79 /3 width=1 by frsupp_fwd_fw, lt_to_le/
  80 qed-.
  81
  82 lemma frsups_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{L1} ≤ #{L2}.
  83 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  84 /2 width=3 by frsupp_fwd_lw/
  85 qed-.
  86
  87 lemma frsups_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{T2} ≤ #{T1}.
  88 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  89 /3 width=3 by frsupp_fwd_tw, lt_to_le/
  90 qed-.
  91
  92 lemma frsups_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
  93 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H
  94 [ * #H1 #H2 destruct
  95   @(ex_intro … (⋆)) //
  96 | /2 width=3 by frsupp_fwd_append/
  97 qed-.
  98
  99 (* Advanced forward lemmas ***************************************************)
 100
 101 lemma lift_frsups_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
 102                          ∀L,K,U2. ⦃L, U1⦄ ⧁* ⦃L @@ K, U2⦄ →
 103                          ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
 104 #T1 #U1 #d #e #HTU1 #L #K #U2 #H elim (frsups_inv_all … H) -H
 105 [ * #H1 #H2 destruct
 106   >(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/
 107 | /2 width=5 by lift_frsupp_trans/
 108 ]
 109 qed-.
 110
 111 (* Advanced inversion lemmas for frsupp **************************************)
 112
 113 lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥.
 114 #J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 //
 115 #L2 #T2 #H elim (frsup_inv_atom1 … H)
 116 qed-.
 117
 118 lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
 119                                ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄.
 120 #b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
 121 [ #L2 #T2 #H
 122   elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
 123 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
 124 ]
 125 qed-.
 126
 127 lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
 128                                ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄.
 129 #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
 130 [ #L2 #T2 #H
 131   elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
 132 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
 133 ]
 134 qed-.