matita/matita/contribs/lambdadelta/basic_2A/etc/frsup/frsups.etc

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  14
  15 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
  16    non associative with precedence 45
  17    for @{ 'RestSupTermStar $L1 $T1 $L2 $T2 }.
  18
  19 include "basic_2/unfold/frsupp.ma".
  20
  21 (* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
  22
  23 definition frsups: bi_relation lenv term ≝ bi_star … frsup.
  24
  25 interpretation "star-iterated restricted structural predecessor (closure)"
  26    'RestSupTermStar L1 T1 L2 T2 = (frsups L1 T1 L2 T2).
  27
  28 (* Basic eliminators ********************************************************)
  29
  30 lemma frsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
  31                   (∀L,L2,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ → R L T → R L2 T2) →
  32                   ∀L2,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L2 T2.
  33 #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
  34 @(bi_star_ind … IH1 IH2 ? ? H)
  35 qed-.
  36
  37 lemma frsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
  38                      (∀L1,L,T1,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → R L T → R L1 T1) →
  39                      ∀L1,T1. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L1 T1.
  40 #L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
  41 @(bi_star_ind_dx … IH1 IH2 ? ? H)
  42 qed-.
  43
  44 (* Basic properties *********************************************************)
  45
  46 lemma frsups_refl: bi_reflexive … frsups.
  47 /2 width=1/ qed.
  48
  49 lemma frsupp_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  50 /2 width=1/ qed.
  51
  52 lemma frsup_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  53 /2 width=1/ qed.
  54
  55 lemma frsups_strap1: ∀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_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ →
  60                      ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
  61 /2 width=4/ qed.
  62
  63 lemma frsups_frsupp_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 lemma frsupp_frsups_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ →
  68                             ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
  69 /2 width=4/ qed.
  70
  71 (* Basic inversion lemmas ***************************************************)
  72
  73 lemma frsups_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ →
  74                       (L1 = L2 ∧ T1 = T2) ∨ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
  75 #L1 #L2 #T1 #T2 * /2 width=1/
  76 qed-.
  77
  78 (* Basic forward lemmas *****************************************************)
  79
  80 lemma frsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
  81 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  82 /3 width=1 by frsupp_fwd_fw, lt_to_le/
  83 qed-.
  84
  85 lemma frsups_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{L1} ≤ ♯{L2}.
  86 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  87 /2 width=3 by frsupp_fwd_lw/
  88 qed-.
  89
  90 lemma frsups_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ♯{T2} ≤ ♯{T1}.
  91 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
  92 /3 width=3 by frsupp_fwd_tw, lt_to_le/
  93 qed-.
  94
  95 lemma frsups_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
  96 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H
  97 [ * #H1 #H2 destruct
  98   @(ex_intro … (⋆)) //
  99 | /2 width=3 by frsupp_fwd_append/
 100 qed-.
 101
 102 (* Advanced forward lemmas ***************************************************)
 103
 104 lemma lift_frsups_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
 105                          ∀L,K,U2. ⦃L, U1⦄ ⧁* ⦃L @@ K, U2⦄ →
 106                          ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
 107 #T1 #U1 #d #e #HTU1 #L #K #U2 #H elim (frsups_inv_all … H) -H
 108 [ * #H1 #H2 destruct
 109   >(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/
 110 | /2 width=5 by lift_frsupp_trans/
 111 ]
 112 qed-.
 113
 114 (* Advanced inversion lemmas for frsupp **************************************)
 115
 116 lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥.
 117 #J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 //
 118 #L2 #T2 #H elim (frsup_inv_atom1 … H)
 119 qed-.
 120
 121 lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
 122                                ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄.
 123 #b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
 124 [ #L2 #T2 #H
 125   elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
 126 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
 127 ]
 128 qed-.
 129
 130 lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
 131                                ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄.
 132 #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
 133 [ #L2 #T2 #H
 134   elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
 135 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
 136 ]
 137 qed-.