matita/matita/contribs/lambda_delta/basic_2/etc/csup/csups.etc

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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 notation "hvbox( ⦃ L1, break T1 ⦄ > * break ⦃ L2 , break T2 ⦄ )"
  16    non associative with precedence 45
  17    for @{ 'SupTermStar $L1 $T1 $L2 $T2 }.
  18
  19 include "basic_2/substitution/csup.ma".
  20 include "basic_2/unfold/csupp.ma".
  21
  22 (* STAR-ITERATED SUPCLOSURE *************************************************)
  23
  24 definition csups: bi_relation lenv term ≝ bi_star … csup.
  25
  26 interpretation "star-iterated structural predecessor (closure)"
  27    'SupTermStar L1 T1 L2 T2 = (csups L1 T1 L2 T2).
  28
  29 (* Basic eliminators ********************************************************)
  30
  31 lemma csups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
  32                  (∀L,L2,T,T2. ⦃L1, T1⦄ >* ⦃L, T⦄ → ⦃L, T⦄ > ⦃L2, T2⦄ → R L T → R L2 T2) →
  33                  ∀L2,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → R L2 T2.
  34 #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
  35 @(bi_star_ind … IH1 IH2 ? ? H)
  36 qed-.
  37
  38 lemma csups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
  39                     (∀L1,L,T1,T. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >* ⦃L2, T2⦄ → R L T → R L1 T1) →
  40                     ∀L1,T1. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → R L1 T1.
  41 #L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
  42 @(bi_star_ind_dx … IH1 IH2 ? ? H)
  43 qed-.
  44
  45 (* Basic properties *********************************************************)
  46
  47 lemma csups_refl: bi_reflexive … csups.
  48 /2 width=1/ qed.
  49
  50 lemma csupp_csups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → ⦃L1, T1⦄ >* ⦃L2, T2⦄.
  51 /2 width=1/ qed.
  52
  53 lemma csup_csups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ⦃L1, T1⦄ >* ⦃L2, T2⦄.
  54 /2 width=1/ qed.
  55
  56 lemma csups_strap1: ∀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 csups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >* ⦃L2, T2⦄ →
  61                     ⦃L1, T1⦄ >* ⦃L2, T2⦄.
  62 /2 width=4/ qed.
  63
  64 lemma csups_csupp_csupp: ∀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 lemma csupp_csups_csupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ >+ ⦃L, T⦄ →
  69                           ⦃L, T⦄ >* ⦃L2, T2⦄ → ⦃L1, T1⦄ >+ ⦃L2, T2⦄.
  70 /2 width=4/ qed.
  71
  72 (* Basic forward lemmas *****************************************************)
  73
  74 lemma csups_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ → #{L2, T2} ≤ #{L1, T1}.
  75 #L1 #L2 #T1 #T2 #H @(csups_ind … H) -L2 -T2 //
  76 /4 width=3 by csup_fwd_cw, lt_to_le_to_lt, lt_to_le/ (**) (* slow even with trace *)
  77 qed-.
  78
  79 (* Advanced inversion lemmas for csupp **************************************)
  80
  81 lemma csupp_inv_atom1_csups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ >+ ⦃L2, T2⦄ →
  82                              ∃∃I,K,V,i. ⇩[0, i] L1 ≡ K.ⓑ{I}V &
  83                              ⦃K, V⦄ >* ⦃L2, T2⦄ & J = LRef i.
  84 #J #L1 #L2 #T2 #H @(csupp_ind … H) -L2 -T2
  85 [ #L2 #T2 #H
  86   elim (csup_inv_atom1 … H) -H * #i #HL12 #H destruct /2 width=7/
  87 | #L #T #L2 #T2 #_ #HT2 * #I #K #V #i #HLK #HVT #H destruct /3 width=8/
  88 ]
  89 qed-.
  90
  91 lemma csupp_inv_bind1_csups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ >+ ⦃L2, T2⦄ →
  92                              ⦃L1, W⦄ >* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ >* ⦃L2, T2⦄.
  93 #b #J #L1 #L2 #W #U #T2 #H @(csupp_ind … H) -L2 -T2
  94 [ #L2 #T2 #H
  95   elim (csup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
  96 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
  97 ]
  98 qed-.
  99
 100 lemma csupp_inv_flat1_csups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ >+ ⦃L2, T2⦄ →
 101                              ⦃L1, W⦄ >* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ >* ⦃L2, T2⦄.
 102 #J #L1 #L2 #W #U #T2 #H @(csupp_ind … H) -L2 -T2
 103 [ #L2 #T2 #H
 104   elim (csup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
 105 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
 106 ]
 107 qed-.