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

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  14
  15 notation "hvbox( ⦃ L1, break T1 ⦄ > + break ⦃ L2 , break T2 ⦄ )"
  16    non associative with precedence 45
  17    for @{ 'SupTermPlus $L1 $T1 $L2 $T2 }.
  18
  19 include "basic_2/substitution/csup.ma".
  20
  21 (* PLUS-ITERATED SUPCLOSURE *************************************************)
  22
  23 definition csupp: bi_relation lenv term ≝ bi_TC … csup.
  24
  25 interpretation "plus-iterated structural predecessor (closure)"
  26    'SupTermPlus L1 T1 L2 T2 = (csupp L1 T1 L2 T2).
  27
  28 (* Basic eliminators ********************************************************)
  29
  30 lemma csupp_ind: ∀L1,T1. ∀R:relation2 lenv term.
  31                  (∀L2,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → R L2 T2) →
  32                  (∀L,T,L2,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_TC_ind … IH1 IH2 ? ? H)
  36 qed-.
  37
  38 lemma csupp_ind_dx: ∀L2,T2. ∀R:relation2 lenv term.
  39                     (∀L1,T1. ⦃L1, T1⦄ > ⦃L2, T2⦄ → R L1 T1) →
  40                     (∀L1,L,T1,T. ⦃L1, T1⦄ > ⦃L, T⦄ → ⦃L, T⦄ >+ ⦃L2, T2⦄ → R L T → R L1 T1) →
  41                     ∀L1,T1. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → R L1 T1.
  42 #L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
  43 @(bi_TC_ind_dx … IH1 IH2 ? ? H)
  44 qed-.
  45
  46 (* Basic properties *********************************************************)
  47
  48 lemma csup_csupp: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ⦃L1, T1⦄ >+ ⦃L2, T2⦄.
  49 /2 width=1/ qed.
  50
  51 lemma csupp_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 csupp_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 (* Basic forward lemmas *****************************************************)
  60
  61 lemma csupp_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ >+ ⦃L2, T2⦄ → #{L2, T2} < #{L1, T1}.
  62 #L1 #L2 #T1 #T2 #H @(csupp_ind … H) -L2 -T2
  63 /3 width=3 by csup_fwd_cw, transitive_lt/
  64 qed-.