matita/matita/contribs/lambda/subterms/booleanize.ma

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  14
  15 include "subterms/carrier.ma".
  16 include "subterms/boolean.ma".
  17
  18 (* BOOLEANIZE (EMPTY OR FILL)  **********************************************)
  19
  20 definition booleanize: bool → subterms → subterms ≝
  21    λb,F. {b}⇑⇓F.
  22
  23 interpretation "make boolean (subterms)"
  24    'ProjectSame b F = (booleanize b F).
  25
  26 notation "hvbox( { term 46 b } ⇕ break term 46 F)"
  27    non associative with precedence 46
  28    for @{ 'ProjectSame $b $F }.
  29
  30 lemma booleanize_inv_vref: ∀j,b0,b,F. {b}⇕ F = {b0}#j →
  31                            ∃∃b1. b = b0 & F = {b1}#j.
  32 #j #b0 #b #F #H
  33 elim (boolean_inv_vref … H) -H #H0 #H
  34 elim (carrier_inv_vref … H) -H /2 width=2/
  35 qed-.
  36
  37 lemma booleanize_inv_abst: ∀U,b0,b,F. {b}⇕ F = {b0}𝛌.U →
  38                            ∃∃b1,T. b = b0 & {b}⇕T = U & F = {b1}𝛌.T.
  39 #U #b0 #b #F #H
  40 elim (boolean_inv_abst … H) -H #C #H0 #H1 #H
  41 elim (carrier_inv_abst … H) -H #b1 #U1 #H3 destruct /2 width=4/
  42 qed-.
  43
  44 lemma booleanize_inv_appl: ∀W,U,b0,b,F. {b}⇕ F = {b0}@W.U →
  45                            ∃∃b1,V,T. b = b0 & {b}⇕V = W & {b}⇕T = U & F = {b1}@V.T.
  46 #W #U #b0 #b #F #H
  47 elim (boolean_inv_appl … H) -H #D #C #H0 #H1 #H2 #H
  48 elim (carrier_inv_appl … H) -H #b1 #W1 #U1 #H3 #H4 destruct /2 width=6/
  49 qed-.
  50 (*
  51 lemma booleanize_lift: ∀b,h,F,d. {b}⇕ ↑[d, h] F = ↑[d, h] {b}⇕ F.
  52 #b #h #M elim M -M normalize //
  53 qed.
  54
  55 lemma booleanize_dsubst: ∀b,G,F,d. {b}⇕ [d ↙ G] F = [d ↙ {b}⇕ G] {b}⇕ F.
  56 #b #N #M elim M -M [2,3: normalize // ]
  57 *)