matita/matita/lib/lambda/subterms/booleanized.ma

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  14
  15 include "lambda/subterms/boolean.ma".
  16
  17 include "lambda/notation/functions/projectsame_2.ma".
  18
  19 include "lambda/xoa/ex_4_3.ma".
  20
  21 (* BOOLEANIZED SUBSET (EMPTY OR FULL) ***************************************)
  22
  23 definition booleanized: bool → subterms → subterms ≝
  24    λb,F. {b}⇑⇓F.
  25
  26 interpretation "booleanized (subterms)"
  27    'ProjectSame b F = (booleanized b F).
  28
  29 lemma booleanized_inv_vref: ∀j,c,b,F. {b}⇕ F = {c}#j →
  30                             ∃∃b1. b = c & F = {b1}#j.
  31 #j #c #b #F #H
  32 elim (boolean_inv_vref … H) -H #H0 #H
  33 elim (carrier_inv_vref … H) -H /2 width=2/
  34 qed-.
  35
  36 lemma booleanized_inv_abst: ∀U,c,b,F. {b}⇕ F = {c}𝛌.U →
  37                             ∃∃b1,T. b = c & {b}⇕T = U & F = {b1}𝛌.T.
  38 #U #c #b #F #H
  39 elim (boolean_inv_abst … H) -H #C #H0 #H1 #H
  40 elim (carrier_inv_abst … H) -H #b1 #U1 #H3 destruct /2 width=4/
  41 qed-.
  42
  43 lemma booleanized_inv_appl: ∀W,U,c,b,F. {b}⇕ F = {c}@W.U →
  44                             ∃∃b1,V,T. b = c & {b}⇕V = W & {b}⇕T = U & F = {b1}@V.T.
  45 #W #U #c #b #F #H
  46 elim (boolean_inv_appl … H) -H #D #C #H0 #H1 #H2 #H
  47 elim (carrier_inv_appl … H) -H #b1 #W1 #U1 #H3 #H4 destruct /2 width=6/
  48 qed-.
  49
  50 lemma booleanized_booleanized: ∀c,b,F. {b}⇕ {c}⇕ F = {b}⇕ F.
  51 normalize //
  52 qed.
  53
  54 lemma booleanized_lift: ∀b,h,F,d. {b}⇕ ↑[d, h] F = ↑[d, h] {b}⇕ F.
  55 normalize //
  56 qed.
  57
  58 lemma booleanized_dsubst: ∀b,G,F,d. {b}⇕ [d ↙ G] F = [d ↙ {b}⇕ G] {b}⇕ F.
  59 normalize //
  60 qed.