matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "ground/xoa/or_5.ma".
  16 include "ground/xoa/ex_3_1.ma".
  17 include "ground/xoa/ex_4_2.ma".
  18 include "ground/xoa/ex_4_3.ma".
  19 include "ground/xoa/ex_5_3.ma".
  20 include "delayed_updating/syntax/term_constructors.ma".
  21 include "delayed_updating/notation/relations/in_predicate_d_phi_1.ma".
  22
  23 (* BY-DEPTH DELAYED (BDD) TERM **********************************************)
  24
  25 inductive bdd: predicate term ≝
  26 | bdd_oref: ∀n. bdd #n
  27 | bdd_iref: ∀t,n. bdd t → bdd 𝛗n.t
  28 | bdd_abst: ∀t. bdd t → bdd 𝛌.t
  29 | bdd_appl: ∀u,t. bdd u → bdd t → bdd @u.t
  30 .
  31
  32 interpretation
  33   "well-formed by-depth delayed (term)"
  34   'InPredicateDPhi t = (bdd t).
  35
  36 (* Basic destructions *******************************************************)
  37
  38 lemma bdd_inv_in_com_gen:
  39       ∀t,p. t ϵ 𝐃𝛗 → p ϵ⬦ t →
  40       ∨∨ ∃∃n. #n = t & 𝗱❨n❩;𝐞 = p
  41        | ∃∃u,q,n. u ϵ 𝐃𝛗 & q ϵ⬦ u & 𝛗n.u = t & 𝗱❨n❩;q = p
  42        | ∃∃u,q. u ϵ 𝐃𝛗 & q ϵ⬦ u & 𝛌.u = t & 𝗟;q = p
  43        | ∃∃v,u,q. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ⬦ u & @v.u = t & 𝗔;q = p
  44        | ∃∃v,u,q. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ⬦ v & @v.u = t & 𝗦;q = p
  45 .
  46 #t #p *
  47 [ #n * /3 width=3 by or5_intro0, ex2_intro/
  48 | #u #n #Hu * #q #Hq #Hp /3 width=7 by ex4_3_intro, or5_intro1/
  49 | #u #Hu * #q #Hq #Hp /3 width=6 by or5_intro2, ex4_2_intro/
  50 | #v #u #Hv #Hu * * #q #Hq #Hp /3 width=8 by ex5_3_intro, or5_intro3, or5_intro4/
  51 ]
  52 qed-.
  53
  54 lemma bdd_inv_in_com_d:
  55       ∀t,q,n. t ϵ 𝐃𝛗 → 𝗱❨n❩;q ϵ⬦ t →
  56       ∨∨ ∧∧ #n = t & 𝐞 = q
  57        | ∃∃u. u ϵ 𝐃𝛗 & q ϵ⬦ u & 𝛗n.u = t
  58 .
  59 #t #q #n #Ht #Hq
  60 elim (bdd_inv_in_com_gen … Ht Hq) -Ht -Hq *
  61 [ #n0 #H1 #H2 destruct /3 width=1 by or_introl, conj/
  62 | #u0 #q0 #n0 #Hu0 #Hq0 #H1 #H2 destruct /3 width=4 by ex3_intro, or_intror/
  63 | #u0 #q0 #_ #_ #_ #H0 destruct
  64 | #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
  65 | #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
  66 ]
  67 qed-.
  68
  69 lemma bdd_inv_in_ini_d:
  70       ∀t,q,n. t ϵ 𝐃𝛗 → 𝗱❨n❩;q ϵ▵ t →
  71       ∨∨ ∧∧ #n = t & 𝐞 = q
  72        | ∃∃u. u ϵ 𝐃𝛗 & q ϵ▵ u & 𝛗n.u = t
  73 .
  74 #t #q #n #Ht * #r #Hq
  75 elim (bdd_inv_in_com_d … Ht Hq) -Ht -Hq *
  76 [ #H1 #H2
  77   elim (eq_inv_list_empty_append … H2) -H2 #H2 #_ destruct
  78   /3 width=1 by or_introl, conj/
  79 | #u #Hu #Hq #H1 destruct
  80   /4 width=4 by ex3_intro, ex_intro, or_intror/
  81 ]
  82 qed-.
  83
  84 lemma pippo:
  85       ∀l,n,p,t. t ϵ 𝐃𝛗 → p,𝗱❨n❩ ϵ▵ t → p,l ϵ▵ t → 𝗱❨n❩ = l.
  86 #l #n #p elim p -p
  87 [ #t #Ht <list_cons_comm <list_cons_comm #Hn #Hl
  88   elim (bdd_inv_in_ini_d … Ht Hn) -Ht -Hn *
  89   [ #H1 #_ destruct
  90     elim (term_in_ini_inv_lcons_oref … Hl) -Hl //
  91   | #u #_ #_ #H1 destruct
  92     elim (term_in_ini_inv_lcons_iref … Hl) -Hl //
  93   ]
  94 | * [ #m ] #p #IH #t #Ht
  95   <list_cons_shift <list_cons_shift #Hn #Hl
  96   [ elim (bdd_inv_in_ini_d … Ht Hn) -Ht -Hn *
  97     [ #_ #H
  98       elim (eq_inv_list_empty_rcons ??? H)
  99     | #u #Hu #Hp #H destruct
 100       elim (term_in_ini_inv_lcons_iref … Hl) -Hl #_ #Hl
 101       @(IH … Hu) //
 102     ]
 103   |
 104   ]
 105 ]