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

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "delayed_updating/syntax/prototerm.ma".
  16 include "delayed_updating/notation/functions/m_hook_1.ma".
  17 include "delayed_updating/notation/functions/hash_1.ma".
  18 include "delayed_updating/notation/functions/tau_2.ma".
  19 include "delayed_updating/notation/functions/tau_3.ma".
  20 include "delayed_updating/notation/functions/lamda_1.ma".
  21 include "delayed_updating/notation/functions/at_2.ma".
  22
  23 (* CONSTRUCTORS FOR PROTOTERM ***********************************************)
  24
  25 definition prototerm_node_0 (l): prototerm ≝
  26            λp. l◗𝐞 = p.
  27
  28 definition prototerm_node_1 (l): prototerm → prototerm ≝
  29            λt,p. ∃∃q. q ϵ t & l◗q = p.
  30
  31 definition prototerm_node_1_2 (l1) (l2): prototerm → prototerm ≝
  32            λt,p. ∃∃q. q ϵ t & l1◗l2◗q = p.
  33
  34 definition prototerm_node_2 (l1) (l2): prototerm → prototerm → prototerm ≝
  35            λt1,t2,p.
  36            ∨∨ ∃∃q. q ϵ t1 & l1◗q = p
  37             | ∃∃q. q ϵ t2 & l2◗q = p.
  38
  39 interpretation
  40   "mark (prototerm)"
  41   'MHook t = (prototerm_node_1 label_m t).
  42
  43 interpretation
  44   "outer variable reference by depth (prototerm)"
  45   'Hash k = (prototerm_node_0 (label_d k)).
  46
  47 interpretation
  48   "inner variable reference by depth (prototerm)"
  49   'Tau k t = (prototerm_node_1_2 (label_d k) label_m t).
  50
  51 interpretation
  52   "inner variable reference by depth with offset (prototerm)"
  53   'Tau k d t = (prototerm_node_1_2 (label_d2 k d) label_m t).
  54
  55 interpretation
  56   "name-free functional abstraction (prototerm)"
  57   'Lamda t = (prototerm_node_1 label_L t).
  58
  59 interpretation
  60   "application (prototerm)"
  61   'At u t = (prototerm_node_2 label_S label_A u t).
  62
  63 (* Basic constructions *******************************************************)
  64
  65 lemma in_comp_iref (t) (q) (k):
  66       q ϵ t → 𝗱k◗𝗺◗q ϵ 𝛕k.t.
  67 /2 width=3 by ex2_intro/ qed.
  68
  69 lemma in_comp_iref2 (t) (q) (k) (d):
  70       q ϵ t → 𝗱❨k,d❩◗𝗺◗q ϵ 𝛕❨k,d❩.t.
  71 /2 width=3 by ex2_intro/ qed.
  72
  73 (* Basic inversions *********************************************************)
  74
  75 lemma in_comp_inv_iref (t) (p) (k):
  76       p ϵ 𝛕k.t →
  77       ∃∃q. 𝗱k◗𝗺◗q = p & q ϵ t.
  78 #t #p #k * #q #Hq #Hp
  79 /2 width=3 by ex2_intro/
  80 qed-.
  81
  82 lemma in_comp_inv_iref2 (t) (p) (k) (d):
  83       p ϵ 𝛕❨k,d❩.t →
  84       ∃∃q. 𝗱❨k,d❩◗𝗺◗q = p & q ϵ t.
  85 #t #p #k #d * #q #Hq #Hp
  86 /2 width=3 by ex2_intro/
  87 qed-.
  88
  89 (* COMMENT
  90 lemma prototerm_in_root_inv_lcons_oref:
  91       ∀p,l,n. l◗p ϵ ▵#n →
  92       ∧∧ 𝗱n = l & 𝐞 = p.
  93 #p #l #n * #q
  94 <list_append_lcons_sn #H0 destruct -H0
  95 elim (eq_inv_list_empty_append … e0) -e0 #H0 #_
  96 /2 width=1 by conj/
  97 qed-.
  98
  99 lemma prototerm_in_root_inv_lcons_iref:
 100       ∀t,p,l,n. l◗p ϵ ▵𝛕n.t →
 101       ∧∧ 𝗱n = l & p ϵ ▵ɱ.t.
 102 #t #p #l #n * #q * #r #Hr
 103 <list_append_lcons_sn #H0 destruct -H0
 104 /4 width=4 by ex2_intro, ex_intro, conj/
 105 qed-.
 106
 107 lemma prototerm_in_root_inv_lcons_mark:
 108       ∀t,p,l. l◗p ϵ ▵ɱ.t →
 109       ∧∧ 𝗺 = l & p ϵ ▵t.
 110 #t #p #l * #q * #r #Hr
 111 <list_append_lcons_sn #H0 destruct
 112 /3 width=2 by ex_intro, conj/
 113 qed-.
 114
 115 lemma prototerm_in_root_inv_lcons_abst:
 116       ∀t,p,l. l◗p ϵ ▵𝛌.t →
 117       ∧∧ 𝗟 = l & p ϵ ▵t.
 118 #t #p #l * #q * #r #Hr
 119 <list_append_lcons_sn #H0 destruct
 120 /3 width=2 by ex_intro, conj/
 121 qed-.
 122
 123 lemma prototerm_in_root_inv_lcons_appl:
 124       ∀u,t,p,l. l◗p ϵ ▵@u.t →
 125       ∨∨ ∧∧ 𝗦 = l & p ϵ ▵u
 126        | ∧∧ 𝗔 = l & p ϵ ▵t.
 127 #u #t #p #l * #q * * #r #Hr
 128 <list_append_lcons_sn #H0 destruct
 129 /4 width=2 by ex_intro, or_introl, or_intror, conj/
 130 qed-.
 131 *)