matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_constructors.ma

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  14
  15 include "delayed_updating/substitution/lift_prototerm_id.ma".
  16 include "delayed_updating/substitution/lift_path_uni.ma".
  17 include "delayed_updating/syntax/prototerm_constructors_eq.ma".
  18 include "ground/relocation/nap.ma".
  19
  20 (* LIFT FOR PROTOTERM *******************************************************)
  21
  22 (* Constructions with constructors for prototerm ****************************)
  23
  24 lemma lift_term_oref_pap (f) (k):
  25       (⧣(f＠⧣❨k❩)) ⇔ 🠡[f]⧣k.
  26 #f #k @conj #p *
  27 [ /2 width=1 by in_comp_lift_path_term/
  28 | #q * #H0 destruct //
  29 ]
  30 qed.
  31
  32 lemma lift_term_iref_pap_sn (f) (t:prototerm) (k:pnat):
  33       (𝛕f＠⧣❨k❩.🠡[⇂*[k]f]t) ⊆ 🠡[f](𝛕k.t).
  34 #f #t #k #p * #q * #r #Hr #H1 #H2 destruct
  35 @(ex2_intro … (𝗱k◗𝗺◗r))
  36 /2 width=1 by in_comp_iref_hd/
  37 qed-.
  38
  39 lemma lift_term_iref_pap_dx (f) (t) (k:pnat):
  40       🠡[f](𝛕k.t) ⊆ 𝛕f＠⧣❨k❩.🠡[⇂*[k]f]t.
  41 #f #t #k #p * #q #Hq #H0 destruct
  42 elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
  43 <lift_path_d_sn <lift_path_m_sn
  44 /3 width=1 by in_comp_iref_hd, in_comp_lift_path_term/
  45 qed-.
  46
  47 lemma lift_term_iref_pap (f) (t) (k:pnat):
  48       (𝛕f＠⧣❨k❩.🠡[⇂*[k]f]t) ⇔ 🠡[f](𝛕k.t).
  49 /3 width=1 by conj, lift_term_iref_pap_sn, lift_term_iref_pap_dx/
  50 qed.
  51
  52 lemma lift_term_iref_nap (f) (t) (n):
  53       (𝛕↑(f＠§❨n❩).🠡[⇂*[↑n]f]t) ⇔ 🠡[f](𝛕↑n.t).
  54 #f #t #n
  55 >tr_pap_succ_nap //
  56 qed.
  57
  58 lemma lift_term_iref_uni (t) (n) (k):
  59       (𝛕(k+n).t) ⇔ 🠡[𝐮❨n❩](𝛕k.t).
  60 #t #n #k
  61 @(subset_eq_trans … (lift_term_iref_pap …))
  62 <tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
  63 /3 width=1 by iref_eq_repl, lift_term_id/
  64 qed.
  65
  66 lemma lift_term_abst (f) (t):
  67       (𝛌.🠡[⫯f]t) ⇔ 🠡[f]𝛌.t.
  68 #f #t @conj #p #Hp
  69 [ elim (in_comp_inv_abst … Hp) -Hp #q #H1 * #r #Hr #H2 destruct
  70   /3 width=1 by in_comp_lift_path_term, in_comp_abst_hd/
  71 | elim Hp -Hp #q #Hq #H0 destruct
  72   elim (in_comp_inv_abst … Hq) -Hq #r #H0 #Hr destruct
  73   /3 width=1 by in_comp_lift_path_term, in_comp_abst_hd/
  74 ]
  75 qed.
  76
  77 lemma lift_term_appl (f) (v) (t):
  78       ＠🠡[f]v.🠡[f]t ⇔ 🠡[f]＠v.t.
  79 #f #v #t @conj #p #Hp
  80 [ elim (in_comp_inv_appl … Hp) -Hp * #q #H1 * #r #Hr #H2 destruct
  81   /3 width=1 by in_comp_lift_path_term, in_comp_appl_sd, in_comp_appl_hd/
  82 | elim Hp -Hp #q #Hq #H0 destruct
  83   elim (in_comp_inv_appl … Hq) -Hq * #r #H0 #Hr destruct
  84   /3 width=1 by in_comp_lift_path_term, in_comp_appl_sd, in_comp_appl_hd/
  85 ]
  86 qed.