matita/matita/contribs/lambdadelta/delayed_updating/unwind/nap.ma

   1 include "ground/relocation/tr_uni_pap.ma".
   2 include "ground/relocation/tr_compose_pap.ma".
   3 include "ground/notation/functions/applysucc_2.ma".
   4 include "ground/arith/nat_lt.ma".
   5 include "ground/arith/nat_plus_rplus.ma".
   6 include "ground/arith/nat_pred_succ.ma".
   7
   8 lemma nlt_npsucc_bi (n1) (n2):
   9       n1 < n2 → npsucc n1 < npsucc n2.
  10 #n1 #n2 #Hn elim Hn -n2 //
  11 #n2 #_ #IH /2 width=1 by plt_succ_dx_trans/
  12 qed.
  13
  14 definition tr_nap (f) (l:nat): nat ≝
  15            ↓(f＠⧣❨↑l❩).
  16
  17 interpretation
  18   "functional non-negative application (total relocation maps)"
  19   'ApplySucc f l = (tr_nap f l).
  20
  21 lemma tr_nap_unfold (f) (l):
  22       ↓(f＠⧣❨↑l❩) = f@↑❨l❩.
  23 // qed.
  24
  25 lemma tr_compose_nap (f2) (f1) (l):
  26       f2@↑❨f1@↑❨l❩❩ = (f2∘f1)@↑❨l❩.
  27 #f2 #f1 #l
  28 <tr_nap_unfold <tr_nap_unfold <tr_nap_unfold
  29 <tr_compose_pap <npsucc_pred //
  30 qed.
  31
  32 lemma tr_uni_nap (n) (m):
  33       m + n = 𝐮❨n❩@↑❨m❩.
  34 #n #m
  35 <tr_nap_unfold
  36 <tr_uni_pap <nrplus_npsucc_sn //
  37 qed.
  38
  39 lemma tr_nap_push (f):
  40       ∀l. ↑(f@↑❨l❩) = (⫯f)@↑❨↑l❩.
  41 #f #l
  42 <tr_nap_unfold <tr_nap_unfold
  43 <tr_pap_push <pnpred_psucc //
  44 qed.
  45
  46 lemma tr_nap_pushs_lt (f) (n) (m):
  47       m < n → m = (⫯*[n]f)@↑❨m❩.
  48 #f #n #m #Hmn
  49 <tr_nap_unfold <tr_pap_pushs_le
  50 /2 width=1 by nlt_npsucc_bi/
  51 qed-.