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".
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/
14 definition tr_nap (f) (l:nat): nat ≝
18 "functional non-negative application (total relocation maps)"
19 'ApplySucc f l = (tr_nap f l).
21 lemma tr_nap_unfold (f) (l):
25 lemma tr_compose_nap (f2) (f1) (l):
26 f2@↑❨f1@↑❨l❩❩ = (f2∘f1)@↑❨l❩.
28 <tr_nap_unfold <tr_nap_unfold <tr_nap_unfold
29 <tr_compose_pap <npsucc_pred //
32 lemma tr_uni_nap (n) (m):
36 <tr_uni_pap <nrplus_npsucc_sn //
39 lemma tr_nap_push (f):
40 ∀l. ↑(f@↑❨l❩) = (⫯f)@↑❨↑l❩.
42 <tr_nap_unfold <tr_nap_unfold
43 <tr_pap_push <pnpred_psucc //
46 lemma tr_nap_pushs_lt (f) (n) (m):
47 m < n → m = (⫯*[n]f)@↑❨m❩.
49 <tr_nap_unfold <tr_pap_pushs_le
50 /2 width=1 by nlt_npsucc_bi/