include "ground/relocation/tr_uni_pap.ma".
include "ground/relocation/tr_compose_pap.ma".
include "ground/relocation/tr_pap_eq.ma".
-include "ground/notation/functions/applysucc_2.ma".
+include "ground/notation/functions/atsection_2.ma".
include "ground/arith/nat_lt.ma".
include "ground/arith/nat_plus_rplus.ma".
include "ground/arith/nat_pred_succ.ma".
interpretation
"functional non-negative application (total relocation maps)"
- 'ApplySucc f l = (tr_nap f l).
+ 'AtSection f l = (tr_nap f l).
lemma tr_nap_unfold (f) (l):
- ↓(f@⧣❨↑l❩) = f@↑❨l❩.
+ ↓(f@⧣❨↑l❩) = f@§❨l❩.
// qed.
lemma tr_pap_succ_nap (f) (l):
- ↑(f@↑❨l❩) = f@⧣❨↑l❩.
+ ↑(f@§❨l❩) = f@⧣❨↑l❩.
// qed.
lemma tr_compose_nap (f2) (f1) (l):
- f2@↑❨f1@↑❨l❩❩ = (f2∘f1)@↑❨l❩.
+ f2@§❨f1@§❨l❩❩ = (f2∘f1)@§❨l❩.
#f2 #f1 #l
<tr_nap_unfold <tr_nap_unfold <tr_nap_unfold
<tr_compose_pap <npsucc_pred //
qed.
lemma tr_uni_nap (n) (m):
- m + n = 𝐮❨n❩@↑❨m❩.
+ m + n = 𝐮❨n❩@§❨m❩.
#n #m
<tr_nap_unfold
<tr_uni_pap <nrplus_npsucc_sn //
qed.
lemma tr_nap_push (f):
- ∀l. ↑(f@↑❨l❩) = (⫯f)@↑❨↑l❩.
+ ∀l. ↑(f@§❨l❩) = (⫯f)@§❨↑l❩.
#f #l
<tr_nap_unfold <tr_nap_unfold
<tr_pap_push <pnpred_psucc //
qed.
lemma tr_nap_pushs_lt (f) (n) (m):
- m < n → m = (⫯*[n]f)@↑❨m❩.
+ m < n → m = (⫯*[n]f)@§❨m❩.
#f #n #m #Hmn
<tr_nap_unfold <tr_pap_pushs_le
/2 width=1 by nlt_npsucc_bi/
qed-.
theorem tr_nap_eq_repl (i):
- stream_eq_repl … (λf1,f2. f1@↑❨i❩ = f2@↑❨i❩).
+ stream_eq_repl … (λf1,f2. f1@§❨i❩ = f2@§❨i❩).
#i #f1 #f2 #Hf
<tr_nap_unfold <tr_nap_unfold
/3 width=1 by tr_pap_eq_repl, eq_f/