include "ground/relocation/tr_uni_pap.ma".
include "ground/relocation/tr_compose_pap.ma".
-include "ground/notation/functions/applysucc_2.ma".
+include "ground/relocation/tr_pap_eq.ma".
+include "ground/relocation/tr_pap_hdtl_eq.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_plus_pplus.ma".
include "ground/arith/nat_pred_succ.ma".
lemma nlt_npsucc_bi (n1) (n2):
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❩.
// 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❩).
+#i #f1 #f2 #Hf
+<tr_nap_unfold <tr_nap_unfold
+/3 width=1 by tr_pap_eq_repl, eq_f/
+qed.
+
+lemma tr_eq_inv_nap_zero_tl_bi (f1) (f2):
+ f1@§❨𝟎❩ = f2@§❨𝟎❩ → ⇂f1 ≗ ⇂f2 → f1 ≗ f2.
+#f1 #f2
+<tr_nap_unfold <tr_nap_unfold #H1 #H2
+/3 width=1 by tr_eq_inv_pap_unit_tl_bi, eq_inv_pnpred_bi/
+qed-.
+
+lemma tr_nap_plus_sn (f) (m) (n):
+ (⫯⇂*[↑n]f)@§❨m❩+f@§❨n❩ = f@§❨m+n❩.
+#f #m @(nat_ind_succ … m) -m [| #m #_ ] #n
+[ <nplus_zero_sn <nplus_zero_sn //
+| <tr_nap_push <nplus_comm in ⊢ (???%);
+ <tr_nap_unfold <tr_nap_unfold <tr_nap_unfold
+ <nsucc_pnpred <nrplus_inj_sn <nrplus_pnpred_dx
+ >nrplus_npsucc_sn <nrplus_inj_dx
+ <pplus_comm in ⊢ (???%); <tr_pap_plus //
+]
+qed.
+
+lemma tr_nap_plus_dx (f) (m) (n):
+ ⇂*[n]f@§❨m❩+(⫯f)@§❨n❩ = f@§❨m+n❩.
+#f #m #n @(nat_ind_succ … n) -n [| #n #_ ]
+[ //
+| <tr_nap_push
+ <tr_nap_unfold <tr_nap_unfold <tr_nap_unfold
+ <nsucc_pnpred <nplus_pnpred_sn
+ >nrplus_npsucc_sn <nrplus_inj_dx
+ <tr_pap_plus //
+]
+qed.