(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_3_2.ma".
include "ground_2/steps/rtc_plus.ma".
include "ground_2/steps/rtc_ist.ma".
(* Properties with test for t-transition counter ****************************)
-lemma ist_plus: â\88\80n1,n2,c1,c2. ð\9d\90\93â¦\83n1,c1â¦\84 â\86\92 ð\9d\90\93â¦\83n2,c2â¦\84 â\86\92 ð\9d\90\93â¦\83n1+n2,c1+c2â¦\84.
+lemma ist_plus: â\88\80n1,n2,c1,c2. ð\9d\90\93â\9dªn1,c1â\9d« â\86\92 ð\9d\90\93â\9dªn2,c2â\9d« â\86\92 ð\9d\90\93â\9dªn1+n2,c1+c2â\9d«.
#n1 #n2 #c1 #c2 #H1 #H2 destruct //
qed.
-lemma ist_plus_O1: â\88\80n,c1,c2. ð\9d\90\93â¦\830,c1â¦\84 â\86\92 ð\9d\90\93â¦\83n,c2â¦\84 â\86\92 ð\9d\90\93â¦\83n,c1+c2â¦\84.
+lemma ist_plus_O1: â\88\80n,c1,c2. ð\9d\90\93â\9dª0,c1â\9d« â\86\92 ð\9d\90\93â\9dªn,c2â\9d« â\86\92 ð\9d\90\93â\9dªn,c1+c2â\9d«.
/2 width=1 by ist_plus/ qed.
-lemma ist_plus_O2: â\88\80n,c1,c2. ð\9d\90\93â¦\83n,c1â¦\84 â\86\92 ð\9d\90\93â¦\830,c2â¦\84 â\86\92 ð\9d\90\93â¦\83n,c1+c2â¦\84.
+lemma ist_plus_O2: â\88\80n,c1,c2. ð\9d\90\93â\9dªn,c1â\9d« â\86\92 ð\9d\90\93â\9dª0,c2â\9d« â\86\92 ð\9d\90\93â\9dªn,c1+c2â\9d«.
#n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by ist_plus/
qed.
-lemma ist_succ: â\88\80n,c. ð\9d\90\93â¦\83n,câ¦\84 â\86\92 ð\9d\90\93â¦\83â\86\91n,c+ð\9d\9f\98ð\9d\9f\99â¦\84.
+lemma ist_succ: â\88\80n,c. ð\9d\90\93â\9dªn,câ\9d« â\86\92 ð\9d\90\93â\9dªâ\86\91n,c+ð\9d\9f\98ð\9d\9f\99â\9d«.
/2 width=1 by ist_plus/ qed.
(* Inversion properties with test for constrained rt-transition counter *****)
lemma ist_inv_plus:
- â\88\80n,c1,c2. ð\9d\90\93â¦\83n,c1 + c2â¦\84 →
- â\88\83â\88\83n1,n2. ð\9d\90\93â¦\83n1,c1â¦\84 & ð\9d\90\93â¦\83n2,c2â¦\84 & n1 + n2 = n.
+ â\88\80n,c1,c2. ð\9d\90\93â\9dªn,c1 + c2â\9d« →
+ â\88\83â\88\83n1,n2. ð\9d\90\93â\9dªn1,c1â\9d« & ð\9d\90\93â\9dªn2,c2â\9d« & n1 + n2 = n.
#n #c1 #c2 #H
elim (plus_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H1 #H2 #H3 #H4 #H5 #H6 destruct
elim (plus_inv_O3 … H1) -H1 #H11 #H12 destruct
/3 width=5 by ex3_2_intro/
qed-.
-lemma ist_inv_plus_O_dx: â\88\80n,c1,c2. ð\9d\90\93â¦\83n,c1 + c2â¦\84 â\86\92 ð\9d\90\93â¦\830,c2â¦\84 â\86\92 ð\9d\90\93â¦\83n,c1â¦\84.
+lemma ist_inv_plus_O_dx: â\88\80n,c1,c2. ð\9d\90\93â\9dªn,c1 + c2â\9d« â\86\92 ð\9d\90\93â\9dª0,c2â\9d« â\86\92 ð\9d\90\93â\9dªn,c1â\9d«.
#n #c1 #c2 #H #H2
elim (ist_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct //
qed-.
lemma ist_inv_plus_SO_dx:
- â\88\80n,c1,c2. ð\9d\90\93â¦\83n,c1 + c2â¦\84 â\86\92 ð\9d\90\93â¦\831,c2â¦\84 →
- â\88\83â\88\83m. ð\9d\90\93â¦\83m,c1â¦\84 & n = ↑m.
+ â\88\80n,c1,c2. ð\9d\90\93â\9dªn,c1 + c2â\9d« â\86\92 ð\9d\90\93â\9dª1,c2â\9d« →
+ â\88\83â\88\83m. ð\9d\90\93â\9dªm,c1â\9d« & n = ↑m.
#n #c1 #c2 #H #H2 destruct
elim (ist_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
/2 width=3 by ex2_intro/
qed-.
-lemma ist_inv_plus_10_dx: â\88\80n,c. ð\9d\90\93â¦\83n,c+ð\9d\9f\99ð\9d\9f\98â¦\84 → ⊥.
+lemma ist_inv_plus_10_dx: â\88\80n,c. ð\9d\90\93â\9dªn,c+ð\9d\9f\99ð\9d\9f\98â\9d« → ⊥.
#n #c #H
elim (ist_inv_plus … H) -H #n1 #n2 #_ #H #_
/2 width=2 by ist_inv_10/