update in ground_2, static_2, basic_2, apps_2, alpha_1

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / steps / rtc_max.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma
index 058b1a3c0163f0660ce5b3af805225558f6257c6..e35ae558a80ef26d7e0e8a4785caf840b99731d7 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma
@@ -12,13 +12,15 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/xoa/ex_3_2.ma".
+include "ground_2/xoa/ex_6_8.ma".
 include "ground_2/steps/rtc_shift.ma".
 
 (* RT-TRANSITION COUNTER ****************************************************)
 
 definition max (c1:rtc) (c2:rtc): rtc ≝ match c1 with [
    mk_rtc ri1 rs1 ti1 ts1 ⇒ match c2 with [
-      mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉
+      mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉
    ]
 ].
 
@@ -28,7 +30,7 @@ interpretation "maximum (rtc)"
 (* Basic properties *********************************************************)
 
 lemma max_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2.
-                 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉 =
+                 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉 =
                  (〈ri1,rs1,ti1,ts1〉 ∨ 〈ri2,rs2,ti2,ts2〉).
 // qed.
 
@@ -59,47 +61,66 @@ qed.
 
 (* Properties with test for constrained rt-transition counter ***************)
 
-lemma isrt_max: â\88\80n1,n2,c1,c2. ð\9d\90\91ð\9d\90\93â¦\83n1, c1â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\83n2, c2â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\83n1â\88¨n2, c1â\88¨c2â¦\84.
+lemma isrt_max: â\88\80n1,n2,c1,c2. ð\9d\90\91ð\9d\90\93â\9dªn1,c1â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dªn2,c2â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dªn1â\88¨n2,c1â\88¨c2â\9d«.
 #n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
 /2 width=3 by ex1_2_intro/
 qed.
 
-lemma isrt_max_O1: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â¦\830, c1â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\83n, c2â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\83n, c1â\88¨c2â¦\84.
+lemma isrt_max_O1: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â\9dª0,c1â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dªn,c2â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dªn,c1â\88¨c2â\9d«.
 /2 width=1 by isrt_max/ qed.
 
-lemma isrt_max_O2: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â¦\83n, c1â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\830, c2â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\83n, c1â\88¨c2â¦\84.
+lemma isrt_max_O2: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â\9dªn,c1â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dª0,c2â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dªn,c1â\88¨c2â\9d«.
 #n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/
 qed.
 
-lemma isrt_max_idem1: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â¦\83n, c1â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\83n, c2â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\83n, c1â\88¨c2â¦\84.
+lemma isrt_max_idem1: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â\9dªn,c1â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dªn,c2â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dªn,c1â\88¨c2â\9d«.
 #n #c1 #c2 #H1 #H2 >(idempotent_max n) /2 width=1 by isrt_max/
 qed.
 
 (* Inversion properties with test for constrained rt-transition counter *****)
 
-lemma isrt_inv_max: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â¦\83n, c1 â\88¨ c2â¦\84 →
-                    â\88\83â\88\83n1,n2. ð\9d\90\91ð\9d\90\93â¦\83n1, c1â¦\84 & ð\9d\90\91ð\9d\90\93â¦\83n2, c2â¦\84 & (n1 ∨ n2) = n.
+lemma isrt_inv_max: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â\9dªn,c1 â\88¨ c2â\9d« →
+                    â\88\83â\88\83n1,n2. ð\9d\90\91ð\9d\90\93â\9dªn1,c1â\9d« & ð\9d\90\91ð\9d\90\93â\9dªn2,c2â\9d« & (n1 ∨ n2) = n.
 #n #c1 #c2 * #ri #rs #H
 elim (max_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
 elim (max_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
 qed-.
 
-lemma isrt_O_inv_max: â\88\80c1,c2. ð\9d\90\91ð\9d\90\93â¦\830, c1 â\88¨ c2â¦\84 â\86\92 â\88§â\88§ ð\9d\90\91ð\9d\90\93â¦\830, c1â¦\84 & ð\9d\90\91ð\9d\90\93â¦\830, c2â¦\84.
+lemma isrt_O_inv_max: â\88\80c1,c2. ð\9d\90\91ð\9d\90\93â\9dª0,c1 â\88¨ c2â\9d« â\86\92 â\88§â\88§ ð\9d\90\91ð\9d\90\93â\9dª0,c1â\9d« & ð\9d\90\91ð\9d\90\93â\9dª0,c2â\9d«.
 #c1 #c2 #H
 elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H
 elim (max_inv_O3 … H) -H #H1 #H2 destruct
 /2 width=1 by conj/
 qed-.
 
-lemma isrt_inv_max_O_dx: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â¦\83n, c1 â\88¨ c2â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\830, c2â¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\83n, c1â¦\84.
+lemma isrt_inv_max_O_dx: â\88\80n,c1,c2. ð\9d\90\91ð\9d\90\93â\9dªn,c1 â\88¨ c2â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dª0,c2â\9d« â\86\92 ð\9d\90\91ð\9d\90\93â\9dªn,c1â\9d«.
 #n #c1 #c2 #H #H2
 elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
 lapply (isrt_inj … Hn2 H2) -c2 #H destruct //
 qed-.
 
+lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓❪n,c1 ∨ c2❫ → eq_t c1 c2 →
+                         ∧∧ 𝐑𝐓❪n,c1❫ & 𝐑𝐓❪n,c2❫.
+#n #c1 #c2 #H #Hc12
+elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
+lapply (isrt_eq_t_trans … Hc1 … Hc12) -Hc12 #H
+<(isrt_inj … H … Hc2) -Hc2
+<idempotent_max /2 width=1 by conj/
+qed-.
+
 (* Properties with shift ****************************************************)
 
 lemma max_shift: ∀c1,c2. ((↕*c1) ∨ (↕*c2)) = ↕*(c1∨c2).
 * #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2
 <shift_rew <shift_rew <shift_rew <max_rew //
 qed.
+
+(* Inversion lemmaswith shift ***********************************************)
+
+lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓❪n,↕*c1 ∨ c2❫ →
+                             ∧∧ 𝐑𝐓❪0,c1❫ & 𝐑𝐓❪n,c2❫.
+#n #c1 #c2 #H
+elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
+elim (isrt_inv_shift … Hc1) -Hc1 #Hc1 * -n1
+/2 width=1 by conj/
+qed-.