update in ground_2, static_2, basic_2, apps_2, alpha_1

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

diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma
index 75a58b25f55b0abc1b7a8e4fc5280c139b0e927f..6993f318d1c7176c7efef18da714a6562a6476ab 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma
@@ -12,15 +12,86 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/steps/rtc.ma".
+include "ground_2/xoa/ex_3_2.ma".
+include "ground_2/xoa/ex_6_8.ma".
+include "ground_2/steps/rtc_isrt.ma".
 
 (* RT-TRANSITION COUNTER ****************************************************)
 
-definition plus (r1:rtc) (r2:rtc): rtc ≝ match r1 with [
-   mk_rtc ri1 rh1 ti1 th1 ⇒ match r2 with [
-      mk_rtc ri2 rh2 ti2 th2 ⇒ 〈ri1+ri2, rh1+rh2, ti1+ti2, th1+th2〉
+definition plus (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〉
    ]
 ].
 
 interpretation "plus (rtc)"
-   'plus r1 r2 = (plus r1 r2).
+   'plus c1 c2 = (plus c1 c2).
+
+(* Basic properties *********************************************************)
+
+(**) (* plus is not disambiguated parentheses *)
+lemma plus_rew: ∀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.
+
+lemma plus_O_dx: ∀c. c = c + 𝟘𝟘.
+* #ri #rs #ti #ts <plus_rew //
+qed.
+
+(* Basic inversion properties ***********************************************)
+
+lemma plus_inv_dx: ∀ri,rs,ti,ts,c1,c2. 〈ri,rs,ti,ts〉 = c1 + c2 →
+                   ∃∃ri1,rs1,ti1,ts1,ri2,rs2,ti2,ts2.
+                   ri1+ri2 = ri & rs1+rs2 = rs & ti1+ti2 = ti & ts1+ts2 = ts &
+                   〈ri1,rs1,ti1,ts1〉 = c1 & 〈ri2,rs2,ti2,ts2〉 = c2.
+#ri #rs #ti #ts * #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2
+<plus_rew #H destruct /2 width=14 by ex6_8_intro/
+qed-.
+
+(* Main Properties **********************************************************)
+
+theorem plus_assoc: associative … plus.
+* #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2 * #ri3 #rs3 #ti3 #ts3
+<plus_rew //
+qed.
+
+(* Properties with test for constrained rt-transition counter ***************)
+
+lemma isrt_plus: ∀n1,n2,c1,c2. 𝐑𝐓❪n1,c1❫ → 𝐑𝐓❪n2,c2❫ → 𝐑𝐓❪n1+n2,c1+c2❫.
+#n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
+/2 width=3 by ex1_2_intro/
+qed.
+
+lemma isrt_plus_O1: ∀n,c1,c2. 𝐑𝐓❪0,c1❫ → 𝐑𝐓❪n,c2❫ → 𝐑𝐓❪n,c1+c2❫.
+/2 width=1 by isrt_plus/ qed.
+
+lemma isrt_plus_O2: ∀n,c1,c2. 𝐑𝐓❪n,c1❫ → 𝐑𝐓❪0,c2❫ → 𝐑𝐓❪n,c1+c2❫.
+#n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by isrt_plus/
+qed.
+
+lemma isrt_succ: ∀n,c. 𝐑𝐓❪n,c❫ → 𝐑𝐓❪↑n,c+𝟘𝟙❫.
+/2 width=1 by isrt_plus/ qed.
+
+(* Inversion properties with test for constrained rt-transition counter *****)
+
+lemma isrt_inv_plus: ∀n,c1,c2. 𝐑𝐓❪n,c1 + c2❫ →
+                     ∃∃n1,n2. 𝐑𝐓❪n1,c1❫ & 𝐑𝐓❪n2,c2❫ & n1 + n2 = n.
+#n #c1 #c2 * #ri #rs #H
+elim (plus_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
+elim (plus_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
+qed-.
+
+lemma isrt_inv_plus_O_dx: ∀n,c1,c2. 𝐑𝐓❪n,c1 + c2❫ → 𝐑𝐓❪0,c2❫ → 𝐑𝐓❪n,c1❫.
+#n #c1 #c2 #H #H2
+elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
+lapply (isrt_inj … Hn2 H2) -c2 #H destruct //
+qed-.
+
+lemma isrt_inv_plus_SO_dx: ∀n,c1,c2. 𝐑𝐓❪n,c1 + c2❫ → 𝐑𝐓❪1,c2❫ →
+                           ∃∃m. 𝐑𝐓❪m,c1❫ & n = ↑m.
+#n #c1 #c2 #H #H2
+elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
+lapply (isrt_inj … Hn2 H2) -c2 #H destruct
+/2 width=3 by ex2_intro/
+qed-.